Relative invariants, difference equations, and the Picard-Vessiot theory (revised version 5)

Katsutoshi Amano

Address: 8-29, Ida-Sammai-cho, Nakahara, Kawasaki, Kanagawa, 211-0037, Japan

E-mail: ma-pfybdb-612019@agate.dti.ne.jp URL: http://www.green.dti.ne.jp/amano/index-eng.html

Revised version of the thesis at the University of Tsukuba, March 2005 The last revision: December 2006

Acknowledgements

First of all I would like to thank my supervisor Professor T. Kimura. He taught me how to learn mathematics from the beginning when I just started afresh my life. About four years ago he suggested to study on archimedean local zeta functions of several variables as my first research task for the Master’s thesis. I can not give enough thanks to his heart-warming encouragement all over the period of this program.

I thank Professor T. Kogiso and Professor K. Sugiyama for being close advisers on prehomogeneous vector spaces.

I also would like to thank Professor M. Takeuchi and Professor A. Masuoka. In June or July 2003, I had a chance to know the existence of Professor Takeuchi’s paper [2] when I was trying to understand [1]. Since I once attended to an introductory part of his lecture on Hopf algebras at 2002, the paper seemed like just what I wanted. Then I was absorbed in it and came to think that the results can be extended to involve [1]. But I could not have developed it in the presented form without the collaboration with Professor Masuoka. He suggested the viewpoint from relative Hopf modules as is described in Section 3.2, added many interesting results, and helped me to complete several proofs.

Finally I would like to thank my parents for supporting my decision to study mathematics as the lifework.

References

[1] M. van der Put, M.F. Singer, “Galois theory of difference equations”, Lecture Notes in Math. 1666, Springer, 1997.

[2] M. Takeuchi, A Hopf algebraic approach to the Picard-Vessiot theory, J. Algebra 122 (1989), 481–509.

This is the revised version of my thesis at the University of Tsukuba and is a compilation of three separate studies. I shall make an excuse for the awful title “Relative invariants, difference equations, and the Picard-Vessiot theory”. Though usually a doctoral thesis is expected to be written in one theme, I could not go this way for institutional reason. To be accepted, I needed to include at least one topic of a paper which had been accepted to publication in a refereed journal. But, when applying for the degree, the paper “Picard-Vessiot extensions of artinian simple module algebras” (see Part 3) was not accepted yet (it was accepted just after I submitted the application for the degree). Hence I needed to include contents of Part 1, and finally I compiled the thesis into three parts to show how my capricious interest had been changed.

Contents

Acknowledgements i References i

Part 1. An equivariant map from (SL_{5}×GL_{4}, (∧^{2}ℂ^{5})⊗ℂ^{4}) to (GL_{4}, Sym^{2}(ℂ^{4})) 1 Introduction of Part 1 1 1.1. SL_{5}-invariant polynomials on Alt_{5}^{⊕4} 2 1.2. Construction of the equivariant map 3 References 8

Part 2. Archimedean local zeta functions which satisfy G_{m}-primitive difference equations 9

Introduction of Part 2 9 2.1. The Ore-Sato theorem 10 2.2. Reductive prehomogeneous vector spaces and its b-functions 15 2.3. Local zeta functions over ℂ 17 2.4. Local zeta functions over ℝ 19 References 22

Part 3. Picard-Vessiot theories for artinian simple module algebras 23 Introduction of Part 3 23 Conventions 29 3.1. Basic notions and results on D-module algebras 30 3.2. Tensor equivalences associated to Hopf subalgebras 35 3.3. Artinian simple D-module algebras 39 3.4. Sweedler’s correspondence theorem 43 3.5. Galois correspondence for Picard-Vessiot extensions 45 3.6. Translation into affine group schemes 52 3.7. Copying and interlacing 53 3.8. Splitting algebras 57 3.9. Liouville extensions 68 3.9.1. Liouville group schemes 68 3.9.2. Finite etale extensions 73 3.9.3. G_{a}-primitive extensions and G_{m}-primitive extensions 74 3.9.4. The solvability theorem 76 References 79 Part 1. An equivariant map from (SL_{5} × GL_{4}, (∧^{2}ℂ^{5}) ⊗ ℂ^{4}) to (GL_{4}, Sym^{2}(ℂ^{4}))

Introduction of Part 1

In this part, we construct an equivariant polynomial map from (SL_{5} × GL_{4}, (∧^{2}ℂ^{5}) ⊗ ℂ^{4}), the prehomogeneous vector space of quadruples of quinary alternating forms, to another prehomogeneous vector space (GL_{4}, Sym^{2}(ℂ^{4})) of quaternary quadratic forms. The presented result was obtained by Kogiso, Fujigami, and the author [1] to get an expression of the irreducible relative invariant of the former space explicitly. (Though it is known that the irreducible relative invariant is a homogeneous polynomial in degree 40 and it can be computed as the determinant of a certain 40 × 40 matrix by [7, §4, Proposition 16], our purpose was to give a more effective calculation.) Then I heard that the equivariant map had further importance as follows. For a field k in general, the structure of the space (SL_{5} ×GL_{4}, (∧^{2}k^{5})⊗k^{4}) has an arithmetic significance by reason of the correspondence between its non-singular orbits and isomorphism classes of separable quintic k-algebras (see [8]). Kable [3, Theorem 5.7] listed all equivariant polynomial maps from this space to any other prehomogeneous vector space and showed those maps could be obtained from two maps; the map presented here is one of the two. Such equivariant maps seem to be used in [4, 5] for arithmetic purposes.

Let Alt_{n} be the set of all skew-symmetric n × n complex matrices (i.e. Alt_{n} = {X ∈ M_{n}(ℂ) |^{t}X = −X}). One sees that the ℂ-vector space (∧^{2}ℂ^{5})⊗ℂ^{4} is isomorphic to Alt_{5}^{⊕4}. The space (SL_{5} × GL_{4}, (∧^{2}ℂ^{5}) ⊗ ℂ^{4}) is identified with (SL_{5} × GL_{4}, ρ = Λ_{2} ⊗ Λ_{1}, Alt_{5}^{⊕4}) in which the representation ρ is defined by

ρ(A, B) : (X_{1}, X_{2}, X_{3}, X_{4}) ⟼ (AX_{1}^{t}A, AX_{2}^{t}A, AX_{3}^{t}A, AX_{4}^{t}A)^{t}B

for (X_{1}, X_{2}, X_{3}, X_{4}) ∈ Alt_{5}^{⊕4} and (A, B) ∈ SL_{5}×GL_{4}. Our construction of the equivariant map is inspired by the method treated in [6, §3], constructing an equivariant map from (SL_{5} × GL_{3}, (∧^{2}ℂ^{5}) ⊗ ℂ^{3}) to (GL_{3}, Sym^{2}(ℂ^{3})). Especially a certain SL_{5}-equivariant bilinear form β : Alt_{5} × Alt_{5} → ℂ^{5}, which is introduced originally in [2], plays an important role. As in [2, 6], we define SL_{5}-invariant polynomials on Alt_{5}^{⊕4} by

[ijklm](X_{1}, X_{2}, X_{3}, X_{4}) := ^{t}β(X_{i}, X_{j})X_{k}β(X_{l}, X_{m})

for X_{1}, X_{2}, X_{3}, X_{4} ∈ Alt_{5} and i, j, k, l, m ∈ {1, 2, 3, 4}. Here each image of β is considered as a column vector. We identify Sym^{2}(ℂ^{4}) with the space of 4 × 4 symmetric matrices.

Then the equivariant map Φ : Alt_{5}^{⊕4} → Sym^{2}(ℂ^{4}), X ↦ (φ_{st}(X)) will be defined like

φ_{st} = ∑_{i,j,k,l,m,i′,j′,k′,l′,m′} c_{stijklmi′j′k′l′m′}[ijklm][i′j′k′l′m′],

where the coefficients c_{stijklmi′j′k′l′m′} are determined suitably.

In §1.1, we define the map β and the polynomials [ijklm], and describe some properties of them which are used to obtain the result. The equivariant map Φ will be defined in §1.2 and we will show the equivariance and the surjectivity of Φ (Proposition 1.2.1 and Theorem 1.2.2).

Notations. For α_{1}, α_{2}, α_{3}, α_{4}, ε ∈ ℂ, let diag(α_{1}, α_{2}, α_{3}, α_{4}) and E_{ε} be the following matrices:

diag(α_{1}, α_{2}, α_{3}, α_{4}) := ( \begin{matrix}α_{1} & 0 & 0 & 0 \\ 0 & α_{2} & 0 & 0 \\ 0 & 0 & α_{3} & 0 \\ 0 & 0 & 0 & α_{4} \end{matrix} ) , E_{ε} := ( \begin{matrix}1 & ε & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ) .

Let 𝔖_{4} be the fourth symmetric group. In 𝔖_{4}, a transposition between i and j is denoted by (i j). One sees each permutation σ ∈ 𝔖_{4} is considered as the 4 × 4 matrix such that its (i, j)-element is 1 or 0 with respect to i = σ(j) or not. So we may apply for regarding one as the other.

1.1. SL_{5}-invariant polynomials on Alt_{5}^{⊕4}

In the beginning, we define a certain SL_{5}-equivariant map β : Alt_{5} × Alt_{5} → ℂ^{5} which is used in [2, 6]. Let Pf be the Pfaffian on Alt_{4}. For X ∈ Alt_{5} and i = 1, ⋯, 5, let X^{(i)} denote the matrix in Alt_{4} which is obtained by deleting i-th row and i-th column from X. For X = (x_{ij}), Y = (y_{ij}) ∈ Alt_{5}, β(X, Y) is defined by

β(X, Y) := ( \begin{matrix}Pf(X^{(1)} + Y^{(1)}) − Pf(X^{(1)}) − Pf(Y^{(1)}) \\ −(Pf(X^{(2)} + Y^{(2)}) − Pf(X^{(2)}) − Pf(Y^{(2)})) \\ Pf(X^{(3)} + Y^{(3)}) − Pf(X^{(3)}) − Pf(Y^{(3)}) \\ −(Pf(X^{(4)} + Y^{(4)}) − Pf(X^{(4)}) − Pf(Y^{(4)})) \\ Pf(X^{(5)} + Y^{(5)}) − Pf(X^{(5)}) − Pf(Y^{(5)}) \end{matrix} )

= ( \begin{matrix}x_{23}y_{45} − x_{24}y_{35} + x_{25}y_{34} + y_{23}x_{45} − y_{24}x_{35} + y_{25}x_{34} \\ x_{34}y_{51} − x_{35}y_{41} + x_{31}y_{45} + y_{34}x_{51} − y_{35}x_{41} + y_{31}x_{45} \\ x_{45}y_{12} − x_{41}y_{52} + x_{42}y_{51} + y_{45}x_{12} − y_{41}x_{52} + y_{42}x_{51} \\ x_{51}y_{23} − x_{52}y_{13} + x_{53}y_{12} + y_{51}x_{23} − y_{52}x_{13} + y_{53}x_{12} \\ x_{12}y_{34} − x_{13}y_{24} + x_{14}y_{23} + x_{12}y_{34} − x_{13}y_{24} + x_{14}y_{23} \end{matrix} ) .

Then, for i, j, k, l, m ∈ {1, 2, 3, 4}, we define a polynomial [ijklm] on Alt_{5}^{⊕4} by

[ijklm](X_{1}, X_{2}, X_{3}, X_{4}) := ^{t}β(X_{i}, X_{j})X_{k}β(X_{l}, X_{m})

for X_{1}, X_{2}, X_{3}, X_{4} ∈ Alt_{5}. They satisfy the following lemmas:

Lemma 1.1.1 ([2, §2, Lemma]). For all i, j, k, l, m ∈ {1, 2, 3, 4}, the polynomial [ijklm] is invariant with respect to SL_{5}, i.e.

[ijklm](AX_{1}^{t}A, AX_{2}^{t}A, AX_{3}^{t}A, AX_{4}^{t}A) = [ijklm](X_{1}, X_{2}, X_{3}, X_{4})

for all A ∈ SL_{5}.

Lemma 1.1.2 ([2, §2, (4)]). If there are only one or two kinds of numbers among {i, j, k, l, m}, then [ijklm] = 0.

Lemma 1.1.3 ([6, Lemma 3.1]). For each i, j, k, l, m ∈ {1, 2, 3, 4},

(i) [ijklm] = [jiklm], [ijklm] = [ijkml], (ii) [ijklm] = −[lmkij],
(iii) [ijklm] + [jkilm] + [kijlm] = 0,  (iv) [iiklm] = −2[kiilm],
(v) [iikli] = −[iilki] = [iklii] = −[ilkii], (vi) [iiilm] = 0, [ijkij] = 0.

Finally in this section, we consider the action of GL_{4} on [ijklm]. GL_{4} is generated by the following three types of matrices: diag(α_{1}, α_{2}, α_{3}, α_{4}), permutation matrices, and E_{ε}. Thus we only need to think on these types. For B ∈ GL_{4} and P a polynomial on Alt_{5}^{⊕4}, let P^{B} denote the polynomial such that P^{B}(X) = P (X^{t}B). Diagonal matrices D = diag(α_{1}, α_{2}, α_{3}, α_{4}) and σ ∈ 𝔖_{4} act on [ijklm] by

\begin{aligned}[ijklm]^{D} & = & α_{i}α_{j}α_{k}α_{l}α_{m}[jiklm], \\ [ijklm]^{σ} & = & [σ^{−1}(i)σ^{−1}(j)σ^{−1}(k)σ^{−1}(l)σ^{−1}(m)]. \end{aligned}

Since [ijklm] are multilinear forms, we see, for i, j, k, l, m ∈ {2, 3, 4},

[ijklm]^{E_{ε}} = [ijklm],
[1ijkl]^{E_{ε}} = [1ijkl] + ε[2ijkl],
[11ijk]^{E_{ε}} = [11ijk] + 2ε[12ijk] + ε^{2}[22ijk],
[11ij1]^{E_{ε}} = [11ij1] + ε(2[12ij1] + [11ij2])
+ε^{2}(2[12ij2] + [22ij1]) + ε^{3}[22ij2], etc.

1.2. Construction of the equivariant map

Our first objective is to define a map Φ : Alt_{5}^{⊕4} → Sym^{2}(ℂ^{4}), X ↦ (φ_{st}(X)), where each φ_{st} is written like

φ_{st} = ∑ c_{stijklmi′j′k′l′m′}[ijklm][i′j′k′l′m′].

Proposition 1.2.1. There exists a polynomial map Φ : Alt_{5}^{⊕4} → Sym^{2}(ℂ^{4}) such that

Φ(ρ(A, B)X) = (det B)^{2}BΦ(X)^{t}B

for X ∈ Alt_{5}^{⊕4} and (A, B) ∈ SL_{5} × GL_{4}.

First we observe that Φ should be determined uniquely from φ_{11} and φ_{12} so that Φ(X) is equivariant with respect to the action of 𝔖_{4}:

(1.1)  φ_{ss} = φ_{11}^{(1 s)} (s = 1, …, 4),  φ_{σ^{−1}(1)σ^{−1}(2)} = φ_{12}^{σ} (σ ∈ 𝔖_{4}).

Furthermore, φ_{12} should also be determined from φ_{11}. To obtain Φ(ρ(A, E_{ε})X) = E_{ε}Φ(X)^{t}E_{ε} (A ∈ SL_{5}), the polynomials φ_{st} should satisfy at least the following:

(1.2)  φ_{11}^{E_{ε}} = φ_{11} + 2εφ_{12} + ε^{2}φ_{22},
(1.3)  φ_{22}^{E_{ε}} = φ_{22},
(1.4)  φ_{33}^{E_{ε}} = φ_{33},
(1.5)  φ_{13}^{E_{ε}} = φ_{13} + εφ_{23},
(1.6)  φ_{34}^{E_{ε}} = φ_{34}.

If we obtain φ_{11}, then φ_{22} = φ_{11}^{(1 2)} and φ_{12} will be determined from (1.2).

Considering the action of diagonal matrices for Φ(X), we start by assuming that each term [ijklm][i′j′k′l′m′] in φ_{11} is constructed by the following numbers:

{i, j, k, l, m, i′, j′, k′, l′, m′} = {1, 1, 1, 1, 2, 2, 3, 3, 4, 4}.

But from Lemma 1.1.2 and Lemma 1.1.3, we need not to think on the all combinations of the above numbers. By choosing combinations and using the method of indeterminate coefficients, it is possible to determine the polynomial φ_{11} so that the equations from (1.2) to (1.6) are satisfied. Indeed, we conclude that the following definitions are suitable:

φ_{11} := 160[31114](3[24132] − 2[21342] − 2[23412])

+160[41112](3[32143] − 2[34213] − 2[31423]) +160[21113](3[43124] − 2[41234] − 2[42314]) +50([11233][11244] + [11322][11344] + [11422][11433]) −288([13241]^{2} + [14321]^{2} + [12431]^{2}) +224([13241][14321] + [14321][12431] + [12431][13241]),

φ_{12} := 400[31114][32224]

−100([21113][22344] + [21114][22433]) −100([12223][11344] + [12224][11433]) +20[11422](4[31423] − [34213] − [32143]) +20[11322](4[41324] − [43214] − [42134]) −25([22144][11233] + [11244][22133]) +368[13241][23142] +112([13241]([21342] + [23412]) + [23142]([12341] + [13421])) +192([14321][23412] + [13421][24312]) −208([14321][21342] + [12431][23412]).

These polynomials satisfy the following properties:

(i) If σ ∈ 𝔖_{4} and σ(1) = 1, then φ_{11}^{σ} = φ_{11}, (ii) If σ ∈ 𝔖_{4} and {σ(1), σ(2)} = {1, 2}, then φ_{12}^{σ} = φ_{12}.

Then we define the map Φ : Alt_{5}^{⊕4} → Sym^{2}(ℂ^{4}), X ↦ (φ_{st}(X)) so that (1.1) is satisfied; the well-definedness follows from (i), (ii). It is easily seen that φ_{st} = φ_{ts} and φ_{st}^{σ} = φ_{σ^{−1}(s)σ^{−1}(t)} for all σ ∈ 𝔖_{4}.

Proof of Proposition 1.2.1. Let D = diag(α_{1}, α_{2}, α_{3}, α_{4}) and let A be an arbitrary element of SL_{5}. Since φ_{st}^{D} = (α_{1}α_{2}α_{3}α_{4})^{2}α_{s}α_{t}φ_{st} for all s, t ∈ {1, 2, 3, 4}, and each φ_{st} is invariant with respect to SL_{5}, we have

Φ(ρ(A, D)X) = (det D)^{2}DΦ(X)^{t}D.

By the definition, it follows

Φ(ρ(A, σ)X) = (φ_{σ^{−1}(s)σ^{−1}(t)}(X)) = σΦ(X)^{t}σ

for all σ ∈ 𝔖_{4}.

The rest of the proof is to show Φ(ρ(A, E_{ε})X) = E_{ε}Φ(X)^{t}E_{ε}, i.e. • φ_{11}^{E_{ε}} = φ_{11} + 2εφ_{12} + ε^{2}φ_{22},

• φ_{1t}^{E_{ε}} = φ_{t1}^{E_{ε}} = φ_{1t} + εφ_{2t} for t = 2, 3, 4,

• φ_{st}^{E_{ε}} = φ_{ts}^{E_{ε}} = φ_{st} for s, t = 2, 3, 4.

Recall that we defined φ_{st} to satisfy the equations from (1.2) to (1.6) (in fact, they are shown directly). By E_{ε}^{2} = E_{2ε} and (1.2), we have

φ_{11}^{E_{ε}^{2}} = φ_{11} + 4εφ_{12} + 4ε^{2}φ_{22}.

On the other hand, by (1.2) and (1.3),

φ_{11}^{E_{ε}^{2}} = (φ_{11} + 2εφ_{12} + ε^{2}φ_{22})^{E_{ε}}
= φ_{11}^{E_{ε}} + 2εφ_{12}^{E_{ε}} + ε^{2}φ_{22}^{E_{ε}}
= φ_{11} + 2εφ_{12} + 2εφ_{12}^{E_{ε}} + 2ε^{2}φ_{22}.

Therefore φ_{12}^{E_{ε}} = φ_{12} + εφ_{22}. Similarly by (1.5),

φ_{13}^{E_{ε}^{2}} = φ_{13} + 2εφ_{23}
= φ_{13} + εφ_{23} + εφ_{23}^{E_{ε}}.

Hence φ_{23}^{E_{ε}} = φ_{23}. By (1.5) and E_{ε}(3 4) = (3 4)E_{ε}, we have

φ_{14}^{E_{ε}} = φ_{13}^{(3 4)E_{ε}} = φ_{13}^{E_{ε}(3 4)} = (φ_{13}^{E_{ε}})^{(3 4)} = φ_{14} + εφ_{24}.

Similarly by (1.4), we have

φ_{44}^{E_{ε}} = φ_{33}^{E_{ε}(3 4)} = φ_{33}^{(3 4)} = φ_{44}.

Now the proof is completed. □

To prove that Φ is surjective, we only need to find five points in Alt_{5}^{⊕4} such that each image has rank 0, 1, 2, 3, 4. For

X_{01} = ( \begin{matrix}0 & 1 & 0 & 0 & 0 \\ −1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & −1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{matrix} ) , X_{02} = ( \begin{matrix}0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & −1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & −1 & 0 \end{matrix} ) ,

X_{03} = ( \begin{matrix}0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ −1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 & 0 \end{matrix} ) , X_{04} = ( \begin{matrix}0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & −1 & 0 & 0 & 0 \\ 0 & 0 & −1 & 0 & 0 \end{matrix} ) ,

Y_{01} = ( \begin{matrix}0 & 1 & 1 & 0 & 0 \\ −1 & 0 & 0 & 0 & 0 \\ −1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{matrix} ) , Y_{02} = ( \begin{matrix}0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & −1 & 0 \end{matrix} ) ,

Y_{03} = ( \begin{matrix}0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 & 0 \end{matrix} ) ,

we have

Φ(X_{01}, X_{02}, X_{03}, X_{04}) = ( \begin{matrix}0 & 0 & −720 & 0 \\ 0 & −480 & 0 & 0 \\ −720 & 0 & 0 & 0 \\ 0 & 0 & 0 & −288 \end{matrix} ) (rank 4),

Φ(Y_{01}, X_{02}, X_{03}, X_{04}) = ( \begin{matrix}−192 & 0 & −192 & −96 \\ 0 & −480 & 0 & 0 \\ −192 & 0 & −192 & −96 \\ −96 & 0 & −96 & −288 \end{matrix} ) (rank 3),

Φ(Y_{01}, Y_{02}, X_{03}, X_{04}) = ( \begin{matrix}−192 & 0 & −192 & −96 \\ 0 & 0 & 0 & 0 \\ −192 & 0 & −192 & −96 \\ −96 & 0 & −96 & −288 \end{matrix} )  (rank 2),

Φ(Y_{01}, Y_{02}, Y_{03}, X_{04}) = ( \begin{matrix}−192 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} )  (rank 1),

Φ(0, 0, 0, 0)  = 0  (rank 0).

Therefore Φ is surjective and especially det Φ(X) ≠ 0 as a polynomial. This fact and Proposition 1.2.1 implies that det Φ(X) is the relative invariant in degree 40.

Theorem 1.2.2. (i) The map Φ : Alt_{5}^{⊕4} → Sym^{2}(ℂ^{4}) is surjective.

(ii) f(X) = det Φ(X) is the irreducible relative invariant of the prehomogeneous vector space (SL_{5} × GL_{4}, Λ_{2} ⊗ Λ_{1}, Alt_{5}^{⊕4}) in degree 40 corresponding to the rational character (det B)^{4}.

References

[1] K. Amano, M. Fujigami, T. Kogiso, Construction of irreducible relative invariant of the prehomogeneous vector space (SL_{5} × GL_{4}, Λ^{2}(ℂ^{5}) ⊗ ℂ^{4}), Linear Algebra Appl. 355 (2002), 215–222.

[2] A. Gyoja, Construction of invariants, Tsukuba J. Math. 14 (1990), 437–457.

[3] A.C. Kable, The concomitants of a prehomogeneous vector space, J. Algebra 271 (2004), 295–311.

[4] A.C. Kable, A. Yukie, On the space of quadruples of quinary alternating forms, J. Pure Appl. Algebra 186 (2004), 277–295.

[5] A.C. Kable, A. Yukie, A construction of quintic rings, Nagoya Math. J. 173 (2004), 163–203.

[6] H. Ochiai, Quotients of some prehomogeneous vector spaces, J. Algebra 192 (1997), 61–73.

[7] M. Sato, T. Kimura, A classification of irreducible prehomogeneous vector spaces and their relative invariants, Nagoya Math. J. 65 (1977), 1–155.

[8] D.J. Wright, A. Yukie, Prehomogeneous vector spaces and field extensions, Invent. math. 110 (1992), 283–314.

Part 2. Archimedean local zeta functions which satisfy G_{m}-primitive difference equations

Introduction of Part 2

Let K be ℂ or ℝ, (G, ρ, V) a reductive prehomogeneous vector space defined over K, and V_{K} the set of K-rational points of V. Let P_{1}(x), ⋯, P_{r}(x) be the basic relative invariants of (G, ρ, V) over K. For a Schwartz function Φ(x) on V_{K} and s = (s_{1}, ⋯, s_{r}) ∈ ℂ^{r}, the integral

Z_{K}(s, Φ) = ∫_{V_{K}} ∏_{i=1}^{r} |P_{i}(x)|_{K}^{s_{i}}Φ(x)dx

is called the archimedean local zeta function associated with (G, ρ, V). When we take Φ(x) as

Φ(x) = \{ \begin{matrix}exp(−2π x^{t}x̄) & (K = ℂ) \\ exp(−π x^{t}x) & (K = ℝ), \end{matrix}

Z_{K}(s, Φ) is denoted by Z_{K}(s) simply.

In the case r = 1, Igusa suggested in [6, §3, Remark] and proved in [7, Chapter 6] the following theorem:

Theorem (Igusa). Let P be the basic relative invariant of (G, ρ, V), assuming r = 1. Let d = deg P and b(s) = c ∏_{j=1}^{d}(s + α_{j}) the b-function of P (x).

(1) When K = ℂ,

Z_{ℂ}(s) = ((2π)^{−d}c)^{s} ∏_{j=1}^{d} \frac{Γ(s + α_{j})}{Γ(α_{j})}.

(2) When K = ℝ and when every term of P (x) is a multilinear form,

Z_{ℝ}(s) = (π^{−d}c)^{\frac{s}{2}} ∏_{j=1}^{d} \frac{Γ((s + α_{j})/2)}{Γ(α_{j}/2)}.

In this part, we extend this theorem to several variables (r ≥ 1). Though basically our proof presented here is an easy modification of the proof given in [7], a careful treatment of the Ore-Sato theorem (see Section 2.1) is needed. The most important point of the proof is the fact that Z_{ℂ}(s) and Z_{ℝ}(2s) satisfies a difference equation in a certain type, called G_{m}-primitive, or hypergeometric, which is written by the b-functions. The proof can be divided into two steps. The first step is to characterize a desired solution of such a difference equation, written as a product of an exponential function and gamma functions.

The next step is to prove that the characterization can be adapted to Z_{ℂ}(s) and Z_{ℝ}(2s). To obtain a suitable difference equation (especially for Z_{ℝ}(2s)), very delicate facts which are seen in the proof of the Ore-Sato theorem are needed. For this reason, we include a detailed proof of the theorem in Section 2.1. The main results will be described in Section 2.3, 2.4 (Theorem 2.3.1 and Theorem 2.4.3).

2.1. The Ore-Sato theorem

Let k be a field of zero characteristic and k(s) = k(s_{1}, …, s_{r}) the rational function field of r variables. Let Ξ be the free abelian group of rank r (Ξ ≃ ℤ^{r}) and τ_{1}, …, τ_{r} a basis of Ξ. Then Ξ acts on k(s) as k-algebra automorphisms by τ_{i}f(s) = f(s + e_{i}) (f(s) ∈ k(s)) where e_{1} = (1, 0, …, 0), …, e_{r} = (0, …, 0, 1), the canonical basis of k^{r}. Let kΞ be the group algebra of Ξ over k and k(s)#kΞ the ring of linear difference operators, which is k(s) ⊗_{k} kΞ with the semi-direct product: (f_{1} ⊗ g) · (f_{2} ⊗ h) = f_{1}(gf_{2}) ⊗ gh (g, h ∈ Ξ). We say that a k(s)#kΞ-module V is G_{m}-primitive, or hypergeometric, iff dim_{k(s)} V = 1. (We will see in Part 3 that the Picard-Vessiot group scheme of such a k(s)#kΞ-module is a closed subgroup scheme of G_{m}.) Let k(s)^{×} = k(s) \ {0}. For a fixed k(s)-basis v of a G_{m}-primitive k(s)#kΞ-module V, we have an associated map b_{v} : Ξ → k(s)^{×}, g ↦ b_{g,v}(s) defined by gv = b_{g,v}(s)v. Since b_{1,v}(s) = 1 and b_{gh,v}(s) = (gb_{h,v}(s))b_{g,v}(s) for all g, h ∈ Ξ, b_{v} is in the set Z^{1}(Ξ; k(s)^{×}) of 1-cocycles. Let v′ be another k(s)-basis of V. Then there exists an f ∈ k(s)^{×} such that v′ = f(s)v. It follows b_{g,v′}(s) = (gf(s))f(s)^{−1}b_{g,v}(s) for all g ∈ Ξ; this implies that both of b_{v′} and b_{v} define the same cohomology class in H^{1}(Ξ; k(s)^{×}) since the map g ↦ (gf(s))f(s)^{−1} is in the set B^{1}(Ξ; k(s)^{×}) of 1-coboundaries. Thus G_{m}-primitive k(s)#kΞ-modules are classified by H^{1}(Ξ; k(s)^{×}).

An explicit description of Z^{1}(Ξ; k(s)^{×}) is given by a result called the Ore-Sato theorem, which was first obtained by Ore [12] for the case r = 2 and by Sato [14] for arbitrary r. Detailed proofs are also seen in [10, §1.1] and [4, §1]. The purpose of this section is to introduce the theorem for later use. Since some delicate facts such as Corollary 2.1.4 are important to us, we follow carefully the discussion given in [14, Appendix]. (Thus our statement of the theorem may be verbose according to the interest of the reader. For a more elegant description of H^{1}(Ξ; ℂ(s)^{×}), [10, Proposition 1.1.4] is recommended. See Remark 2.1.5.) k(s)^{×} has a natural ℤΞ-module structure as follows:

(∑_{i} n_{i}g_{i}) f(s) = ∏_{i}(g_{i}f(s))^{n_{i}} (n_{i} ∈ ℤ, g_{i} ∈ Ξ).

We easily see k(s)^{×} ≃ k^{×} × k(s)^{×}/k^{×} as ℤΞ-modules. Moreover, by decomposing to irreducible polynomials, we have the following ℤΞ-module isomorphisms:

k(s)^{×} ⟶^{∼} k^{×} ⊕ ⨁_{f} ℤΞf ⟶^{∼} k^{×} ⊕ ⨁_{f} ℤ(Ξ/Ξ_{f}),

where f runs over a set of irreducible polynomials which are not translated into one another up to constant multiple by the action of ℤΞ, and Ξ_{f} := {g ∈ Ξ | gf = (const.)f} = {g ∈ Ξ | gf = f}. ℤ(Ξ/Ξ_{f}) is the group ring of Ξ/Ξ_{f} over ℤ.

Since Ξ is a finitely generated group, we have

Z^{1}(Ξ; k(s)^{×}) ≃ Z^{1}(Ξ; k^{×}) ⊕ ⨁_{f} Z^{1}(Ξ; ℤ(Ξ/Ξ_{f})).

One sees that Z^{1}(Ξ; k^{×}) is equal to the character group Hom(Ξ, k^{×}). The structure of each Z^{1}(Ξ; ℤ(Ξ/Ξ_{f})) is described by the following two lemmas.

Lemma 2.1.1. Let f be an irreducible polynomial in k[s] = k[s_{1}, …, s_{r}].

(i) Ξ/Ξ_{f} is a free abelian group of rank Ξ/Ξ_{f} > 0.

(ii) Take an arbitrary α ∈ Z^{1}(Ξ; ℤ(Ξ/Ξ_{f})). Then α(g) = 0 for all g ∈ Ξ_{f}.

(iii) If rank Ξ/Ξ_{f} ≥ 2, then H^{1}(Ξ; ℤ(Ξ/Ξ_{f})) = 0.

Proof. (i) Since f is not a constant, Ξ_{f} ≠ Ξ. Suppose g ∈ Ξ and g^{n} ∈ Ξ_{f} for some positive integer n. There is an m ∈ ℤ^{r} such that gf(s) = f(s + m). Consider P (t) := f(s + tnm) − f(s) as a polynomial in k(s)[t]. Since P (l) = g^{ln}f(s) − f(s) = 0 for all l ∈ ℤ and since k(s) is an infinite field, we have P ≡ 0. Hence 0 = P (1/n) = gf(s) − f(s), and so g ∈ Ξ_{f}.

(ii) Take a g′ ∈ Ξ which is not in Ξ_{f}. Let g̃′ be the image of g′ in Ξ/Ξ_{f}. Then for all g ∈ Ξ_{f}, we have

α(gg′) = α(g′) + α(g) = g̃′α(g) + α(g′).

Thus (g̃′ −1)α(g) = 0. Since ℤ(Ξ/Ξ_{f}) is an integral domain by part (i), we have α(g) = 0.

(iii) By part (ii), we have H^{1}(Ξ; ℤ(Ξ/Ξ_{f})) ≃ H^{1}(Ξ/Ξ_{f}; ℤ(Ξ/Ξ_{f})). Let λ_{1}, …, λ_{l} be a basis of Ξ/Ξ_{f}. For all α ∈ Z^{1}(Ξ/Ξ_{f}; ℤ(Ξ/Ξ_{f})), we have

α(λ_{i}λ_{j}) = λ_{i}α(λ_{j}) + α(λ_{i}) = λ_{j}α(λ_{i}) + α(λ_{j})

and hence (λ_{i} − 1)α(λ_{j}) = (λ_{j} − 1)α(λ_{i}) for i, j = 1, …, l. Since ℤ(Ξ/Ξ_{f}) is the Laurent polynomial ring, there exists an a ∈ ℤ(Ξ/Ξ_{f}) such that

α(λ_{i}) = (λ_{i} − 1)a (i = 1, …, l).

Moreover, since

0 = α(1) = α(λ_{i}^{−1}λ_{i}) = λ_{i}^{−1}α(λ_{i}) + α(λ_{i}^{−1}) = (1 − λ_{i}^{−1})a + α(λ_{i}^{−1}),

we have α(λ_{i}^{−1}) = (λ_{i}^{−1} − 1)a. Therefore

α(g) = ga − a (g ∈ Ξ/Ξ_{f}),

concluding α ∈ B^{1}(Ξ/Ξ_{f}; ℤ(Ξ/Ξ_{f})), follows by induction. Indeed, if α(g′) = g′a − a for a g′ ∈ Ξ/Ξ_{f}, then

\begin{matrix}α(λ_{i}g′) = λ_{i}(g′a − a) + (λ_{i} − 1)a = λ_{i}g′a − a, \\ α(λ_{i}^{−1}g′) = λ_{i}^{−1}(g′a − a) + (λ_{i}^{−1} − 1)a = λ_{i}^{−1}g′a − a \end{matrix}

for i = 1, …, l. □

Lemma 2.1.2. Let f be an irreducible polynomial in k[s]. Suppose that rank Ξ/Ξ_{f} = 1 and let λ be a basis of Ξ/Ξ_{f}.

(i) There exist a linear form μ(s) = n_{1}s_{1}+⋯+n_{r}s_{r} (n_{1}, …, n_{r} ∈ ℤ) and an irreducible polynomial h(t) ∈ k[t] of one variable t, such that f(s) = h(μ(s)) and λf(s) = h(μ(s)+1). Moreover, there exists an m ∈ ℤ^{r} such that μ(m) = 1, i.e. the greatest common divisor of non-zero coefficients of μ is 1.

(ii) For every α ∈ Z^{1}(Ξ; ℤ(Ξ/Ξ_{f})), there is an η ∈ ℤ(Ξ/Ξ_{f}) such that

α(τ_{1}^{m_{1}} ⋯ τ_{r}^{m_{r}}) = \{ \begin{matrix}η ∑_{ν=0}^{μ(m)−1}λ^{ν} & (μ(m) ≥ 1), \\ 0 & (μ(m) = 0), \\ −η ∑_{ν=μ(m)}^{−1}λ^{ν} & (μ(m) ≤ −1), \end{matrix}

for all m = (m_{1}, …, m_{r}) ∈ ℤ^{r}.

(iii) H^{1}(Ξ; ℤ(Ξ/Ξ_{f})) ≃ ℤ.

Proof. (i) Let λ_{1}, …, λ_{r} be a basis of Ξ such that the image of λ_{1} in Ξ/Ξ_{f} is λ and λ_{2}, …, λ_{r} ∈ Ξ_{f}. Then there exists an invertible r × r matrix (n_{ij})_{i,j} in GL_{r}(ℤ) such that τ_{i} = λ_{1}^{n_{i1}} ⋯ λ_{r}^{n_{ir}} for i = 1, …, r. Put μ(s) := n_{11}s_{1} + ⋯ + n_{r1}s_{r} and take linear forms s′_{1}(= μ(s)), …, s′_{r} by (s′_{1}, …, s′_{r}) = (s_{1}, …, s_{r})(n_{ij})_{i,j}. Then we have λ_{j}s′_{i} = s′_{i} +δ_{ij}, where δ_{ij} denotes Kronecker’s delta. There exists an irreducible polynomial h(t_{1}, …, t_{r}) ∈ k[t_{1}, …, t_{r}] such that f(s) = h(s′_{1}, …, s′_{r}). But

h(s′_{1}, s′_{2} + m_{2}, …, s′_{r} + m_{r}) = λ_{2}^{m_{2}} ⋯ λ_{r}^{m_{r}}f(s) = f(s) = h(s′_{1}, …, s′_{r})

for all m_{2}, …, m_{r} ∈ ℤ, since λ_{2}, …, λ_{r} ∈ Ξ_{f}. Thus it follows that h is actually a polynomial of one variable t_{1}. The second part obviously follows by the definition of μ.

(ii) By part (ii) of Lemma 2.1.1, we have Z^{1}(Ξ; ℤ(Ξ/Ξ_{f})) ≃ Z^{1}(Ξ/Ξ_{f}; ℤ(Ξ/Ξ_{f})). For an arbitrary α ∈ Z^{1}(Ξ/Ξ_{f}; ℤ(Ξ/Ξ_{f})), set η = α(λ). Then the assertion follows by induction on μ(m), since the image of τ_{1}^{m_{1}} ⋯ τ_{r}^{m_{r}} in Ξ/Ξ_{f} is λ^{μ(m)}.

(iii) We see that Z^{1}(Ξ/Ξ_{f}; ℤ(Ξ/Ξ_{f})) → ℤ(Ξ/Ξ_{f}), α ↦ α(λ) is an isomorphism. Then B^{1}(Ξ/Ξ_{f}; ℤ(Ξ/Ξ_{f})) is isomorphic to ℤ(Ξ/Ξ_{f})(λ − 1) under this isomorphism. Hence H^{1}(Ξ/Ξ_{f}; ℤ(Ξ/Ξ_{f})) ≃ ℤ, via the map ε : ℤ(Ξ/Ξ_{f}) → ℤ, ∑_{i} z_{i}λ^{i} ↦ ∑_{i} z_{i}. □

By translating the lemmas above, we have the following theorem.

Theorem 2.1.3 (Ore-Sato). Let Ξ → k(s)^{×}, g ↦ b_{g}(s) be a 1-cocycle in Z^{1}(Ξ; k(s)^{×}). Write τ^{m} = τ_{1}^{m_{1}} ⋯ τ_{r}^{m_{r}} for m = (m_{1}, …, m_{r}) ∈ ℤ^{r}. Then b is written as the following form:

(2.1) b_{τ^{m}}(s) = c(τ^{m}) ∏_{j=1}^{N} ζ_{j} \frac{f_{j}(s + m)}{f_{j}(s)} ∏_{i=1}^{l} η_{i} · \{ \begin{matrix}∏_{ν=0}^{μ_{i}(m)−1}h_{i}(μ_{i}(s) + ν) & (μ_{i}(m) ≥ 1) \\ 1 & (μ_{i}(m) = 0) \\ ∏_{ν=μ_{i}(m)}^{−1} \frac{1}{h_{i}(μ_{i}(s) + ν)} & (μ_{i}(m) ≤ −1) \end{matrix}

for all m ∈ ℤ^{r}. Here, c : Ξ → k^{×} is a character, h_{1}, …, h_{l} ∈ k[t] are irreducible polynomials of one variable, μ_{1}, …, μ_{l} ∈ ℤs_{1} + ⋯ + ℤs_{r} are non-zero linear forms whose each greatest common divisor of non-zero coefficients is 1, η_{i} ∈ ℤ(Ξ/Ξ_{μ_{i}(s)}) (i = 1, …, l), f_{1}, …, f_{N} are irreducible polynomials in k[s] which satisfy that rank Ξ/Ξ_{f_{j}} > 1 (j = 1, …, N) and f_{1}(s), …, f_{N}(s), h_{1}(μ_{1}(s)), …, h_{l}(μ_{l}(s)) are not translated into one another up to constant multiple by the action of ℤΞ, and ζ_{j} ∈ ℤ(Ξ/Ξ_{f_{j}}) (j = 1, …, N).

Let ε : ℤ(Ξ/Ξ_{μ_{i}(s)}) → ℤ, ∑_{i} z_{i}λ^{i} ↦ ∑_{i} z_{i} be the map described in the last sentence in the proof of Lemma 2.1.2. We normalize each η_{i}, μ_{i}, h_{i} in (2.1) so that ε(η_{i}) > 0 (i = 1, …, l) (when ε(η_{i}) < 0, replace η_{i}, μ_{i}, h_{i}(t) with −η_{i}, −μ_{i}, h_{i}(−t − 1)). Put f(s) = ∏_{j=1}^{N} ζ_{j}f_{j}(s).

Corollary 2.1.4. In Theorem 2.1.3, assume b_{τ_{1}}(s), …, b_{τ_{r}}(s) are polynomials. Take the expression (2.1) with f(s) = 1 (which necessarily holds) and η_{i}, μ_{i}, h_{i} normalized as above. Then they satisfy the following conditions (1), (2).

(1) All coefficients of each μ_{i} are non-negative integers.

(2) For each i = 1, …, l, let λ_{i} be the basis of Ξ/Ξ_{μ_{i}(s)} such that λ_{i}μ_{i}(s) = μ_{i}(s) + 1. Write η_{i} = ∑_{j=n}^{n′} z_{ij}λ_{i}^{j} (z_{ij} ∈ ℤ, z_{in}, z_{in′} ≠ 0). Then h_{i}(μ_{i}(s) + n)^{z_{in}} and h_{i}(μ_{i}(s) + n′ + μ_{i}(m) − 1)^{z_{in′}} are not canceled in the product ∏_{j=n}^{n′} ∏_{ν=0}^{μ_{i}(m)−1} h_{i}(μ_{i}(s) + j + ν)^{z_{ij}} (0 ≠ m ∈ ℤ_{≥0}^{r}). Hence z_{in} and z_{in′} are positive integers.

Proof. (1) Necessarily μ_{i}(e_{1})ε(η_{i}), …, μ_{i}(e_{r})ε(η_{i}) are non-negative integers for i = 1, …, l by the assumption.

(2) This is easily seen. □

Remark 2.1.5. (i) The 1-cocycle given by (2.1) and the following one both define the same cohomology class in H^{1}(Ξ; k(s)^{×}):

τ^{m} ↦ c(τ^{m}) ∏_{i=1}^{l} \{ \begin{matrix}∏_{ν=0}^{μ_{i}(m)−1}h_{i}(μ_{i}(s) + ν)^{ε(η_{i})} & (μ_{i}(m) ≥ 1) \\ 1 & (μ_{i}(m) = 0) \\ ∏_{ν=μ_{i}(m)}^{−1}h_{i}(μ_{i}(s) + ν)^{−ε(η_{i})} & (μ_{i}(m) ≤ −1). \end{matrix}

In [4, §1], η_{i}, μ_{i}, h_{i} are normalized to be ε(η_{i}) < 0. Another normalization is given in [10, Proposition 1.1.4] with a restriction on μ_{i} instead of ε(η_{i}). To describe the group structure of cohomology classes, such descriptions are more elegant. But, in this article, the Ore-Sato theorem should be introduced in the presented form for a reason which will arise later (especially in Lemma 2.4.2). Here we are interested in Z^{1}(Ξ; k(s)^{×}) rather than H^{1}(Ξ; k(s)^{×}).

(ii) The assumption in Corollary 2.1.4 does not imply that each η_{i}h_{i}(μ_{i}(s)) is a polynomial; for example, let r = 2 and consider the 1-cocycle defined by (2.1) with c = 1, f(s) = 1, l = 1, h_{1}(t) = t, μ_{1}(s) = 2s_{1} + 3s_{2}, η_{1} = 1 − λ + λ^{2} (here λ is the basis of Ξ/Ξ_{μ_{1}(s)} represented by τ_{1}^{−1}τ_{2}). In addition, we observe that the conditions (1), (2) in the corollary are not sufficient for the assumption; consider the 1-cocycle where η_{1} in the above example is replaced with η_{1} = 1 − λ − λ^{2} + λ^{3} + λ^{4}. (But these two examples define the same cohomology class in H^{1}(Ξ; k(s)^{×})).

Let k = ℂ. Take a 1-cocycle b ∈ Z^{1}(Ξ; ℂ(s)^{×}) and keep the notation in Theorem 2.1.3. Write h_{i} = t + α_{i} (α_{i} ∈ ℂ) and c(τ^{m}) = c_{1}^{m_{1}} ⋯ c_{r}^{m_{r}} (c_{1}, …, c_{r} ∈ ℂ^{×}). Then the G_{m}-primitive difference equation associated with b has a solution

γ(s) = c_{1}^{s_{1}} ⋯ c_{r}^{s_{r}}f(s) ∏_{i=1}^{l} η_{i}Γ(μ_{i}(s) + α_{i}).

Assume b_{τ_{1}}, …, b_{τ_{r}} are polynomials. In this case, γ(s) has no zeros by Corollary 2.1.4. There are several methods to characterize γ(s) among other solutions; by asymptotic behavior or by log convexity: M. Fujigami [3] gave a generalization of the Bohr-Mollerup theorem [1, Theorem 2.1]. Further methods may be suggested in [1, §6].

2.2. Reductive prehomogeneous vector spaces and its b-functions

Let V be an n-dimensional ℂ-vector space and G a connected reductive linear algebraic group over ℂ. Suppose (G, ρ, V) is a reductive prehomogeneous vector space. Then, by the definition, there exists a proper algebraic subset S of V such that V \S is a single G-orbit. Let S_{0} denote the union of the irreducible components of S with codimension 1. We always assume that S_{0} is not empty. Since ρ(G) is connected reductive, it is self-adjoint with respect to a ℂ-basis of V by the theorem of Mostow [11]. By such a basis, we identify the coordinate ring ℂ[V] of V with ℂ[x] = ℂ[x_{1}, …, x_{n}], the polynomial ring of n variables. Let P_{1}, …, P_{r} ∈ ℂ[x] be irreducible polynomials which define the irreducible components of S_{0}. These polynomials are relative invariants and every relative invariant is uniquely expressed in the form cP_{1}(x)^{m_{1}} ⋯ P_{r}(x)^{m_{r}} (c ∈ ℂ^{×}, (m_{1}, …, m_{r}) ∈ ℤ^{r}). In this sense, P_{1}, …, P_{r} are called the basic relative invariants of (G, ρ, V) [9, Definition 2.10]. They are necessarily homogeneous polynomials [9, Corollary 2.7]. Let P̄_{1}, …, P̄_{r} be the polynomials obtained by complex conjugation of coefficients of P_{1}, …, P_{r} respectively. By the choice of the basis of V, those are the basic relative invariants of the dual prehomogeneous vector space (G, ρ^{∗}, V^{∗}), when we identify ℂ[V^{∗}] with ℂ[x] by the dual basis (see [5, Lemma 1.5] or [9, Proposition 2.21]).

In the following, we consider P_{1}(x)^{s_{1}} ⋯ P_{r}(x)^{s_{r}} as a many-valued holomorphic function on ℂ^{r} × (V \ S_{0}). Write grad_{x} = (∂/∂x_{1}, …, ∂/∂x_{r}). For each m = (m_{1}, …, m_{r}) ∈ ℤ_{≥0}^{r},

\frac{P̄_{1}(grad_{x})^{m_{1}} ⋯ P̄_{r}(grad_{x})^{m_{r}}(P_{1}(x)^{s_{1}+m_{1}} ⋯ P_{r}(x)^{s_{r}+m_{r}})}{P_{1}(x)^{s_{1}} ⋯ P_{r}(x)^{s_{r}}} ∈ ℂ[s_{1}, …, s_{r}; x_{1}, …, x_{n}]

is (absolutely) invariant under the action of G. Since every absolute invariant in ℂ[x] is a constant [9, Proposition 2.4], it is independent of x. Thus there exists a polynomial b_{m}(s) ∈ ℂ[s_{1}, …, s_{r}] such that

P̄_{1}(grad_{x})^{m_{1}} ⋯ P̄_{r}(grad_{x})^{m_{r}}(P_{1}(x)^{s_{1}+m_{1}} ⋯ P_{r}(x)^{s_{r}+m_{r}}) = b_{m}(s)P_{1}(x)^{s_{1}} ⋯ P_{r}(x)^{s_{r}}.

Moreover, it is known that the degree of b_{m}(s) (on s) is equal to the degree of ∏_{i=1}^{r} P_{i}(x)^{m_{i}} (on x); this follows from an easy modification of the proof given in [5, Lemma 1.7] or [9, Proposition 2.22].

Definition 2.2.1. The polynomials b_{m}(s) are called the b-functions of P_{1}, …, P_{r}.

By calculating

P̄_{1}(grad_{x})^{m_{1}+m′_{1}} ⋯ P̄_{r}(grad_{x})^{m_{r}+m′_{r}}(P_{1}(x)^{s_{1}+m_{1}+m′_{1}} ⋯ P_{r}(x)^{s_{r}+m_{r}+m′_{r}})

in two ways, we have

b_{m+m′}(s) = b_{m′}(s + m)b_{m}(s)

for all m = (m_{1}, …, m_{r}), m′ = (m′_{1}, …, m′_{r}) ∈ ℤ_{≥0}^{r}. Hence the map ℤ_{≥0}^{r} → ℂ(s)^{×}, m ↦ b_{m}(s) is uniquely extended to a 1-cocycle b : ℤ^{r} → ℂ(s)^{×}. By the Ore-Sato theorem (Theorem 2.1.3), each b_{m}(s) (0 ≠ m = (m_{1}, …, m_{r}) ∈ ℤ_{≥0}^{r}) is written as

(2.2)  b_{m}(s) = c_{1}^{m_{1}} ⋯ c_{r}^{m_{r}} ∏_{i=1}^{l} ∏_{j} ∏_{ν=0}^{μ_{i}(m)−1}(μ_{i}(s) + α_{i} + j + ν)^{z_{ij}},

where the notations are taken as in §2.1 and η_{i} = ∑_{j} z_{ij}λ_{i}^{j} satisfying the conditions in Corollary 2.1.4. We take them so that (μ_{1}(s) + α_{1}), …, (μ_{l}(s) + α_{l}) are not translated into one another by the action of Ξ. It is known that each α_{i} + j (with z_{ij} > 0) is a positive rational number (due to M. Kashiwara [8]). Moreover, we have c_{1}, …, c_{r} ∈ ℝ_{>0} since

b_{m}(0) = P̄_{1}(grad_{x})^{m_{1}} ⋯ P̄_{r}(grad_{x})^{m_{r}}(P_{1}(x)^{m_{1}} ⋯ P_{r}(x)^{m_{r}}) ∈ ℝ_{>0}

for all m = (m_{1}, …, m_{r}) ∈ ℤ_{≥0}^{r}. Let d_{k} be the degree of P_{k}(x) for k = 1, …, r. Then the observation just before Definition 2.2.1 implies that

(2.3)  d_{k} = ∑_{i=1}^{l} μ_{i}(e_{k})ε(η_{i}) (k = 1, …, r),

where e_{1} = (1, 0, …, 0), …, e_{r} = (0, …, 0, 1) and ε(η_{i}) = ∑_{j} z_{ij}.

2.3. Local zeta functions over ℂ

We identify V with ℂ^{n} by the basis fixed in §2.2. Let dx denote the Haar measure on V normalized to satisfy

∫_{V} exp(−2πx^{t}x̄)dx = 1,

where x = (x_{1}, …, x_{r}) ∈ ℂ^{n} and ^{t}x̄ denotes the transposition of the complex conjugate of x. Let | · |_{ℂ} be the valuation of ℂ defined by |z|_{ℂ} = zz̄ = |z|^{2} for z ∈ ℂ. The integral

Z_{ℂ}(s) = ∫_{V} |P_{1}(x)|_{ℂ}^{s_{1}} ⋯ |P_{r}(x)|_{ℂ}^{s_{r}} exp(−2πx^{t}x̄)dx

converges when s ∈ {(s_{1}, …, s_{r}) ∈ ℂ^{r} | Re(s_{1}), …, Re(s_{r}) > 0}, and hence Z_{ℂ}(s) is a holomorphic function on this region. Our purpose is to show that Z_{ℂ}(s) is equal to

γ_{ℂ}(s) := ∏_{k=1}^{r}((2π)^{−d_{k}}c_{k})^{s_{k}} ∏_{i=1}^{l} ∏_{j} (\frac{Γ(μ_{i}(s) + α_{i} + j)}{Γ(α_{i} + j)})^{z_{ij}}

with the notations in (2.2).

Theorem 2.3.1. Z_{ℂ}(s) has a meromorphic continuation to ℂ^{r} and Z_{ℂ}(s) = γ_{ℂ}(s).

Actually, the first part of this theorem has been known (see [2]) and our aim is to obtain the second part. To prove this theorem, we need the following lemma:

Lemma 2.3.2. For s = (s_{1}, …, s_{r}) ∈ ℂ^{r} and m = (m_{1}, …, m_{r}) ∈ ℤ_{≥0}^{r}, we have

P̄_{1}(grad_{x})^{m_{1}} ⋯ P̄_{r}(grad_{x})^{m_{r}}(|P_{1}(x)|_{ℂ}^{s_{1}} ⋯ |P_{r}(x)|_{ℂ}^{s_{r}}P_{1}(x)^{m_{1}} ⋯ P_{r}(x)^{m_{r}})

= b_{m}(s)|P_{1}(x)|_{ℂ}^{s_{1}} ⋯ |P_{1}(x)|_{ℂ}^{s_{1}}

on V \ S_{0}.

Proof. Locally we choose the branching of the value of log P_{i}(x) and log \overline{P_{i}(x)} = log P̄_{i}(x̄) so that |P_{i}(x)|_{ℂ}^{s_{i}} = P̄_{i}(x̄)^{s_{i}}P_{i}(x)^{s_{i}} holds. Since P̄_{j}(grad_{x}) and P̄_{i}(x̄)^{s_{i}} commute as differential operators for i, j = 1, ⋯, r, we have

P̄_{1}(grad_{x})^{m_{1}} ⋯ P̄_{r}(grad_{x})^{m_{r}}(|P_{1}(x)|_{ℂ}^{s_{1}} ⋯ |P_{r}(x)|_{ℂ}^{s_{r}}P_{1}(x)^{m_{1}} ⋯ P_{r}(x)^{m_{r}})

= P̄_{1}(grad_{x})^{m_{1}} ⋯ P̄_{r}(grad_{x})^{m_{r}}(P̄_{1}(x̄)^{s_{1}} ⋯ P̄_{r}(x̄)^{s_{r}}P_{1}(x)^{s_{1}+m_{1}} ⋯ P_{r}(x)^{s_{r}+m_{r}})
= P̄_{1}(x̄)^{s_{1}} ⋯ P̄_{r}(x̄)^{s_{r}} P̄_{1}(grad_{x})^{m_{1}} ⋯ P̄_{r}(grad_{x})^{m_{r}}(P_{1}(x)^{s_{1}+m_{1}} ⋯ P_{r}(x)^{s_{r}+m_{r}})
= P̄_{1}(x̄)^{s_{1}} ⋯ P̄_{r}(x̄)^{s_{r}}b_{m}(s)P_{1}(x)^{s_{1}} ⋯ P_{r}(x)^{s_{r}}
= b_{m}(s)|P_{1}(x)|_{ℂ}^{s_{1}} ⋯ |P_{1}(x)|_{ℂ}^{s_{1}}.
□

Proof of Theorem 2.3.1. By Lemma 2.3.2, we have

b_{m}(s)Z_{ℂ}(s)
= ∫_{V} {(∏_{i=1}^{r} P̄_{i}(grad_{x})^{m_{i}}) (∏_{j=1}^{r} |P_{j}(x)|_{ℂ}^{s_{j}}P_{j}(x)^{m_{j}})} exp(−2πx^{t}x̄)dx
= ∫_{V} (∏_{j=1}^{r} |P_{j}(x)|_{ℂ}^{s_{j}}P_{j}(x)^{m_{j}}) (∏_{i=1}^{r} P̄_{i}(− grad_{x})^{m_{i}}) exp(−2πx^{t}x̄)dx
= (2π)^{∑_{k=1}^{r} d_{k}m_{k}} ∫_{V} ∏_{j=1}^{r} |P_{j}(x)|_{ℂ}^{s_{j}}P_{j}(x)^{m_{j}} ∏_{i=1}^{r} P̄_{i}(x̄)^{m_{i}} exp(−2πx^{t}x̄)dx
= (2π)^{∑_{k=1}^{r} d_{k}m_{k}} ∫_{V} ∏_{j=1}^{r} |P_{j}(x)|_{ℂ}^{s_{j}+m_{j}} exp(−2πx^{t}x̄)dx
= (2π)^{∑_{k=1}^{r} d_{k}m_{k}}Z_{ℂ}(s + m)

for (s_{1}, …, s_{r}) ∈ {(s_{1}, …, s_{r}) ∈ ℂ^{r} | Re(s_{1}), …, Re(s_{r}) > 0} and m = (m_{1}, …, m_{r}) ∈ ℤ_{≥0}^{r}. Hence Z_{ℂ}(s) satisfies the following equation:

(2.4)  Z_{ℂ}(s + m) = (2π)^{− ∑_{k=1}^{r} d_{k}m_{k}}b_{m}(s)Z_{ℂ}(s).

Then Z_{ℂ}(s) has a meromorphic continuation to ℂ^{r} by this equation (as in [2]). Furthermore, (2.4) implies that both Z_{ℂ}(s) and γ_{ℂ}(s) satisfy the same (G_{m}-primitive) difference equation. Therefore C(s) := Z_{ℂ}(s)/γ_{ℂ}(s) is a holomorphic and periodic function with periods e_{1}, …, e_{r}. To show that C(s) is a constant function, we investigate the asymptotic behavior of C(s) on the strip S = {(s_{1}, …, s_{r}) ∈ ℂ^{r} | 1 ≤ Re(s_{i}) ≤ 2 (i = 1, ⋯, r)}.

Put S^{n−1}(ℂ) = {x ∈ V | x^{t}x̄ = 1} (≃ SO(2n, ℝ)/SO(2n − 1, ℝ) as real manifolds). We identify V \ \{0\} with ℝ_{>0} × S^{n−1}(ℂ) via x ↦ (ξ, u) = (\sqrt{x^{t}x̄}, x/\sqrt{x^{t}x̄}). Take the SO(2n, ℝ)-invariant measure du on S^{n−1}(ℂ) such that dx = 2^{n}ξ^{2n−1}dξdu on V \ {0}. Let

ψ(s) = 2^{n−1} ∫_{S^{n−1}(ℂ)} |P_{1}(u)|_{ℂ}^{s_{1}} ⋯ |P_{r}(u)|_{ℂ}^{s_{r}}du.

Then we have

Z_{ℂ}(s) = ∫_{V \{0}} |P_{1}(x)|_{ℂ}^{s_{1}} ⋯ |P_{r}(x)|_{ℂ}^{s_{r}} exp(−2πx^{t}x̄)dx
= 2ψ(s) ∫_{0}^{∞} ξ^{2(∑_{k=1}^{r} d_{k}s_{k}+n)−1} exp(−2πξ^{2})dξ
= (2π)^{− ∑_{k=1}^{r} d_{k}s_{k}−n}ψ(s)Γ(d_{1}s_{1} + ⋯ + d_{r}s_{r} + n)

when Re(s_{i}) > 0 (i = 1, ⋯, r). Since

|(2π)^{− ∑_{k=1}^{r} d_{k}s_{k}−n}ψ(s)| ≤ (2π)^{− ∑_{k=1}^{r} d_{k} Re(s_{k})−n}2^{n−1} ∫_{S^{n−1}(ℂ)} |P_{1}(u)|_{ℂ}^{Re(s_{1})} ⋯ |P_{r}(u)|_{ℂ}^{Re(s_{r})}du,

|(2π)^{− ∑_{k=1}^{r} d_{k}s_{k}−n}ψ(s)| is bounded in S. Let a_{1}, …, a_{r} be arbitrary positive real numbers and t_{0} a real variable. Then the well-known asymptotic behavior of the gamma function and (2.3) imply that

C(1 + \sqrt{−1}a_{1}t_{0}, …, 1 + \sqrt{−1}a_{r}t_{0}) = o(exp(|t_{0}|)) (|t_{0}| → ∞).

Unless C(s) is a constant function, this is impossible. Thus the proof is completed since C(0) = 1. □

2.4. Local zeta functions over ℝ

In this section, we assume that the prehomogeneous vector space (G, ρ, V) is defined over ℝ (in the sense of [13, §1] or [9, §2.1]) and replace P_{1}, …, P_{r} and b as follows. Since S_{0} is defined over ℝ (see [13, Lemma 1.1]), we can take irreducible polynomials P_{1}, …, P_{r} ∈ ℝ[x] which define the ℝ-irreducible components of S_{0} (possibly r becomes smaller). Here, we are assuming the basis of V is fixed so that the ℝ-rational points of ρ(G) is self-adjoint with respect to the induced ℝ-basis of V_{ℝ}, the ℝ-rational points of V. Those P_{1}, …, P_{r} are often called the basic relative invariants of (G, ρ, V) over ℝ. They are also considered as the basic relative invariants of (G, ρ^{∗}, V^{∗}) over ℝ. Then there exist polynomials b_{m}(s) such that

P_{1}(grad_{x})^{m_{1}} ⋯ P_{r}(grad_{x})^{m_{r}}(P_{1}(x)^{s_{1}+m_{1}} ⋯ P_{r}(x)^{s_{r}+m_{r}}) = b_{m}(s)P_{1}(x)^{s_{1}} ⋯ P_{r}(x)^{s_{r}}

for m = (m_{1}, …, m_{r}) ∈ ℤ_{≥0}^{r}. All properties on b_{m}(s) described in §2.2 also hold. Keep the notations in (2.2) and in (2.3).

Let dx denote the Lebesgue measure on V_{ℝ} identified with ℝ^{r}. As in the previous section, the integral

Z_{ℝ}(s) = ∫_{V_{ℝ}} |P_{1}(x)|^{s_{1}} ⋯ |P_{r}(x)|^{s_{r}} exp(−πx^{t}x)dx

converges when (s_{1}, …, s_{r}) ∈ {(s_{1}, …, s_{r}) ∈ ℂ^{r} | Re(s_{1}), …, Re(s_{r}) > 0}, and hence Z_{ℝ}(s) is a holomorphic function on this region. From now on, we assume the following:

Assumption 2.4.1. Every term of P_{i}(x) is a multilinear form on x for i = 1, …, r, i.e. each P_{i}(x) is of the form:

P_{i}(x) = ∑_{1≤j_{1}<···<j_{d_{i}}≤n} a_{j_{1}···j_{d_{i}}}x_{j_{1}} ⋯ x_{j_{d_{i}}}  (i = 1, …, r).

We will see that Z_{ℝ}(2s) satisfies a certain G_{m}-primitive difference equation in such a case. By the assumption above, we see

P_{i}(grad_{x}) exp(−πx^{t}x) = (−2π)^{d_{i}}P_{i}(x) exp(−πx^{t}x) (i = 1, ⋯, r).

Thus

b_{e_{i}}(s)Z_{ℝ}(s) = ∫_{V_{ℝ}}{P_{i}(grad_{x})(|P_{1}(x)|^{s_{1}} ⋯ |P_{r}(x)|^{s_{r}}P_{i}(x))} exp(−πx^{t}x)dx
= (−1)^{d_{i}} ∫_{V_{ℝ}}(|P_{1}(x)|^{s_{1}} ⋯ |P_{r}(x)|^{s_{r}}P_{i}(x))P_{i}(grad_{x}) exp(−πx^{t}x)dx
= (2π)^{d_{i}} ∫_{V_{ℝ}} |P_{1}(x)|^{s_{1}} ⋯ |P_{r}(x)|^{s_{r}}P_{i}(x)P_{i}(x) exp(−πx^{t}x)dx
= (2π)^{d_{i}} ∫_{V_{ℝ}} |P_{1}(x)|^{s_{1}} ⋯ |P_{i}(x)|^{s_{i}+2} ⋯ |P_{r}(x)|^{s_{r}} exp(−πx^{t}x)dx
= (2π)^{d_{i}}Z_{ℝ}(s + 2e_{i}),

for i = 1, …, r. Hence Z_{ℝ}(s) satisfies the equation

(2.5)  Z_{ℝ}(s + 2e_{i}) = (2π)^{−d_{i}}b_{e_{i}}(s)Z_{ℝ}(s) (i = 1, …, r).

We can obtain a meromorphic continuation of Z_{ℝ}(s) to ℂ^{r}. By considering Z_{ℝ}(s + 2e_{j} + 2e_{k}) in two ways, we have

(2.6)  b_{e_{j}}(s)b_{e_{k}}(s + 2e_{j}) = b_{e_{j}}(s + 2e_{k})b_{e_{k}}(s) (j, k = 1, ⋯, r).

Lemma 2.4.2. When Assumption 2.4.1 holds, μ_{i}(e_{j}) (i = 1, ⋯, l, j = 1, ⋯, r) are equal to either 0 or 1 and hence each η_{i} · (μ_{i}(s) + α_{i}) is a polynomial.

Proof. For each i = 1, ⋯, l, take any j, k such that μ_{i}(e_{j}), μ_{i}(e_{k}) > 0 and write η_{i} = ∑_{u=κ}^{κ′} z_{iu}λ_{i}^{u} (z_{iκ}, z_{iκ′} ≠ 0). By (2.6), we have

∏_{u=κ}^{κ′} ( ∏_{v=0}^{μ_{i}(e_{j})−1}(μ_{i}(s) + α_{i} + u + v) ∏_{w=0}^{μ_{i}(e_{k})−1}(μ_{i}(s) + 2μ_{i}(e_{j}) + α_{i} + u + w))^{z_{iu}}

= ∏_{u=κ}^{κ′} ( ∏_{v=0}^{μ_{i}(e_{j})−1}(μ_{i}(s) + 2μ_{i}(e_{k}) + α_{i} + u + v) ∏_{w=0}^{μ_{i}(e_{k})−1}(μ_{i}(s) + α_{i} + u + w))^{z_{iu}} .

In each side, the constant terms of the factors

\begin{matrix}(μ_{i}(s) + 2μ_{i}(e_{j}) + α_{i} + κ′ + μ_{i}(e_{k}) − 1), \\ (μ_{i}(s) + 2μ_{i}(e_{k}) + α_{i} + κ′ + μ_{i}(e_{j}) − 1) \end{matrix}

are maximal respectively; recall that these factors are not canceled (Corollary 2.1.4). Hence the two factors coincide and we have μ_{i}(e_{j}) = μ_{i}(e_{k}). Therefore the all non-zero numbers among μ_{i}(e_{1}), …, μ_{i}(e_{r}) coincide. This proves the lemma since the greatest common divisor of them is 1. □

Theorem 2.4.3. When Assumption 2.4.1 holds, we have

Z_{ℝ}(s) = ∏_{k=1}^{r}(π^{−d_{i}}c_{i})^{\frac{s_{i}}{2}} ∏_{i=1}^{l} ∏_{j} (\frac{Γ((μ_{i}(s) + α_{i} + j)/2)}{Γ((α_{i} + j)/2)})^{z_{ij}} .

Proof. Let β : ℤ^{r} → ℂ(s)^{×} be the map given by

β_{m}(s) = ∏_{k=1}^{r}(π^{−d_{k}}c_{k})^{m_{k}} ∏_{i=1}^{l} ∏_{j} \{ \begin{matrix}∏_{ν=0}^{μ_{i}(m)−1}(μ_{i}(s) + \frac{α_{i} + j}{2} + ν)^{z_{ij}} & (μ_{i}(m) ≥ 1) \\ 1 & (μ_{i}(m) = 0) \\ ∏_{ν=μ_{i}(m)}^{−1}(μ_{i}(s) + \frac{α_{i} + j}{2} + ν)^{−z_{ij}} & (μ_{i}(m) ≤ −1). \end{matrix}

We see that β is a 1-cocycle since Z_{ℝ}(2s) satisfies the G_{m}-primitive difference equation

Z_{ℝ}(2(s + m)) = β_{m}(s)Z_{ℝ}(2s)

by (2.5) and Lemma 2.4.2. In addition, the second assertion of Lemma 2.4.2 implies that all z_{ij} is non-negative. Let

γ_{ℝ}(s) := ∏_{k=1}^{r}(π^{−d_{i}}c_{i})^{\frac{s_{i}}{2}} ∏_{i=1}^{l} ∏_{j} (\frac{Γ((μ_{i}(s) + α_{i} + j)/2)}{Γ((α_{i} + j)/2)})^{z_{ij}} .

Then both γ_{ℝ}(2s) and Z_{ℝ}(2s) satisfy the same difference equation.

Put S^{n−1}(ℝ) = {x ∈ V_{ℝ} | x^{t}x = 1} (≃ SO(n, ℝ)/SO(n − 1, ℝ)). We identify V_{ℝ} \ {0} with ℝ_{>0} × S^{n−1}(ℝ) via x ↦ (ξ, u) = (\sqrt{x^{t}x}, x/\sqrt{x^{t}x}). Take the SO(n, ℝ)-invariant measure du on S^{n−1}(ℝ) such that dx = ξ^{n−1}dξdu on V_{ℝ} \ {0}. Let

ψ(s) = \frac{1}{2} ∫_{S^{n−1}(ℝ)} |P_{1}(u)|^{s_{1}} ⋯ |P_{r}(u)|^{s_{r}}du.

Then we have

Z_{ℝ}(s) = ∫_{V_{ℝ}\{0}} |P_{1}(x)|^{s_{1}} ⋯ |P_{r}(x)|^{s_{r}} exp(−πx^{t}x)dx

= 2ψ(s) ∫_{0}^{∞} ξ^{∑_{k=1}^{r} d_{k}s_{k}+n−1} exp(−πξ^{2})dξ

= π^{(− ∑_{k=1}^{r} d_{k}s_{k}−n)/2}ψ(s)Γ (\frac{d_{1}s_{1} + ⋯ + d_{r}s_{r} + n}{2})

when Re(s_{i}) > 0 (i = 1, ⋯, r). Therefore, similarly to the proof of Theorem 2.3.1, we obtain that Z_{ℝ}(2s)/γ_{ℝ}(2s) = 1. □

References

[1] E. Artin, translated by M. Butler, “The gamma function”, Holt, Rinehart and Winston, 1964.

[2] I.N. Bernshtein, The analytic continuation of generalized functions with respect to a parameter, Funct. Anal. Appl. 6 (1972), 273–285.

[3] M. Fujigami, A generalization of the Bohr-Mollerup theorem and the Γ-factors of local functional equations (Japanese), Sūrikaisekikenkyūsho Kōkyūroku, Kyoto Univ., 1238 (2001), 20–29.

[4] I.M. Gel’fand, M.I. Graev, V.S. Retakh, General hypergeometric systems of equations and series of hypergeometric type, Russian Math. Surveys 47:4 (1992), 1–88.

[5] A. Gyoja, Theory of prehomogeneous vector spaces without regularity condition, Publ. Res. Inst. Math. Sci., Kyoto Univ., 27 (1991), 861–922.

[6] J. Igusa, On functional equations of complex powers, Invent. math. 85 (1986), 1–29

[7] J. Igusa, “An introduction to the theory of local zeta functions”, Studies in Advanced Mathematics 14, AMS/IP, 2000.

[8] M. Kashiwara, B-functions and holonomic systems—rationality of roots of b-functions, Invent. math. 38 (1976), 33–53.

[9] T. Kimura, translated by M. Nagura and T. Niitani, “Introduction to prehomogeneous vector spaces”, Translations of Mathematical Monographs 215, AMS, 2003.

[10] F. Loeser, C. Sabbah, Equations aux différences finies et déterminants d’intégrales de fonctions multiformes, Comment. Math. Helvetici 66 (1991), 458–503.

[11] G.D. Mostow, Self-adjoint groups, Ann. of Math. 62 (1955), 44–55.

[12] O. Ore, Sur la forme des fonctions hypergéométriques de plusieurs variables, J. Math. Pures et Appl. 9 (1930), 311–326.

[13] F. Sato, Zeta functions in several variables associated with prehomogeneous vector spaces I: Functional equations, Tôhoku Math. J. (2) 34 (1982), 437–483.

[14] M. Sato, note by T. Shintani, translated by M. Muro, Theory of prehomogeneous vector spaces (algebraic part)—the English translation of Sato’s lecture from Shintani’s note, Nagoya Math. J. 120 (1990), 1–34.

Part 3. Picard-Vessiot theories for artinian simple module algebras

Introduction of Part 3

The purpose of this part is to develop a unified Picard-Vessiot theory, including Picard-Vessiot theories for differential equations and for difference equations. The presented result was obtained by the author and Masuoka. The contents from Section 3.1 to Section 3.8 are made by retouching [2] including some omitted proofs. Section 3.9 was recently published as another paper [1], in which some descriptions, such as the definition of 𝒢-extension, etc, are improved than this article (but a sentence “...the smoothness assumption; namely, it is spanned by divided power sequences...” in Introduction of [1] may be a little mistake).

In the usual sense, the “Picard-Vessiot theory” means a Galois theory for linear ordinary differential equations. See [21] for modern treatment. For example, consider the following differential equation over ℂ:

(3.1)  y′′(x) − y′(x) − y(x) = 0.

Let ℂ[∂] (∂ = d/dx) be the ring of differential operators with constant coefficients. The differential module (ℂ[∂]-module) associated with the equation (3.1) is

ℂ[∂]/ℂ[∂](∂^{2} − ∂ − 1) ≃ (ℂ[∂]/ℂ[∂](∂ − \frac{1+\sqrt{5}}{2})) ⊕ (ℂ[∂]/ℂ[∂](∂ − \frac{1−\sqrt{5}}{2})).

Thus the space of solutions is given by the 2-dimensional ℂ-vector space ℂα + ℂβ with α = e^{\frac{1+\sqrt{5}}{2} x}, β = e^{\frac{1−\sqrt{5}}{2} x}. A differential field (i.e. a field given a derivation) L including this space is called a splitting field for the equation. If L is minimal with this property, it is called a minimal splitting field. For the equation above, L = ℂ(α, β) is a minimal splitting field. Like Galois extensions, L/ℂ is then an extension of a special type, called a Picard-Vessiot extension. For such an extension, we can take a Galois group, called the differential Galois group (or the Picard-Vessiot group) as an algebraic group defined by Aut_{ℂ[∂],ℂ-alg}(L) =: G(L/ℂ), where Aut_{ℂ[∂],ℂ-alg} denotes the ℂ[∂]-linear and ℂ-algebra automorphisms. We have G(L/ℂ) = G_{m} × G_{m} in this case, and we can obtain the Galois correspondence between closed subgroups of G(L/ℂ) and intermediate differential fields of L/ℂ. For example, the differential field ℂ(α) corresponds to {1} × G_{m} (or G_{m} × {1} according to the choice of the group action).

An analogous theory for difference equations is also known. Białynicki-Birula [4] and Franke [8] first developed such a theory for inversive difference fields, i.e. fields given an automorphism (though Białynicki-Birula’s paper was intended for more general theory, not only for difference fields). A definition of Picard-Vessiot extensions of inversive difference fields and Galois correspondences were obtained there. But the theory had a difficulty on the existence of suitable solution fields. For example, consider the Fibonacci recurrence

(3.2)  a(n + 2) − a(n + 1) − a(n) = 0.

Let ℂ[τ, τ^{−1}] (τ : n ↦ n+1) be the ring of difference operators with constant coefficients. The difference module (ℂ[τ, τ^{−1}]-module) associated with the equation (3.2) is

\begin{matrix}ℂ[τ, τ^{−1}]/ℂ[τ, τ^{−1}](τ^{2} − τ − 1) \\ ≃ (ℂ[τ, τ^{−1}]/ℂ[τ, τ^{−1}](τ − \frac{1+\sqrt{5}}{2})) ⊕ (ℂ[τ, τ^{−1}]/ℂ[τ, τ^{−1}](τ − \frac{1−\sqrt{5}}{2})). \end{matrix}

Let S_{ℂ} denote the ring of complex sequences (see [20, Example 3]). The space of solutions in S_{ℂ} is given by the 2-dimensional ℂ-vector space ℂα + ℂβ with α = \{(\frac{1+\sqrt{5}}{2})^{n}\}, β = \{(\frac{1−\sqrt{5}}{2})^{n}\} ∈ S_{ℂ}. But one can not take any splitting field which becomes a Picard-Vessiot extension for this equation. If a subring in S_{ℂ} contains α, β, then it has a zero divisor:

(αβ − 1)(αβ + 1) = 0.

On the other hand, if we take any inversive difference field which includes a 2-dimensional ℂ-vector space of solutions of (3.2), then it necessarily contains a new constant (see [20, p. 2]). However, overcoming this difficulty, the Picard-Vessiot theory for difference equations in modern sense was developed by van der Put and Singer [20] with the notion of Picard-Vessiot rings, as follows. Consider the Laurent polynomial ring ℂ[x, y, (xy)^{−1}] as an inversive difference ring by

τx = \frac{1 + \sqrt{5}}{2}x, τy = \frac{1 − \sqrt{5}}{2}y.

On sees that ⟨(xy − 1)(xy + 1)⟩ ⊂ ℂ[x, y, (xy)^{−1}] is a maximal difference ideal. Put

A = ℂ[x, y, (xy)^{−1}]/⟨(xy − 1)(xy + 1)⟩.

Then A is a Picard-Vessiot ring for the equation (3.2) in their sense (see [20, Definition 1.4]). The total quotient ring of a Picard-Vessiot ring is called the total Picard-Vessiot ring [20, Definition 1.22]. For the equation (3.2), we have the following total Picard-Vessiot ring:

(3.3)  \begin{matrix}Q(A) & ≃ & ℂ(α) × ℂ(α) \\ x & ↦ & (α, α) \\ y & ↦ & (α^{−1}, −α^{−1}). \end{matrix}

Then the difference Galois group for the equation can be defined by G(Q(A)/ℂ) := Aut_{ℂ[τ,τ^{−1}],ℂ-alg}(Q(A)). In this case, we have G(Q(A)/ℂ) ≃ G_{m} × ℤ/2ℤ. We obtain the Galois correspondence between closed subgroups of G(Q(A)/ℂ) and intermediate difference subrings of Q(A)/ℂ such that every non-zero divisor is invertible (see [20, Theorem 1.29]). For example, ℂ(α) (= ℂ(α)(1, 1)) corresponds to {1} × ℤ/2ℤ and ℂ × ℂ corresponds to G_{m} × {1}.

A unified approach to both differential and difference cases was first attempted by Białynicki-Birula [4], though it was a theory for field extensions. Including the case that the solution algebras can have zero divisors, André [3] gave such a unified approach from the viewpoint of non-commutative differential geometry with the theory of tannakian categories [5, 6]. Alternatively we develop a unified Picard-Vessiot theory by a different way based on Takeuchi’s Hopf algebraic approach: Takeuchi [27] beautifully clarified the heart of the Picard-Vessiot theory in the generalized context of C-ferential fields, intrinsically defining PV extensions and the minimal splitting fields of C-ferential modules. By replacing linear algebraic groups with affine group schemes (or equivalently commutative Hopf algebras), he succeeded in removing from many of the results the assumptions of finite generation, zero characteristic and algebraic closedness. For a cocommutative coalgebra C with a specific grouplike 1_{C}, a C-ferential field [27] is a field given a unital, measuring action by C; the concept includes differential fields, Δ-fields [13], fields with higher derivations [18], and difference fields (even non-inversive ones are included). However, it was also a theory for field extensions and the assumption that the tensor bialgebra T (C^{+}) [27, p. 485] is a Birkhoff-Witt coalgebra (see [27, p. 504] or Assumption 3.3.1), is required for the existence theorem of minimal splitting algebras.

In this article, we consider module algebras over a cocommutative, pointed smooth Hopf algebra D. Thus D is of the form D = D^{1}#RG over a fixed field, say R, where G is the group of grouplikes in D, and the irreducible component D^{1} containing 1 is a Birkhoff-Witt coalgebra. An inversive difference ring which includes R in its constants is precisely a D-module algebra when D^{1} = R and G is the free group with one generator. Differential rings are also within our scope, though only in characteristic zero because of the smoothness assumption. Precisely a differential ring which includes R (of zero characteristic) in its constants is a D-module algebra when D^{1} = R[∂] with a primitive ∂ and G is trivial. Algebras with higher derivations of infinite length fit in the assumption, in arbitrary characteristic. An algebra (over R) with R-linear higher derivations d_{0} = id, d_{1}, d_{2}, … of infinite length is precisely a module algebra over the Hopf algebra R⟨d_{1}, d_{2}, …⟩, which denotes the (non-commutative) free algebra generated by d_{1}, d_{2}, …, and in which 1, d_{1}, d_{2}, … form a divided power sequence. This Hopf algebra becomes a Birkhoff-Witt coalgebra in arbitrary characteristic.

Throughout this article, D-module algebras are all supposed to be commutative. A Dmodule algebra K is said to be artinian simple (AS) if it is artinian as a ring and simple as a D-module algebra. The last condition means that K has no non-trivial D-stable ideal. For example, the total Picard-Vessiot ring considered in (3.3) is an AS ℂ[τ, τ^{−1}]- module algebra. Of course differential fields over ℂ are AS ℂ[∂]-module algebras. In this sense we can generalize and unify the Picard-Vessiot theories for differential and difference equations, involving the theory of van der Put and Singer [20].

Let L be an AS D-module algebra. If P ⊂ L is a maximal ideal, then one will see that L_{1} := L/P is a module field over the Hopf subalgebra D(G_{P}) := D^{1}#RG_{P}, where G_{P} denotes the subgroup (necessarily of finite index) of the stabilizers of P. Moreover, L can recover from L_{1}, so as

L = D ⊗_{D(G_{P})} L_{1} = ⨁_{g∈G/G_{P}} g ⊗ L_{1},

where the product in L recovers from the component-wise product (g ⊗ a)(g ⊗ b) = g ⊗ ab in the last direct sum; see Section 3.3. (For example, when D = ℂ[τ, τ^{−1}] and L = Q(A) ≃ ℂ(α) × ℂ(α) as above, take P = ⟨xy − 1⟩ ⊂ Q(A). Then G_{P} = {g^{2} | g ∈ G} ⟶^{∼} 2ℤ under the group isomorphism G ≃ ℤ.) The D-invariants L^{D} = {a ∈ L | da = ε(d)a for all d ∈ D} (where ε denotes the counit of D) in L form a subfield, such that L^{D} ≃ L_{1}^{D(G_{P})}. Following [27], we say that an inclusion K ⊂ L of AS D-module algebras is a Picard-Vessiot (PV) extension iff K^{D} = L^{D} and there exists a (necessarily unique) D-module algebra K ⊂ A ⊂ L such that the total quotient ring Q(A) equals L, and H := (A ⊗_{K} A)^{D} generates the left (or right) A-module A ⊗_{K} A. Then H has a natural structure of a commutative Hopf algebra over K^{D} (= L^{D}), with which A/K is a right H-Galois extension; see Proposition 3.5.2. (In the example (3.3), we have H = ℂ[z_{1}, z_{2}]/⟨z_{1}^{2}z_{2}^{2}−1⟩ with grouplikes z_{1} = x⊗x^{−1}, z_{2} = y⊗y^{−1}.) If an inclusion K ⊂ L of AS D-module algebras is a PV extension, then the induced inclusion K/P ∩ K ⊂ L/P of D(G_{P})-module fields is a PV extension, where P is an arbitrary maximal ideal of L. The converse holds true if G_{P} is normal in G_{P ∩K}; see Proposition 3.7.4 and Theorem 3.7.5.

As our main theorems we prove:

Galois Correspondence (Theorem 3.5.4): Given a PV extension L/K of AS D-module algebras, there is a 1-1 correspondence between the intermediate AS D-module algebras K ⊂ M ⊂ L and the Hopf ideals I in the associated Hopf algebra H; L/M is then a PV extension with the associated Hopf algebra H/I (Proposition 3.5.7). This has the obvious interpretation in terms of the affine group scheme G(L/K) = SpecH corresponding to H, and G(L/K) is isomorphic to the automorphism group scheme Aut_{D,K-alg}(A) (see Section 3.6).

Characterization (Theorem 3.8.7): An inclusion K ⊂ L of AS D-module algebras with K^{D} = L^{D} is a finitely generated PV extension iff L/K is a minimal splitting algebra for some K#D-module V of finite K-free rank, say n; this means that dim_{L^{D}} Hom_{K#D}(V, L) = n and L is “minimal” with this property (see Proposition 3.8.3).

Tensor Equivalence (Theorem 3.8.13): If this is the case, the symmetric tensor category M_{fin}^{H} of finite-dimensional right comodules over the associated Hopf algebra H (or equivalently that category Rep_{G(L/K)} of finite-dimensional linear representations of G(L/K)) is equivalent to the abelian, rigid tensor full subcategory {{V }} “generated” by V, in the tensor category (_{K#D}M, ⊗_{K}, K) of K#D-modules; cf. [21, Theorem 2.33].

Unique Existence (Theorem 3.8.11): Suppose that K^{D} is an algebraically closed field. For every K#D-module V of finite K-free rank, there is a unique (up to isomorphism) minimal splitting algebra L/K which is a (finitely generated) PV extension.

One cannot overestimate the influence of the article [27] by Takeuchi on this article of ours. Especially the main theorems above except the third are very parallel to results in [27], including their proofs. A C-ferential field is equivalent to a module field over the tensor bialgebra T (C^{+}). We remark that even if K, L are fields, the first two theorems above do not imply the corresponding results in [27] since the notion of C-ferential fields is more general than D-module fields in the sense of ours. The fourth only generalizes [27, Theorems 4.5, 4.6] in which T (C^{+}) is supposed to be of Birkhoff-Witt type.

The last section (Section 3.9) treats the solvability theory for Liouville extensions. The notion of Liouville extensions of differential fields first appeared in Kolchin’s historical work on the Picard-Vessiot theory [11]. An extension of differential fields (of zero characteristic) is called Liouville iff it contains no new constants and it is obtained by iterating integrations, exponentiations, and algebraic extensions. It was shown that a Picard-Vessiot extension of differential fields is Liouville iff the connected component of its differential Galois group is solvable. By the Lie-Kolchin triangularization theorem and others [11, Ch. I], we can characterize several types of Liouville extensions in matrix theoretical way. For example, a Liouville extension is obtained only by iterating integrations iff its differential Galois group is unipotent. For the case of an arbitrary characteristic, Okugawa [18] studied the Picard-Vessiot theory for fields with higher derivations of infinite length, and obtained similar results on Liouville extensions.

Liouville extensions of difference fields were first studied by Franke [8]. In the context of [20], Hendriks and Singer [9] studied on Liouville solutions of difference equations with rational function coefficients. They defined the notion of “Liouvillian sequences” and showed that a linear difference equation can be solved in terms of such sequences iff the difference Galois group is solvable. (Moreover, they gave an algorithm to find such Liouville solutions, using Petkovšek’s algorithm [19].)

In the last section, we define the notion of Liouville extensions of AS D-module algebras and prove a solvability theorem in the unified context.

When we study Liouville extensions in terms of affine group schemes, we will meet the following difficulty: the Lie-Kolchin triangularization theorem can not be extended generally to affine group schemes (see [29, Ch. 10]). Certainly there are gaps between the triangulability and the connected solvability, even if the base field is algebraically closed. So we need some intermediate notions and have to study how they are related each other. In Section 3.9.1, we define “Liouville group schemes” so that it is suitable for Liouville extensions defined later, and study how strong the definition is. An algebraic affine group scheme G over a field k is called (k-)Liouville (cf. [13, p. 374]) iff there exists a normal chain of closed subgroup schemes G = G_{0} ▷ G_{1} ▷ ⋯ ▷ G_{r} = {1} such that each G_{i−1}/G_{i} (i = 1, …, r) is at least one of the following types: finite etale, a closed subgroup scheme of G_{a}, or a closed subgroup scheme of G_{m}. When k is algebraically closed, G is Liouville iff the connected component G^{∘} is solvable (Proposition 3.9.5). But in general it does not holds; we show this fact by examples. For connected affine group schemes, we will see the condition to be Liouville is properly stronger than the solvability but weaker than the triangulability.

Let L ⊃ K be an inclusion of AS D-module algebras. For finitely many elements x_{1}, …, x_{n} ∈ L, let K⟨x_{1}, …, x_{n}⟩ denote the smallest AS D-module subalgebra in L including both K and x_{1}, …, x_{n}. L/K is called a G_{a}-primitive extension (resp., a G_{m}- primitive extension) iff there is an x ∈ L such that d(x) ∈ K for all d ∈ D^{+} = Ker ε (resp., x is a non-zero divisor of L (which is necessarily invertible) and d(x)x^{−1} ∈ K for all d ∈ D) and L = K⟨x⟩. We say that L/K is a finite etale extension iff L is a separable K-algebra in the sense of [7], i.e. L is a projective L ⊗_{K} L-module. Then we define Liouville extension as such a finitely generated extension L/K that L^{D} = K^{D} and there exists a sequence of AS D-module algebras K = L_{0} ⊂ L_{1} ⊂ ⋯ ⊂ L_{r} = L such that each L_{i}/L_{i−1} (i = 1, …, r) is at least one of the following types: G_{a}-primitive extension, G_{m}-primitive extension, or finite etale extension. As the last one of the main theorems, we will show the following:

Solvability (Theorem 3.9.17): Let L/K be a finitely generated PV extension. Then the following are equivalent:

(a) L/K is a Liouville extension.

(b) There exists a Liouville extension F/K such that L ⊂ F.

(c) G(L/K) is Liouville. When k is algebraically closed, these are equivalent to:

(d) G(L/K)^{∘} is solvable.

Moreover we will characterize ten types of Liouville extensions just being compatible with [11, §24–27]; see Definition 3.9.15, Corollary 3.9.19 and its following paragraph.

Conventions. Throughout this part, we always work over an arbitrarily fixed field R. All vector spaces, algebras and coalgebras are defined at least over R. All algebras are associative and have the identity element. All modules over an algebra are unital, left modules unless otherwise stated. All separable algebras are taken in the sense of [7]; see also [29, Ch. 6].

The notation Hom_{R} (resp. End_{R}) with a ring R always denotes the set of all Rlinear maps (resp. R-linear endomorphisms), but the unadorned Hom may indicate group homomorphisms or homomorphisms of group schemes. Algebra (resp. coalgebra) maps are always dentoed by Alg (resp. Coalg). The notation Aut indicates automorphisms in some sense; for example, Aut_{D, K-alg} means D-linear and K-algebra automorphisms. Aut in the bold style indicates an associated group functor as in [29, (7.6)]. Coalgebra structures are denoted by (Δ, ε). If we need to specify a coalgebra (or a coring) C, the notation (Δ_{C}, ε_{C}) is also used. For a coalgebra C, C^{+} denotes Ker ε. The antipode of a Hopf algebra is denoted by S. We use the sigma notation (see [23, §1.2, pp. 10–11] or [16, §1.4, pp. 6–7]):

Δ(c) = ∑_{(c)} c_{(1)} ⊗ c_{(2)} etc.

When (M, λ) is a right (resp. left) C-comodule, λ(m) (m ∈ M) is denoted by the sigma notation

λ(m) = ∑_{(m)} m_{(0)} ⊗ m_{(1)} ∈ M ⊗_{R} C (resp. λ(m) = ∑_{(m)} m_{(−1)} ⊗ m_{(0)} ∈ C ⊗_{R} M).

By “a symmetric tensor category (𝔄, ⊗, I)” we mean that (𝔄, ⊗) is a symmetric tensor (or monoidal) category [16, §10.4, p. 199] with a fixed unit object I. We can define algebras, coalgebras, etc., in (𝔄, ⊗, I) naturally by commutative diagrams. For an algebra A in (𝔄, ⊗, I), left A-modules (resp. right A-modules, resp. (A, A)-bimodules) in (𝔄, ⊗, I) can also be defined and the category of them is denoted by _{A}𝔄 (resp. 𝔄_{A}, resp. _{A}𝔄_{A}). For a ring R, _{R}M (resp. M_{R}) denotes the category of left (resp. right) R-modules. For a coalgebra C, M^{C} (resp. ^{C}M) denotes the category of right (resp. left) C-comodules. Moreover, further notations, such as_{A}M^{H}, _{A}^{H}M, etc., which indicate categories of relative Hopf modules are used as in [16, §8.5].

3.1. Basic notions and results on D-module algebras

Let D be a cocommutative bialgebra. An algebra A is called a D-module algebra (see [23, §7.2, p. 153] or [16, §4.1]) iff A has a D-module structure D ⊗_{R} A → A and the action of D measures A to A. The last condition means that

ρ_{A} : A → Hom_{R}(D, A), a ↦ [d ↦ da]

is an algebra map, where Hom_{R}(D, A) is considered as an algebra with the convolution product (see [23, pp. 69–70] or [16, §1.4]); or in other words, using the sigma notation,

d(ab) = ∑_{(d)}(d_{(1)}a)(d_{(2)}b), d(1) = ε(d)1

hold for all d ∈ D and a, b ∈ A. Throughout this part, we assume D-module algebras are commutative unless otherwise stated. Note that the algebra Hom_{R}(D, A) is commutative in our situation, and it has a D-module algebra structure given by

(3.4)  (dφ)(c) = φ(cd) (c, d ∈ D, φ ∈ Hom_{R}(D, A)).

One sees ρ_{A} is an injective D-module algebra map. For a D-module algebra A, the smash product A#D means the algebra which is A ⊗_{R} D with the semi-direct product:

(a#c)(b#d) = ∑_{(c)} a(c_{(1)}b)#c_{(2)}d

(see [23, pp. 155–156] or [16, §4.1]). For A#D-modules V, W ∈ _{A#D}M, we have an A#D-module structure on V ⊗_{A} W given by

(a#d)(v ⊗ w) = a ∑_{(d)} d_{(1)}v ⊗ d_{(2)}w (a ∈ A, d ∈ D, v ∈ V, w ∈ W).

Thus we have an abelian symmetric tensor category [6, Definition 1.15] (_{A#D}M, ⊗_{A}, A) with the canonical symmetry V ⊗_{A} W → W ⊗_{A} V, v ⊗ w ↦ w ⊗ v. For a D-module V,

V^{D} := {v ∈ V | dv = ε(d)v for all d ∈ D}

is called the constants (or the D-invariants) of V. Especially A^{D} becomes an algebra. If V is an A#D-module, we see Hom_{A#D}(A, V) ⟶^{∼} V^{D}, φ ↦ φ(1) is an A^{D}- module isomorphism and in particular End_{A#D}(A) ≃ A^{D} as algebras. The functor (−)^{D} : _{A#D}M → _{A^{D}}M is an exact functor since A is a projective A#D-module (indeed, A#D ≃ A ⊕ (A ⊗_{R} D^{+}) as A#D-modules via a#d ↦ (aε(d), a ⊗ (d − ε(d)))). Let B be a D-module algebra including A as a D-module subalgebra, V an A#D-module, and W a B#D-module. If D is a Hopf algebra with the antipode S, then Hom_{A}(V, W) has a B#D-module structure given by the D-conjugation:

(3.5)  ((b#d)φ)(v) = b ∑_{(d)} d_{(1)}(φ(S(d_{(2)})v) (v ∈ V)

for b ∈ B, d ∈ D, and φ ∈ Hom_{A}(V, W); see [27, Proposition 1.8]. We see Hom_{A}(V, W)^{D} = Hom_{A#D}(V, W). Especially Hom_{A} is an internal Hom of (_{A#D}M, ⊗_{A}, A) in such a case (see [6, p. 109] for the definition of internal Hom).

The following proposition, like Schur’s lemma, is very important:

Proposition 3.1.1. Let 𝔄 be an abelian subcategory of _{ℤ}M (the category of additive groups) whose inclusion functor is additive. An object X in 𝔄 is simple iff

(a) the endomorphism ring E := End_{𝔄}(X) is a division ring, and (b) for every object Y in 𝔄, the evaluation map

ev : Hom_{𝔄}(X, Y) ⊗_{E} X → Y

is injective.

Proof. (“If” part.) Let Y be a subobject of X. Since Hom_{𝔄}(X, Y) is a right ideal of E = End_{𝔄}(X), Hom_{𝔄}(X, Y) equals 0 or E by (a). If Hom_{𝔄}(X, Y) = E, we have Y = X since X ≃ E⊗_{E}X → Y is injective. If Hom_{𝔄}(X, Y) = 0, then E → Hom_{𝔄}(X, X/Y) is injective. Since all E-modules are flat, we have that X ≃ E ⊗_{E} X → Hom_{𝔄}(X, X/Y)⊗_{E} X → X/Y is injective and hence Y = 0.

(“Only if” part.) (a) For 0 ≠ f ∈ E, Im(f) is a non-zero subobject of X and hence Im(f) = X, Ker(f) = 0. Thus f is invertible.

(b) Since X is simple, each 0 ≠ f ∈ Hom_{𝔄}(X, Y) is monic and hence Im(f) is isomorphic to X. It suffices to prove that the sum ∑_{i=1}^{r} Im(f_{i}) ⊂ Y is direct if f_{1}, …, f_{r} are E-linearly independent in Hom_{𝔄}(X, Y). To prove this, we shall use induction on r. When r = 1, the assertion is clear. When r > 1, suppose that the assertion is true for {f_{1}, …, f_{r−1}}. Seeking a contradiction, assume Im(f_{r}) ∩ ∑_{i=1}^{r−1} Im(f_{i}) ≠ 0. Since Im(f_{r}) is simple, we have Im(f_{r}) ⊂ ⨁_{i=1}^{r−1} Im(f_{i}). Then there exist φ_{i} ∈ E (i = 1, …, r − 1) such that the diagram

Im(f_{r}) ⟶^{inclusion} ⨁_{i=1}^{r−1} Im(f_{i}) ⟶^{projection} Im(f_{i})
f_{r}↑  ↑^{f_{i}}
X ⟶^{φ_{i}} X

commutes for i = 1, …, r − 1 since f_{i} : X → Im(f_{i}) is invertible. Then f_{r} = f_{1} ∘ φ_{1} + ⋯ + f_{r−1} ∘ φ_{r−1}. This contradicts that f_{1}, …, f_{r} are E-linearly independent. □

Remark 3.1.2. I heard the above proposition from Professor A. Masuoka. Though it seems well-known, an explicit citation was not found as far as I searched. It is said that Professor Masuoka knew this by a comment from Professor T. Breziński on [15, Theorem 1.1 and the Theorem on p. 232].

Definition 3.1.3. A D-module algebra A is called simple iff it is simple in _{A#D}M, i.e. A has no non-trivial D-stable ideal.

The next corollary follows immediately from Propositoin 3.1.1.

Corollary 3.1.4. A D-module algebra A is simple iff (a) A^{D} is a field, and

(b) for every A#D-module Y, the map

\begin{aligned}& Y^{D} ⊗_{A^{D}} A → Y, & y ⊗ a ↦ ay \\ (or & A ⊗_{A^{D}} Y^{D} → Y, & a ⊗ y ↦ ay) \end{aligned}

is injective.

Proof. Recall that End_{A#D}(A) ≃ A^{D} and Hom_{A#D}(A, Y) ≃ Y^{D}. The evaluation map is identified with the map in (b) above. □

Let A be a D-module algebra and ρ_{A} : A → Hom_{R}(D, A) the associated algebra map. Then Hom_{R}(D, A) has two A-module structures A⊗_{R}Hom_{R}(D, A) → Hom_{R}(D, A), given by (I) a ⊗ φ ↦ ρ_{A}(a) ∗ φ = φ ∗ ρ_{A}(a), and (II) a ⊗ φ ↦ aφ = [d ↦ aφ(d)]. The structure given by (II) can be considered through the following algebra isomorphism:

σ : A ⟶^{∼} Hom_{R}(D, A)^{D}, a ↦ aε_{D},

whose inverse is φ ↦ φ(1). Here we are taking the D-module structure on Hom_{R}(D, A) in the sense of (3.4). As in [27, Corollary 1.4], the next lemma follows from Corollary 3.1.4.

Lemma 3.1.5. If A is simple, then the following map:

β : A ⊗_{A^{D}} A → Hom_{R}(D, A), a ⊗ b ↦ ρ_{A}(b) ∗ (aε_{D}) = aρ_{A}(b)

is a two-sided A-linear (left through σ, right through ρ_{A}) injection.

Proof. Consider Y = Hom_{R}(D, A) as an A#D-module by the A-module structure given by (I) and by the D-module structure in the sense of (3.4):

(a#d)φ = ρ_{A}(a) ∗ (dφ) (a ∈ A, d ∈ D, φ ∈ Y).

This is well-defined:

(a′#d′)((a#d)φ)) = ρ_{A}(a′) ∗ ∑_{(d′)}(d′_{(1)}ρ_{A}(a)) ∗ (d′_{(2)}dφ)

= ρ_{A}(a′) ∗ ∑_{(d′)}ρ_{A}(d′_{(1)}a) ∗ (d′_{(2)}dφ) = ∑_{(d′)}ρ_{A}(a′(d′_{(1)}a)) ∗ (d′_{(2)}dφ)
= ((a′#d′)(a#d))φ.

Then β is injective since A ≃ Y^{D} through σ. □

This lemma has an application when one needs to think of the A^{D}-linear dependence of elements in A. Takeuchi generalized the Wronskian (and Casoratian) criterion as follows:

Proposition 3.1.6. ([27, Proposition 1.5]) Let K be a D-module field. Then a_{1}, …, a_{n} ∈ K are K^{D}-linearly independent iff there exist h_{1}, …, h_{n} ∈ D such that det(h_{i}(a_{j}))_{i,j} ≠ 0.

Proof. We include the proof for convenience.

(“If” part.) If ∑_{j=1}^{n} c_{j}a_{j} = 0 (c_{1}, …, c_{n} ∈ K^{D}), then ∑_{j=1}^{n} c_{j}h_{i}(a_{j}) = 0 for i = 1, …, n. Since the matrix (h_{i}(a_{j}))_{i,j} is invertible, we have c_{1} = ⋯ = c_{n} = 0.

(“Only if” part.) Put W = K^{D}a_{1} + ⋯ + K^{D}a_{n}, an n-dimensional K^{D}-vector subspace of K. Consider the K-linear injection

β : K ⊗_{K^{D}} W → Hom_{R}(D, K) ≃ Hom_{K}(K ⊗_{R} D, K), b ⊗ w ↦ [d ↦ b(dw)] to which β in Lemma 3.1.5 restricts. Let {d_{α}}_{α∈Λ} be an R-basis of D and d_{α}^{∨} be the dual of d_{α} in Hom_{R}(D, K). Then Hom_{K}(K ⊗_{R} D, K) = ∏_{α∈Λ} Kd_{α}^{∨} as a K-vector space. Notice that ρ_{A}(a) = ∑_{α∈Λ}(d_{α}a)d_{α}^{∨} (a ∈ K). Since ρ_{A}(a_{1}), …, ρ_{A}(a_{n}) are K-linearly independent and since K is a field, we obtain a K-basis v_{1}, …, v_{n} of K ⊗_{K^{D}} W such that

β(v_{1}) = h_{1}^{∨} + ∑_{\begin{matrix}α∈Λ \\ d_{α}≠h_{1} \end{matrix}} c_{1,α}d_{α}^{∨}
β(v_{2}) =  h_{2}^{∨} + ∑_{\begin{matrix}α∈Λ \\ d_{α}≠h_{1},h_{2} \end{matrix}} c_{2,α}d_{α}^{∨}
⋮
β(v_{n}) =  h_{n}^{∨} + ∑_{\begin{matrix}α∈Λ \\ d_{α}≠h_{1},...,h_{n} \end{matrix}} c_{n,α}d_{α}^{∨}

for some n elements h_{1}, …, h_{n} ∈ {d_{α}}_{α∈Λ} by sweeping-out. Consider the transposed K-linear map of β:

γ : K ⊗_{R} D → Hom_{K}(K ⊗_{K^{D}} W, K), a ⊗ d ↦ [b ⊗ w ↦ ab(dw)].

Let v_{1}^{∨}, …, v_{n}^{∨} be the dual basis of v_{1}, …, v_{n}. Then we have ^{t}(γ(h_{1}), …, γ(h_{n})) = T^{t}(v_{1}^{∨}, …, v_{n}^{∨}) with a strictly lower triangular matrix T ∈ GL_{n}(K). Thus γ(h_{1}), …, γ(h_{n}) form a K-basis for Hom_{K}(K ⊗_{K^{D}} W, K) ≃ Hom_{K^{D}}(W, K). The K-isomorphism

\begin{matrix}K^{n} & ⟶^{∼} & Hom_{K^{D}}(W, K) & ⟶^{∼} & K^{n} \\ (c_{i})_{i} & ↦ & ∑_{i=1}^{n}c_{i}γ(h_{i}) & ↦ & (∑_{i=1}^{n}c_{i}h_{i}(a_{j}))_{j} \end{matrix}

is precisely the right multiplication of matrix (h_{i}(a_{j}))_{i,j}. It follows that the matrix has an inverse. □

Remark 3.1.7. In the above proof, we see that Ker γ is a left ideal of K#D. If D = R[∂] with one primitive ∂, then Ker γ is generated by a monic differential operator of order n. Thus we can take h_{1} = 1, h_{2} = ∂, …, h_{n} = ∂^{n−1} in such a case. Namely we have the Wronskian criterion in the usual sense. Similarly we also have the ordinary Casoratian criterion for difference fields.

3.2. Tensor equivalences associated to Hopf subalgebras

In what follows we assume that D is a cocommutative Hopf algebra. Let C be a Hopf subalgebra of D. A coalgebra in the tensor category (_{D}M, ⊗_{R}, R) is called a D-module coalgebra. D is a D-module coalgebra, and D̄ := D/DC^{+} is its quotient. The R-abelian category D̄ ; DM of left (D̄, D)-Hopf modules is defined as follows (see [26, pp. 454–455] or [16, §8.5]):

\underline{Objects}. An object of D̄ ; DM is a left D-module M which is also a left D̄-comodule with a structure λ_{M}, say, such that

λ_{M}(dm) = Δ(d)λ_{M}(m) = ∑_{(d)} ∑_{(m)} d_{(1)}m_{(−1)} ⊗ d_{(2)}m_{(0)} ∈ D̄ ⊗_{R} M

for all d ∈ D and m ∈ M.

\underline{Morphisms}. Morphisms of D̄ ; DM are D-module and D̄-comodule maps.

Given objects M, N in D̄ ; DM, let M□D̄N denote the cotensor product; this is by definition the equalizer of the two D̄-colinear maps

M ⊗ N\overrightarrow{⟶}D̄ ⊗ M ⊗ N

given by the structure maps of M, N, or in other words,

M□D̄N = {∑_{i} x_{i} ⊗ y_{i} | ∑_{i} ∑_{(x_{i})}(x_{i})_{(−1)} ⊗ (x_{i})_{(0)} ⊗ y_{i} = ∑_{i} ∑_{(y_{i})}(y_{i})_{(−1)} ⊗ x_{i} ⊗ (y_{i})_{(0)}} .

This is a D-submodule of M ⊗ N, and is further an object in D̄ ; DM. We see that D̄ ; DM = (D̄ ; DM, □D̄, D̄) is a symmetric tensor category. Indeed, the associativity (M□D̄N)□D̄L ⟶^{∼} M□D̄(N□D̄L) and the symmetry M□D̄N ⟶^{∼} N□D̄M are induced by those of (_{D}M, ⊗_{R}) naturally. We have isomorphisms

\begin{matrix}M□D̄D̄ ⟶^{∼} M, ∑_{i}m_{i} ⊗ a_{i} ↦ ∑_{i}m_{i}ε(a_{i}), \\ D̄□D̄M ⟶^{∼} M, ∑_{i}a_{i} ⊗ m_{i} ↦ ∑_{i}ε(a_{i})m_{i}, \end{matrix}

whose inverses are obtained by λ_{M}. Thus D̄ is a unit object.

For an object V in _{C}M, define

Φ(V) = D ⊗_{C} V.

This is naturally an object in D̄ ; DM. We thus have an R-linear functor

Φ : _{C}M → D̄ ; DM.

Proposition 3.2.1. Φ is an equivalence of symmetric tensor categories.

Proof. By [26, Theorem 2 and 4], Φ is a category equivalence; its quasi-inverse N ↦ Ψ(N) is given by

Ψ(N) = {n ∈ N | λ_{N}(n) = ¯1 ⊗ n in D̄ ⊗ N},

where λ_{N} : N → D̄ ⊗ N is the structure map on N. It is easy to see that

\begin{matrix}Ψ(M) ⊗ Ψ(N) → Ψ(M□D̄N), m ⊗ n ↦ m ⊗ n, \\ R → Ψ(D̄), 1 ↦ ¯1 \end{matrix}

are isomorphisms in _{C}M. We see that the isomorphisms, as tensor structures, make Ψ an equivalence of symmetric tensor categories. □

Let D^{1} denote the irreducible component in D containing 1; this is the largest irreducible Hopf subalgebra. If the characteristic ch R of R is zero, then D^{1} = U(𝔤), the universal envelope of the Lie algebra 𝔤 = P (D) of all primitives in D; see [23, Ch. XIII] or [16, §5.6]. Let G = G(D) denote the group of all grouplikes in D.

In what follows we suppose:

Assumption 3.2.2. D is pointed, so that

D = D^{1}#RG,

the smash product with respect to the conjugate action by G on D^{1}; see [23, Theorem 8.1.5] or [16, Corollary 5.6.4].

In the following, we take as C a Hopf subalgebra of the form

C = D(G_{1}) := D^{1}#RG_{1},

where G_{1} ⊂ G is a subgroup of finite index. The equivalence Φ will be denoted by

(3.6)  Φ_{G_{1}} : _{D(G_{1})}M ⟶^{≈} D̄ ; DM,

if one needs to specify G_{1}.

The vector space R(G/G_{1}) freely spanned by the set G/G_{1} of left cosets is a quotient left D-module coalgebra of D along the map D = D^{1}#RG → R(G/G_{1}) which is given by the counit ε : D^{1} → R and the natural projection G → G/G_{1}. Since the map induces an isomorphism D̄ ⟶^{∼} R(G/G_{1}), left D̄-comodules are identified with (G/G_{1})-graded vector spaces: for N ∈ D̄M,

N = ⨁_{s∈G/G_{1}} N_{s} (N_{s} = {n ∈ N | λ_{N}(n) = s ⊗ n}).

An object in D̄ ; DM is a left D-module N = ⨁_{s∈G/G_{1}} N_{s} which satisfy that gN_{s} ⊂ N_{gs} (g ∈ G, s ∈ G/G_{1}). If M = ⨁_{s∈G/G_{1}} M_{s} is another object in D̄ ; DM, then

M□D̄N = ⨁_{s∈G/G_{1}} M_{s} ⊗ N_{s}.

We have D = ⨁_{g∈G/G_{1}} gD(G_{1}).

Notation 3.2.3. Here and in what follows, g ∈ G/G_{1} means that g lies in a fixed system of those representatives in G for the left cosets G/G_{1} which include the neutral element 1 in G.

The neutral component N_{1} = Ψ(N) in N is a D(G_{1})-submodule. We have the identification

Φ(N_{1}) = ⨁_{g∈G/G_{1}} g ⊗ N_{1}.

Here D acts on the right-hand side so that if d ∈ D^{1},

d(g ⊗ n) = g ⊗ (g^{−1}dg)n (n ∈ N_{1}),

and if h ∈ G,

h(g ⊗ n) = g′ ⊗ tn (n ∈ N_{1}),

where g′ is a representative and t ∈ G_{1} such that hg = g′t. Hence, by Proposition 3.2.1, we have an isomorphism Φ(N_{1}) = ⨁_{g∈G/G_{1}} g ⊗ N_{1} ⟶^{∼} N in D̄ ; DM, given by g ⊗ n ↦ gn.

An algebra A in (D̄ ; DM, □D̄, D̄) is precisely such a D-module algebra that is the direct product ∏_{s∈G/G_{1}} A_{s} of D^{1}-module algebras A_{s} (s ∈ G/G_{1}), satisfying gA_{s} ⊂ A_{gs} (g ∈ G). It is identified with Φ(A_{1}) = ⨁_{g∈G/G_{1}} g ⊗ A_{1}, which is endowed with the component-wise product. We observe that e_{g} = g ⊗ 1 ∈ Φ(A_{1}) are orthogonal central idempotents.

Let A = Φ(A_{1}) be as above. An A_{1}-module V in _{D(G_{1})}M is precisely a module over the algebra A_{1}#D(G_{1}) of smash product: _{A_{1}}(_{D(G_{1})}M) = _{A_{1}#D(G_{1})}M. Φ(V) is naturally an A-module in D̄ ; DM; this is in particular an A#D-module.

Proposition 3.2.4. Let A_{1} be a D(G_{1})-module algebra and A = Φ(A_{1}). The functor

Φ : _{A_{1}#D(G_{1})}M → _{A#D}M

is an equivalence of R-abelian categories.

Proof. By Proposition 3.2.1, it suffices to prove that the category _{A}(D̄ ; DM) of A-modules in D̄ ; DM is isomorphic to _{A}(_{D}M) = _{A#D}M. Given N in _{A#D}M, define N_{g} = e_{g}N (e_{g} = g ⊗ 1 ∈ A = Φ(A_{1}), g ∈ G/G_{1}). Then N = ⨁_{g∈G/G_{1}} N_{g} so that N is in _{A}(D̄ ; DM). This gives the desired isomorphism. □

This proposition can be extended as follows:

Proposition 3.2.5. Let A = Φ(A_{1}) be as above. The functor

Φ : (_{A_{1}}(_{D(G_{1})}M)_{A_{1}}, ⊗_{A_{1}}, A_{1}) → (_{A}(D̄ ; DM)_{A}, ⊗_{A}, A)

is a tensor equivalence.

Proof. For V, W ∈ _{A_{1}}(_{D(G_{1})}M)_{A_{1}}, we easily see

Φ(V) ⊗_{A} Φ(W) ≃ ∑_{g∈G/G_{1}} g ⊗ (V ⊗_{A_{1}} W) = Φ(V ⊗_{A_{1}} W)

in (_{A}(D̄ ; DM)_{A}, ⊗_{A}, A). □

We see that the functor Φ preserves constants and simple module algebras:

Lemma 3.2.6. (i) Let V be a D(G_{1})-module. Then an isomorphism V^{D(G_{1})} ⟶^{∼} Φ(V)^{D} is given by v ↦ ∑_{g∈G/G_{1}} g ⊗ v.

(ii) Let A_{1} be a commutative D(G_{1})-module algebra. Then A_{1} is a simple D(G_{1})-module algebra iff Φ(A_{1}) is a simple D-module algebra.

Proof. (i) If ∑_{g} g⊗v_{g} ∈ Φ(V)^{D}, one sees first v_{1} ∈ V^{C}, and then v_{g} = v_{1} for all g ∈ G/G_{1}. (ii) This directly follows from Proposition 3.2.4. □

3.3. Artinian simple D-module algebras

Let D = D^{1}#RG be a cocommutative pointed Hopf algebra as in the previous section. In what follows we further assume:

Assumption 3.3.1. The irreducible Hopf algebra D^{1} is of Birkhoff-Witt type.

This means that every primitive element of D lies in a divided power sequence of infinite length; an infinite sequence {1 = d_{0}, d_{1}, …, d_{n}, … } in D^{1} is called a divided power sequence if Δ(d_{n}) = ∑_{i=0}^{n} d_{i}⊗d_{n−i} (see [23, p. 268]). This assumption is necessarily satisfied if ch R = 0 (for each primitive ∂ ∈ P (D), {1, ∂, ∂^{2}/2, …, ∂^{n}/n!, … } is a divided power sequence of infinite length). If ch R = p > 0, this is equivalent to the Verschiebung map D^{1} → R^{1/p} ⊗ D^{1} being surjective; see [10]. The assumption is also equivalent to saying that D is smooth as a cocommutative coalgebra.

Moreover this implies that, for a commutative algebra A, the A-algebra Hom_{R}(D^{1}, A) with the convolution product is the projective limit of power series A-algebras (see [23, p. 278]). Thus, if A is a domain (resp. reduced), then Hom_{R}(D^{1}, A) is also a domain (resp. reduced). Furthermore, Hom_{R}(D, A) is isomorphic to the direct product of A-algebras isomorphic to Hom_{R}(D^{1}, A) indexed by G:

Hom_{R}(D, A) ⟶^{∼} Hom_{R}(RG, Hom_{R}(D^{1}, A)) ⟶^{∼} ∏_{g∈G} Hom_{R}(D^{1}, A)
φ ↦  [g ↦ [d ↦ φ(dg)]]  ↦ ([d ↦ φ(dg)])_{g}.

Hence, if A is reduced, then Hom_{R}(D, A) is also reduced. (These facts implies that D^{1} and D are convolutionally reduced in the sense of [28, Definition 5.2].)

As in [27, p. 505], we have the following:

Lemma 3.3.2. Let A be a D-module algebra and ρ_{A} : A → Hom_{R}(D, A) the algebra map associated with the structure on A.

(i) If J ⊂ A is an ideal, then ρ_{A}^{−1}(Hom_{R}(D, J)) is a D-stable ideal, which is maximal among D-stable ideals included in J. Therefore J is a D-stable ideal iff J = ρ_{A}^{−1}(Hom_{R}(D, J)).

(ii) If I ⊂ A is a D-stable ideal, then also the radical \sqrt{I} is a D-stable ideal.

(iii) If P ⊂ A is a prime ideal, then (ρ_{A}^{1})^{−1}(Hom_{R}(D^{1}, P)) is a prime D^{1}-stable ideal. Here ρ_{A}^{1} : A → Hom_{R}(D^{1}, A) is the algebra map associated with the D^{1}-module algebra structure on A.

Proof. (i) This is easily seen.

(ii) Since the algebra Hom_{R}(D, A/\sqrt{I}) ≃ Hom_{R}(D, A)/ Hom_{R}(D, \sqrt{I}) is reduced, we have Hom_{R}(D, \sqrt{I}) is a radical ideal of Hom_{R}(D, A). Hence its pull-back ρ_{A}^{−1}(Hom_{R}(D, \sqrt{I})) is also a radical ideal. By part (i), it includes I. On the other hand, ρ_{A}^{−1}(Hom_{R}(D, \sqrt{I})) is included in \sqrt{I}. Therefore ρ_{A}^{−1}(Hom_{R}(D, \sqrt{I})) = \sqrt{I}.

(iii) Since Hom_{R}(D^{1}, A/P) ≃ Hom_{R}(D^{1}, A)/ Hom_{R}(D^{1}, P) is a domain, Hom_{R}(D^{1}, P) is a prime ideal. Thus its pull-back (ρ_{A}^{1})^{−1}(Hom_{R}(D^{1}, P)) is also prime. □

Let K be a D-module algebra and Ω(K) the set of all minimal prime ideals in K. Then G acts on Ω(K). Let G_{Ω(K)} denote the normal subgroup consisting of those elements in G which stabilize every P ∈ Ω(K).

Proposition 3.3.3. Suppose that K is noetherian as a ring and simple as a D-module algebra. Then Ω(K) is a finite set.

(i) The action of G on Ω(K) is transitive, so that the subgroups G_{P} of stabilizers of P ∈ Ω(K) are conjugate to each other.

(ii) Every P ∈ Ω(K) is D^{1}-stable, so that K/P is a D(G_{P})-module domain. This is simple as a D(G_{Ω(K)})-module algebra.

(iii) Let P ∈ Ω(K), and set K_{1} = K/P. Then we have a natural isomorphism of D-module algebras,

K ≃ Φ_{G_{P}} (K_{1}).

Proof. (ii) Let ρ : K → Hom_{R}(D^{1}, K) be the algebra map associated with the D^{1}-module algebra structure on K. Put P′ = ρ^{−1}(Hom_{R}(D^{1}, P)). By Lemma 3.3.2 (iii), P′ is a D^{1}-stable prime ideal included in P. Then we have P = P′ by the minimality of P. Hence P is D^{1}-stable. (This also follows from [28, Theorem 5.9 (2)].)

Let P ⊂ J ⊊ K be a D(G_{Ω(K)})-stable ideal. Then, ⋂_{g∈G/G_{Ω(K)}} gJ is D-stable, and hence is zero. Since P is prime, there exists g such that gJ ⊂ P, and so P ⊂ J ⊂ g^{−1}P. By the minimality of g^{−1}P, we have P = J (= g^{−1}P). Thus K/P is a simple D(G_{Ω(K)})- module algebra.

(i) Let P ∈ Ω(K). We see

(3.7)  ⋂_{g∈G} gP = ⋂_{Q∈Ω(K)} Q = 0,

since the intersections are both D-stable. The first equality implies {gP | g ∈ G} = Ω(K); this proves (i).

(iii) By (i), g ↦ gP gives a bijection G/G_{P} ⟶^{∼} Ω(K). If Q and Q′ in Ω(K) are distinct, then (Q ⊊) Q + Q′ = K, by (ii). This together with (3.7) proves that the natural map gives an isomorphism,

K ⟶^{∼} ∏_{Q∈Ω(K)} K/Q = ∏_{g∈G/G_{P}} K/gP.

Obviously, Φ_{G_{P}} (K_{1}) is isomorphic to the last direct product. □

For a commutative ring K in general, we say that K is total iff every non-zero divisor in K is invertible.

Corollary 3.3.4. Let K be a noetherian simple D-module algebra as above. Then the following are equivalent.

(a) K is total;

(b) K is artinian as a ring;

(c) The Krull dimension Kdim(K) = 0, or in other words Ω(K) equals the set of all maximal ideals in K.

If these conditions are satisfied, every K#D-module is free as a K-module.

Proof. Each condition is equivalent to that for any/some P ∈ Ω(K), K/P is a field. The last assertion holds true by part (iii) of the last proposition and by Proposition 3.2.4. □

Definition 3.3.5. A D-module algebra K is said to be AS iff it is artinian and simple. By the corollary above, this is equivalent to that K is total, noetherian and simple.

For later use we prove some results. The following lemma is a particular case of [28, Theorem 3.4].

Lemma 3.3.6. Let A be a D-module algebra, and let T ⊂ A be a G-stable multiplicative subset which contains no zero-divisors. The D-module algebra structure on A can be uniquely extended to the localization T^{−1}A of A by T. (D^{1} may not be of Birkhoff-Witt type.)

Proof. Let ρ : A → Hom_{R}(D, A) ⊂ Hom_{R}(D, T^{−1}A) be the algebra map associated with the D-module algebra structure on A. For each t ∈ T, we see ρ(t)(g) = g(t) are invertible in T^{−1}A for all g ∈ G. Hence ρ(t) (t ∈ T) is invertible in Hom_{R}(D, T^{−1}A) by [23, Corollary 9.2.4]. This implies that ρ is uniquely extended to an algebra map ρ̃ : T^{−1}A → Hom(D, T^{−1}A) so that ρ̃(1/t) ∗ ρ(t) = ε (t ∈ T); cf. the proof of [27, Proposition 1.9]. We have thus obtained the measuring action

d(a/t) = ρ̃(a/t)(d) (d ∈ D, a ∈ A, t ∈ T)

by D on T^{−1}A. It remains to prove that this makes T^{−1}A a D-module. We have only to see that

cd(1/t) = c(d(1/t)) (c, d ∈ D, t ∈ T).

This holds, since the two maps D ⊗ D → T^{−1}A, given by c ⊗ d ↦ cd(1/t) and c ⊗ d ↦ c(d(1/t)) coincide, being the convolution-inverse of c ⊗ d ↦ cdt.

For convenience, we describe how to extend the action of D explicitly. Let t ∈ T. The action of D on 1/t is given by:

g(1/t) = 1/g(t) (g ∈ G),
d(1/t) = \frac{ε(d)}{t} − \frac{dt − ε(d)t}{t^{2}} + \frac{1}{t^{3}} ∑_{(d)}(d_{(1)}t − ε(d_{(1)})t)(d_{(2)}t − ε(d_{(2)})t)

− \frac{1}{t^{4}} ∑_{(d)}(d_{(1)}t − ε(d_{(1)})t)(d_{(2)}t − ε(d_{(2)})t)(d_{(3)}t − ε(d_{(3)})t) + ⋯ (d ∈ D^{1}).

We observe that the right hand side of d(1/t) (d ∈ D^{1}) in the equation is a finite sum by the coradical filtration; see the proof of [23, Lemma 9.2.3] or [16, Lemma 5.2.10]. □

Lemma 3.3.7. Let L be an AS D-module algebra, and let K ⊂ L be a D-module subalgebra. If K is total, then K is AS.

Proof. Given an element x ≠ 0 in L = ∏_{P ∈Ω(L)} L/P, define the support of x by

(3.8)  supp(x) = {P ∈ Ω(L) | x ∉ P }.

One sees that x is a non-zero divisor iff supp(x) = Ω(L).

Choose an element x ≠ 0 in K with minimal support. Then for g ∈ G, the supports supp(x) and supp(gx) are either equal or disjoint, according to x(gx) being non-zero or zero. By Proposition 3.3.3 (i), we have those elements x, g_{1}x, …, g_{r}x in K with disjoint supports, whose sum is a non-zero divisor. Let y be the inverse of the sum; this is indeed in K, since K is total. We see that e := xy is a (primitive) idempotent in K with supp(e) = supp(x). By the minimality of the support, each non-zero element in eK has supp(x) as its support, and hence has an inverse in eK, just as x above. We have K = ∏_{i=1}^{r} g_{i}eK, the direct product of the fields g_{i}eK; this proves the lemma. □

Corollary 3.3.8. Let A be a D-module subalgebra in an AS D-module algebra L. (i) Every non-zero divisor x in A has full support: supp(x) = Ω(L) (see (3.8)). (ii) Let K = Q(A) denote the total quotient ring of A; this is realized in L by (i). Then

K is an AS D-module subalgebra of L.

Proof. Let T be the set of all non-zero divisors in A. Then, K = T^{−1}A.

(i) Choose an x ∈ T such that supp(x) is minimal in {supp(t) | t ∈ T }. If supp(x) ≠ Ω(L), then there is a g ∈ G such that supp(gx) ∩ supp(x) = ∅, which implies x(gx) = 0, a contradiction.

(ii) Let ρ_{L} : L → Hom(D, L) be the algebra map associated to the D-module algebra structure on L. It restricts to ρ : A → Hom(D, A) associated to A. If t ∈ T, ρ_{L}(1/t) is the inverse of ρ(t) in Hom(D, L), and hence is contained in Hom(D, T^{−1}A) by the proof of Lemma 3.3.6. This implies that K (= T^{−1}A) is a D-module subalgebra of L. K is AS by Lemma 3.3.7. □

3.4. Sweedler’s correspondence theorem

Let K ⊂ A be an inclusion of D-module algebras. Then A ⊗_{K} A has a coalgebra structure in the symmetric tensor category (_{A}(_{D}M)_{A}, ⊗_{A}, A) given by

\begin{matrix}Δ : A ⊗_{K} A → (A ⊗_{K} A) ⊗_{A} (A ⊗_{K} A), a ⊗ b ↦ (a ⊗ 1) ⊗ (1 ⊗ b), \\ ε : A ⊗_{K} A → A, a ⊗ b ↦ ab. \end{matrix}

In particular A ⊗_{K} A is an A-coring (a coalgebra in (_{A}M_{A}, ⊗_{A}, A), the category of (A, A)-bimodules); see [24]. The next theorem is an analog on AS D-module algebras of Sweedler’s correspondence theorem [24, Theorem 2.1], which play a key role to obtain the Galois correspondence later.

Theorem 3.4.1. Let K ⊂ L be an inclusion of AS D-module algebras. Let C_{L/K} be the set of all D-stable coideals of L ⊗_{K} L and A_{L/K} the set of all intermediate AS D-module algebras of L/K.

(i) For M ∈ A_{L/K}, we have J_{M} := Ker(L ⊗_{K} L ↠ L ⊗_{M} L) ∈ C_{L/K}.

(ii) For J ∈ C_{L/K}, let π : L ⊗_{K} L ↠ L ⊗_{K} L/J be the canonical surjection. Then M_{J} := {a ∈ L | aπ(1 ⊗ 1) = π(1 ⊗ 1)a} ∈ A_{L/K}.

(iii) The correspondence

\begin{aligned}C_{L/K} & ↔ & A_{L/K} \\ J & → & M_{J} \\ J_{M} & ← & M \end{aligned}

is bijective.

Proof. (i) Both L ⊗_{K} L and L ⊗_{M} L are coalgebras in (_{L}(_{D}M)_{L}, ⊗_{L}, L) and obviously L ⊗_{K} L ↠ L ⊗_{M} L is a D-linear L-coring map. Hence its kernel J_{M} is a D-stable coideal of L ⊗_{K} L.

(ii) We easily see that M_{J} is a subalgebra of L which contains K. For any d ∈ D and a ∈ M_{J}, we have d(a)π(1 ⊗ 1) = d(aπ(1 ⊗ 1)) = d(π(1 ⊗ 1)a) = π(1 ⊗ 1)d(a). Thus M_{J} is a D-module subalgebra of L. Let t be a non-zero divisor in M_{J}. By Corollary 3.3.8 (i), t is invertible in L. We see t^{−1}π(1 ⊗ 1) = t^{−1}π(1 ⊗ 1)tt^{−1} = t^{−1}tπ(1 ⊗ 1)t^{−1} = π(1 ⊗ 1)t^{−1} and hence t^{−1} ∈ M_{J}. This implies that M_{J} is total. Therefore M_{J} is an intermediate AS D-module algebra of L/K by Lemma 3.3.7.

(iii) Take an M ∈ A_{L/K}. For all a ∈ M, we have a ⊗ 1 − 1 ⊗ a ∈ J_{M}. Hence M ⊂ M_{J_{M}}. By the definition of M_{J_{M}}, one sees M_{J_{M}} ⊗_{M} M_{J_{M}} ≃ M_{J_{M}} ⊗_{M} M. Since M_{J_{M}} is an M#D-module, it is a free M-module (see Corollary 3.3.4). Hence M_{J_{M}} = M.

Conversely, take J ∈ C_{L/K}. Let ξ : L ⊗_{M_{J}} L → L ⊗_{K} L/J, a ⊗ b ↦ aπ(1 ⊗ 1)b, which is a surjective D-linear L-coring map. Then we have J_{M_{J}} ⊂ J by chasing the following commutative diagram:

0 ⟶ J_{M_{J}} ⟶ L ⊗_{K} L ⟶ L ⊗_{M_{J}} L ⟶ 0 (exact)
∥ ↓^{ξ}
0 ⟶ J ⟶ L ⊗_{K} L ⟶ L ⊗_{K} L/J ⟶ 0 (exact).

If we prove that ξ is injective, then J = J_{M_{J}} follows.

For a fixed P ∈ Ω(M_{J}), put M′ = M_{J}/P (= Ψ_{G_{P}} (M_{J})), L′ = L/P L (= Ψ_{G_{P}} (L)), and C = Ψ_{G_{P}} (L ⊗_{K} L/J) = (L ⊗_{K} L/J)e_{1} = e_{1}(L ⊗_{K} L/J)e_{1} (where e_{1} ∈ M_{J} is the primitive idempotent such that M′ = M_{J}e_{1}). Then C is a coalgebra in (_{L′}(_{D(G_{P})}M)_{L′}, ⊗_{L′}, L′) by Corollary 3.2.5. It suffices to prove that ξ′ = Ψ_{G_{P}} (ξ) : L′ ⊗_{M′} L′ → C is injective.

Regarding C merely as an L′-coring, let 𝔄 be the category of right C-comodules in (_{L′}M_{L′}, ⊗_{L′}, L′). Then 𝔄 is an abelian category since C is a left free L′-module. L′ has a natural C-comodule structure given by

λ : L′ → L′ ⊗_{L′} C ≃ C, a ↦ π(1 ⊗ 1)a (= e_{1}π(1 ⊗ 1)e_{1}a).

We see End_{𝔄}(L′) ⟶^{∼} M′, f ↦ f(1) is an algebra isomorphism. (f ∈ End_{L′}(L′) is a Ccomodule map iff f(1)π(1 ⊗ 1) = π(1 ⊗ 1)f(1).) On the other hand, Hom_{𝔄}(L′, C) ⟶^{∼} L′, f ↦ (ε ∘ f)(1) is an M′-module isomorphism whose inverse is given by a ↦ [b ↦ aπ(1 ⊗ 1)b]. Indeed, for f ∈ Hom_{𝔄}(L′, C),

(ε ∘ f)(1)π(1 ⊗ 1)b = (ε ∘ f)(1)λ(b) = (ε ∘ f)(1)∑_{(b)}b_{(0)} ⊗ b_{(1)}

= ∑_{(b)}(ε ∘ f)(b_{0}) ⊗ b_{(1)} = ((ε ⊗ id) ∘ (f ⊗ id))(λ(b)) = (ε ⊗ id)(Δ_{C}(f(b))) = f(b).

We will show that L′ is simple in 𝔄, concluding that ξ′ is injective by Proposition 3.1.1. Every simple subobject of L′ is of the form eL′, where e is an idempotent of L′. Since λ is D(G_{P})-linear, we see that g(eL′) is also a simple object for each g ∈ G. Each g(eL′) coincides or trivially intersects with eL′ since g(eL′) ∩ eL′ is also a right C-comodule. It follows from Proposition 3.3.3 (i) that L′ is semisimple in 𝔄. But the endomorphism ring End_{𝔄}(L′) ≃ M′ is a field. This implies that L′ is a simple object in 𝔄. □

Example 3.4.2. Let R = ℚ and D = ℚ[τ, τ^{−1}], the ring of linear difference operators. Take α = \{(\frac{1+\sqrt{5}}{2})^{n}\} ∈ S_{ℂ} as in Introduction. Then ℚ = ℚ(1, 1) ⊂ L = ℚ(\sqrt{5}, α) × ℚ(\sqrt{5}, α) (where τ(1, 0) = (0, 1), τ(0, 1) = (1, 0), L^{D} = ℚ(\sqrt{5}) = ℚ(\sqrt{5})(1, 1)) is an inclusion of AS D-module algebras. Write e_{1} = (1, 0), e_{2} = (0, 1) ∈ L. Then A_{L/ℚ} and C_{L/ℚ} correspond as the following:

A_{L/ℚ}  C_{L/ℚ}
L  Span_{L,L}\{\sqrt{5} ⊗ 1 − 1 ⊗ \sqrt{5}, e_{1} ⊗ e_{2}, e_{2} ⊗ e_{1}, α ⊗ 1 − 1 ⊗ α\}
ℚ(\sqrt{5}, α)  Span_{L,L}\{\sqrt{5} ⊗ 1 − 1 ⊗ \sqrt{5}, α ⊗ 1 − 1 ⊗ α\}
ℚ(\sqrt{5}) × ℚ(\sqrt{5})  Span_{L,L}\{e_{1} ⊗ e_{2}, e_{2} ⊗ e_{1}, \sqrt{5} ⊗ 1 − 1 ⊗ \sqrt{5}\}
ℚ(α)  Span_{L,L}{α ⊗ 1 − 1 ⊗ α}
ℚ(\sqrt{5})  Span_{L,L}\{\sqrt{5} ⊗ 1 − 1 ⊗ \sqrt{5}\}
ℚ × ℚ  Span_{L,L}{e_{1} ⊗ e_{2}, e_{2} ⊗ e_{1}}
ℚ  0

We will see that L/ℚ is not a Picard-Vessiot extension but L/ℚ(\sqrt{5}) is.

3.5. Galois correspondence for Picard-Vessiot extensions

Let K ⊂ A be an inclusion of D-module algebras. Then A ⊗_{K} A has an algebra structure naturally and become a D-module algebra since D is cocommutative. Thus (A ⊗_{K} A)^{D} is a K^{D}-subalgebra of A ⊗_{K} A.

Definition 3.5.1. Let K ⊂ L be an inclusion of AS D-module algebras. We say that L/K is a Picard-Vessiot, or PV, extension if the following conditions are satisfied:

(a) K^{D} = L^{D}; this will be denoted by k.

(b) There exists a D-module subalgebra A ⊂ L including K, such that the total quotient ring Q(A) of A equals L, and the k-subalgebra H := (A⊗_{K} A)^{D} generates the left (or equivalently right) A-module A ⊗_{K} A: A · H = A ⊗_{K} A (or H · A = A ⊗_{K} A).

Proposition 3.5.2. Let L/K be a PV extension of AS D-module algebras and take A, H as in the condition (b) above.

(i) The product map μ : A⊗_{k} H → A⊗_{K} A, μ(a⊗h) = a·h is a D-linear isomorphism. (ii) The A-coring structure maps Δ, ε of A ⊗_{K} A induce k-algebra maps Δ_{H} : H →

H ⊗_{k} H, ε_{H} : H → k. Then (H, Δ_{H}, ε_{H}) becomes a commutative Hopf algebra over k. The antipode is induced from the twist map tw : A ⊗_{K} A → A ⊗_{K} A, a ⊗ b ↦ b ⊗ a.

(iii) The k-algebra map θ : A → A⊗_{k}H, θ(a) = μ^{−1}(1⊗a) makes A a right H-comodule. A/K is necessarily a right H-Galois extension [16, Sect. 8.1] in the sense that

Aθ : A ⊗_{K} A → A ⊗_{k} H, _{A}θ(a ⊗ b) = aθ(b)

is an isomorphism. A Hopf algebra structure on H with this property is unique. (iv) Such an algebra A that satisfies the condition (b) above is unique.

Proof. (i) By Corollary 3.1.4, the natural map L ⊗_{k} (L ⊗_{K} A)^{D} → L ⊗_{K} A is injective. Since the map μ is its restriction, it is injective. On the other hand, μ is surjective by the condition (b). Note that this can be uniquely extended to an isomorphism L ⊗_{k} H ⟶^{∼} L ⊗_{K} A.

(ii) Since A^{D} = k by the condition (a), ε maps H into k. The twofolds of μ:

φ : A ⊗_{k} H ⊗_{k} H ⟶^{μ⊗id} A ⊗_{K} A ⊗_{k} H ⟶^{id ⊗μ} A ⊗_{K} A ⊗_{K} A

is a D-linear isomorphism. This induces an algebra isomorphism φ_{1} := id ⊗μ|_{H⊗_{k}H} : H ⊗_{k} H ⟶^{∼} (A⊗_{K} A⊗_{K} A)^{D}. Similarly the threefolds of μ induces an algebra isomorphism

φ_{2} : H ⊗_{k} H ⊗_{k} H ⟶^{∼} (A⊗_{K} A⊗_{K} A⊗_{K} A)^{D}. Since Δ : A⊗_{K} A → (A⊗_{K} A)⊗_{A}(A⊗_{K} A) ≃

A ⊗_{K} A ⊗_{K} A maps H into (A ⊗_{K} A ⊗_{K} A)^{D}, a k-algebra map Δ_{H} : H → H ⊗_{k} H is induced by Δ_{H} = φ_{1}^{−1} ∘ Δ|_{H}. We see

φ_{2} ∘ (Δ_{H} ⊗ id_{H}) ∘ Δ_{H} = (Δ ⊗ id) ∘ φ_{1} ∘ Δ_{H} = (Δ ⊗ id_{A}) ∘ Δ|_{H},

φ_{2} ∘ (id_{H} ⊗Δ_{H}) ∘ Δ_{H} = (id ⊗Δ) ∘ φ_{1} ∘ Δ_{H} = (id_{A} ⊗Δ) ∘ Δ|_{H}.

Then we have (Δ_{H} ⊗id)∘Δ_{H} = (id ⊗Δ_{H})∘Δ_{H} by the coassociativity of Δ. The counitary property is easily seen. Therefore (H, Δ_{H}, ε_{H}) is a commutative bialgebra. Since D is cocommutative, tw maps H into H. Put S = tw |_{H}. For w = ∑_{i}(a_{i} ⊗ a′_{i}) ⊗ (b′_{i} ⊗ b_{i}) ∈ H ⊗_{k} H, we see

φ_{1}(w) = ∑_{i} a_{i} ⊗ a′_{i}b′_{i} ⊗ b_{i}, m((id ⊗S)(w)) = ∑_{i} a_{i}b_{i} ⊗ a′_{i}b′_{i}

where m denotes the multiplication of H. Thus, for h = ∑_{i} a_{i} ⊗ b_{i} ∈ H,

m((id ⊗S)(Δ_{H}(h))) = m((id ⊗S)(φ_{1}^{−1}(∑_{i} a_{i} ⊗ 1 ⊗ b_{i}))) = ∑_{i} a_{i}b_{i} ⊗ 1.

This implies id ∗S = m ∘ (id ⊗S) ∘ Δ_{H} = uε where u : k → H is the unit map of H. We have S ∗ id = uε similarly. Therefore S is the antipode of H.

(iii) We see

(φ ∘ (θ ⊗ id) ∘ θ)(a) = 1 ⊗ 1 ⊗ a = (Δ ∘ μ ∘ θ)(a) = (φ ∘ (id ⊗Δ_{H}) ∘ θ)(a)

for all a ∈ A. Thus we have (θ ⊗ id) ∘ θ = (id ⊗Δ_{H}) ∘ θ. On the other hand,

(id ⊗ε_{H}) ∘ θ = ε ∘ μ ∘ θ = id .

Therefore (A, θ) is an H-comodule. The map _{A}θ, being μ^{−1}, is an isomorphism. Since this interprets θ into the natural right A⊗_{K} A-comodule structure A → A⊗_{A} (A⊗_{K} A) ≃ A ⊗_{K} A, a ↦ 1 ⊗ a on A, we see the described uniqueness of the structure on H.

(iv) This follows in the same way as [27, Lemma 2.5]. We include the proof for convenience. If A, B satisfy the condition (b), then also AB satisfy it. Thus we may assume A ⊂ B. Put H_{A} = (A ⊗_{K} A)^{D}, H_{B} = (B ⊗_{K} B)^{D}, the corresponding Hopf algebras. Then H_{A} is a Hopf subalgebra of H_{B}. Hence H_{B}/H_{A} is a faithfully flat extension (see [25, Theorem 3.1] or [29, Ch. 14]). The extension (L ⊗_{K} B)/(L ⊗_{K} A) is identified with (L ⊗_{k} H_{B})/(L ⊗_{k} H_{A}) through the μ-isomorphism. It follows that B/A is a faithfully flat extension since L is a free K-module. Hence aA = aB ∩ A for all a ∈ A since the canonical map A/aA → B ⊗_{A} (A/aA) ≃ B/aB is injective. For any b ∈ B, there exists a non-zero divisor a ∈ A such that ab ∈ A. Since ab ∈ aB ∩ A = aA and since a is a non-zero divisor, b ∈ A follows. Therefore we have A = B. □

Definition 3.5.3. A (resp., H) is called the principal algebra (resp., the Hopf algebra) for L/K. To indicate these we say that (L/K, A, H) is a PV extension. The associated affine group scheme G(L/K) := SpecH is called the PV group scheme for L/K.

Theorem 3.5.4. Let L/K be a PV extension of AS D-module algebras with the Hopf algebra H. Let A_{L/K} be the set of intermediate AS D-module algebras of L/K and HI_{H} the set of all Hopf ideals of H. Then A_{L/K} and HI_{H} correspond bijectively as follows:

\begin{matrix}A_{L/K} → HI_{H}, M ↦ H ∩ Ker(L ⊗_{K} L → L ⊗_{M} L), \\ HI_{H} → A_{L/K}, I ↦ \{x ∈ L | x ⊗ 1 − 1 ⊗ x ∈ I · (L ⊗_{K} L)\}. \end{matrix}

This theorem is obtained as the composite of 1-1 correspondences given by Theorem 3.4.1 and the next proposition. For a commutative algebra A (resp. a D-module algebra B), let I(A) (resp. I_{D}(B)) denote the set of all ideals of A (resp. D-stable ideals of B).

Proposition 3.5.5. Let (L/K, A, H) be a PV extension.

(i) I(H) and I_{D}(L ⊗_{K} L) correspond bijectively as follows:

\begin{matrix}I(H) → I_{D}(L ⊗_{K} L), I ↦ I · (L ⊗_{K} L), \\ I_{D}(L ⊗_{K} L) → I(H), J ↦ J ∩ H. \end{matrix}

(ii) Under the correspondence, J is a D-stable coideal iff I is a Hopf ideal.

Proof. This follows in the same way as [27, Proposition 2.6]. We include the proof for convenience.

(i) Since L is the total quotient ring of A, we have I_{D}(L ⊗_{K} L) ⊂ I_{D}(A ⊗_{K} A). Furthermore, I_{D}(L ⊗_{K} A) ∩ I_{D}(A ⊗_{K} L) = I_{D}(L ⊗_{K} L) in I_{D}(A ⊗_{K} A). Considering the μ-isomorphism, we claim the map

I(H) → I_{D}(A ⊗_{k} H) ⟶^{∼} I_{D}(A ⊗_{K} A), I ↦ A ⊗_{k} I ↦ I · (A ⊗_{K} A)

is injective with the image I_{D}(L ⊗_{k} H) ≃ I_{D}(L ⊗_{K} A). The injectivity is clear. Since A⊗_{k}I = (L⊗_{k}I)∩(A⊗_{k}H), the image is contained in I_{D}(L⊗_{k}H). Then it suffices to prove that every D-stable ideal of L⊗_{k}H is written as L⊗_{k}I by some I ∈ I(H). Let 𝔞 ⊂ L⊗_{k}H be a D-stable ideal and take the canonical map φ : H ⟶^{∼} (L ⊗_{k} H)^{D} ↠ ((L ⊗_{k} H)/𝔞)^{D}. Put I = Ker φ = 𝔞 ∩ H, an ideal of H. Since L ⊗_{k} ((L ⊗_{k} H)/𝔞)^{D} → (L ⊗_{k} H)/𝔞 is injective (Corollary 3.1.4), we have 𝔞 = L ⊗_{k} I by chasing the following diagram:

0 ⟶ L ⊗_{k} I ⟶ L ⊗_{k} H ⟶^{id⊗φ} L ⊗_{k} ((L ⊗_{k} H)/𝔞)^{D} ⟶ 0
∥  ↓
0 ⟶ 𝔞 ⟶ L ⊗_{k} H ⟶  (L ⊗_{k} H)/𝔞  ⟶ 0.

Then the claim is proved. By symmetry, we see the image of I(H) → I_{D}(A ⊗_{K} A) is also equal to I_{D}(A⊗_{K} L). It follows I_{D}(L⊗_{K} A) = I_{D}(A⊗_{K} L) = I_{D}(L⊗_{K} L) in I_{D}(A⊗_{K} A), proving (i).

(ii) By the similar discussion to (i), we have that I(H ⊗_{k} H) and I_{D}(L ⊗_{K} L ⊗_{K} L) correspond bijectively. If I ↔ J in (i), then I ⊗_{k} H ↔ J ⊗_{K} L and H ⊗_{k} I ↔ L ⊗_{K} J.

Therefore, Δ_{H}(I) ⊂ I ⊗_{k} H + H ⊗_{k} I iff ΔJ ⊂ J ⊗_{K} L + L ⊗_{K} J. On the other hand, Ker((L⊗_{K}L)⊗_{L}(L⊗_{K}L) → (L⊗_{K}L/J)⊗_{L}(L⊗_{K}L/J)) = J⊗_{L}(L⊗_{K}L)+(L⊗_{K}L)⊗_{L}J = J ⊗_{K} L+L⊗_{K} J holds since J, L⊗_{K} L, and J ⊗_{K} L/J are free L-modules. It follows that I is a biideal of H iff J is a D-stable coideal of L ⊗_{K} L. It is known that every biideal of a commutative Hopf algebra over a field is a Hopf ideal (see [17, Theorem 1 (iv)]). □

Actually, I_{D}(A ⊗_{K} A) = I_{D}(L ⊗_{K} L) holds in the above (replace L with A in the proof of (i) and use Proposition 3.5.9 (i)).

Example 3.5.6. In Example 3.4.2, if we put K = ℚ(\sqrt{5}), then L/K is a PV extension. The principal algebra and the Hopf algebra are given by A = K[α, α^{−1}] × K[α, α^{−1}] and H = K[g_{1}, g_{2}] with grouplikes g_{1} = α ⊗ α^{−1}, g_{2} = (e_{1} − e_{2}) ⊗ (e_{1} − e_{2}). In this case, A_{L/K}, C_{L/K}, HI_{H} correspond as follows:

A_{L/K} C_{L/K} HI_{H}
L  Span_{L,L}{e_{1} ⊗ e_{2}, e_{2} ⊗ e_{1}, α ⊗ 1 − 1 ⊗ α} H^{+} = ⟨g_{1} − 1, g_{2} − 1⟩
K(α)  Span_{L,L}{α ⊗ 1 − 1 ⊗ α}  ⟨g_{1} − 1⟩
K × K  Span_{L,L}{e_{1} ⊗ e_{2}, e_{2} ⊗ e_{1}}  ⟨g_{2} − 1⟩
K  0  0

Proposition 3.5.7. Let (L/K, A, H) be a PV extension. Suppose A_{L/K} ∋ M ↔ I ∈ HI_{H} in Theorem 3.5.4.

(i) (L/M, AM, H/I) is a PV extension.

(ii) A^{co H/I} = {a ∈ A | θ(a) − a ⊗ 1 ∈ A ⊗_{k} I} = A ∩ M, and the μ-isomorphism

A ⊗_{k} H ⟶^{∼} A ⊗_{K} A induces an isomorphism A ⊗_{k} H^{co H/I} ⟶^{∼} A ⊗_{K} (A ∩ M).

Proof. (i) We have an isomorphism L ⊗_{k} H/I ⟶^{μ̄} L ⊗_{M} AM by considering the following diagram:

0 ⟶ L ⊗_{k} I ⟶ L ⊗_{k} H ⟶ L ⊗_{k} H/I ⟶ 0
↓^{μ} ↓^{μ} ↓μ̄
0 ⟶ I · (L ⊗_{K} A) ⟶ L ⊗_{K} A ⟶ L ⊗_{M} AM ⟶ 0.

Restrict the diagram as

0 ⟶ AM ⊗_{k} I ⟶ AM ⊗_{k} H ⟶ AM ⊗_{k} H/I ⟶ 0
↓^{μ} ↓^{μ} ↓μ̄
0 ⟶ I · (AM ⊗_{K} A) ⟶ AM ⊗_{K} A ⟶ AM ⊗_{M} AM ⟶ 0.

Since AM ⊗_{K} A = (AM ⊗_{K} K)·H, we have AM ⊗_{M} AM = AM ·(AM ⊗_{M} AM)^{D} through the surjection AM ⊗_{K} A → AM ⊗_{M} AM. On the other hand, μ̄ induces an isomorphism H/I ⟶^{∼} (AM ⊗_{M} AM)^{D} of Hopf algebras.

(ii) This follows by considering the next diagram:

[figure: 0; A ⊗_{K} (A ∩ M); A ⊗_{K} A; A ⊗_{K} AM ⊗_{M} AM; 0; A ⊗_{K} A^{co H/I}; A ⊗_{K} A; A ⊗_{K} A ⊗_{k} H/I; 0; A ⊗_{k} H^{co H/I}; A ⊗_{k} H; A ⊗_{k} H ⊗_{k} H/I.]
□

Let H be a commutative Hopf algebra over k. It is known that normal Hopf ideals I of H and Hopf subalgebras H_{1} ⊂ H correspond bijectively by H_{1} = H^{co H/I} = ^{co H/I}H and I = HH_{1}^{+} (see [25]). Let (V, λ) be a right H-comodule in general. If we put V_{1} = λ^{−1}(V ⊗_{k} H_{1}) (= V^{co H/I}), then we have λ(V_{1}) ⊂ V_{1} ⊗_{k} H_{1}. Indeed, write λ(v) = ∑_{i} v_{i} ⊗ h_{i} ∈ V ⊗_{k} H_{1} for v ∈ V_{1}, where h_{i} are k-linearly independent. Then

∑_{i} ∑_{(v_{i})}(v_{i})_{(0)} ⊗ (v_{i})_{(1)} ⊗ h_{i} = ∑_{i} ∑_{(h_{i})} v_{i} ⊗ (h_{i})_{(1)} ⊗ (h_{i})_{(2)} ∈ V ⊗_{k} H_{1} ⊗_{k} H_{1}.

This implies v_{i} ∈ V_{1}. As in [27, Theorem 2.9], we have the following proposition.

Proposition 3.5.8. Let (L/K, A, H) be a PV extension and H_{1} ⊂ H a Hopf subalgebra. Put I = HH_{1}^{+} and A_{1} = θ^{−1}(A ⊗_{k} H_{1}) = A^{co H/I}. Let L_{1} be the total quotient ring of A_{1} so that L_{1} is an intermediate AS D-module algebra of L/K.

(i) (L_{1}/K, A_{1}, H_{1}) is a PV extension.

(ii) I is the Hopf ideal of H which corresponds to L_{1}.

(iii) H_{1} ↦ L_{1} gives a 1-1 correspondence between the Hopf subalgebras of H and the intermediate AS D-module algebras which are PV extensions over K.

Proof. (i) Since θ(A_{1}) ⊂ A_{1} ⊗_{k} H_{1}, we have μ(A_{1} ⊗_{k} H_{1}) ⊃ A_{1} ⊗_{K} A_{1}. Consider A ⊗_{K} A as a right H-comodule by the structure map id ⊗θ. Then the inclusion H ↪ A ⊗_{K} A is an H-comodule map; recall that, for h = ∑_{i} a_{i} ⊗_{K} b_{i} ∈ H = (A ⊗_{K} A)^{D},

Δ(h) = ∑_{i} a_{i} ⊗_{K} μ^{−1}(1 ⊗_{K} b_{i}) = ∑_{i} a_{i} ⊗_{K} θ(b_{i}).

Thus H_{1} ⊂ A ⊗_{K} θ^{−1}(A ⊗_{k} H_{1}) = A ⊗_{K} A_{1}. Also we have H_{1} = S(H_{1}) ⊂ tw(A ⊗_{K} A_{1}) = A_{1} ⊗_{K} A. Hence H_{1} ⊂ A_{1} ⊗_{K} A_{1} and so μ(A_{1} ⊗_{k} H_{1}) ⊂ A_{1} ⊗_{K} A_{1}. This implies that μ : A ⊗_{k} H ⟶^{∼} A ⊗_{K} A induces a D-linear isomorphism A_{1} ⊗_{k} H_{1} ⟶^{∼} A_{1} ⊗_{K} A_{1}. Therefore H_{1} = (A_{1} ⊗_{K} A_{1})^{D} and A_{1} ⊗_{K} A_{1} = A_{1} · H_{1}.

(ii) Let M be the intermediate AS D-module algebra of L/K which corresponds to I. Then M ⊃ A ∩ M = A^{co H/I} = A_{1} by Proposition 3.5.7 (ii). Since L_{1} is the smallest AS D-module subalgebra of L which includes A_{1}, we have L_{1} ⊂ M. Let I′ be the Hopf ideal which corresponds to L_{1}. Then I = HH_{1}^{+} ⊂ H ∩ Ker(L ⊗_{K} L ↠ L ⊗_{L_{1}} L) = I′ since the counit ε : H_{1} → k is a restriction of mult : A_{1} ⊗_{K} A_{1} → A_{1}. Thus we have L_{1} ⊃ M.

(iii) Let L_{1} be an intermediate AS D-module algebra of L/K such that (L_{1}/K, A_{1}, H_{1}) is a PV extension. Since A ⊗_{K} A = A · H and A_{1} ⊗_{K} A_{1} = A_{1} · H_{1}, we have A_{1}A ⊗_{K} A_{1}A = A_{1}A · H_{1}H. This implies that A_{1}A is the principal algebra for L/K and hence A_{1}A = A by Proposition 3.5.2 (iv). Thus A_{1} ⊂ A and H_{1} ⊂ H, a Hopf subalgebra. Since the μ-isomorphism A_{1} ⊗_{k} H_{1} ⟶^{∼} A_{1} ⊗_{K} A_{1} induces a left A-module isomorphism A ⊗_{k} H_{1} ⟶^{∼} A ⊗_{K} A_{1}, we have A_{1} = θ^{−1}(A ⊗_{k} H_{1}). This proves (iii). □

Finally in this section, we prove two important properties on principal algebras which are used later.

Proposition 3.5.9. Let (L/K, A, H) be a PV extension.

(i) A is simple as a D-module algebra.

(ii) A contains all primitive idempotents in L.

Proof. (i) The following proof is essentially the same as that of [27, Theorem 2.11].

Let 0 ≠ I ⊂ A be a D-stable ideal. Since L ⊗_{K} I ∈ I_{D}(L ⊗_{K} A), there exists an ideal 𝔞 ∈ I(H) such that L ⊗_{K} I = 𝔞 · (L ⊗_{K} A) by the proof of Proposition 3.5.5 (i). But IL = L since L is simple and hence L ⊗_{K} IL = L ⊗_{K} L. This implies that the D-stable ideal of L ⊗_{K} L which corresponds to 𝔞 is L ⊗_{K} L. Thus 𝔞 = H. Therefore L ⊗_{K} I = H · (L ⊗_{K} A) = L ⊗_{K} A, concluding I = A.

(ii) Since L is a localization of A, we have Ω(L) ⊂ Ω(A) via P ↦ P ∩ A. We see A ⊂ ∏_{P ∈Ω(L)} A/P ∩ A. It remains to prove that if P ≠ Q in Ω(L), then the sum J := P ∩ A + Q ∩ A equals A. If J ⊊ A on the contrary, one sees as in the proof of Proposition 3.3.3 (ii) that J = P ∩ A = Q ∩ A by Part (i), and so P = Q. □

Remark 3.5.10. Before the version four of this article, the following assertion was written in this place (as a lemma):

Let K be a finite product of fields and A a commutative K-algebra. Let T be a multiplicative subset of A which has no zero divisors and put L = T^{−1}A. Then every separable K-subalgebra of L is included in A.

But this is false (the sentence “the proof can be reduced to the case that A is a domain” in the proof was wrong). The following is a counter example. Let A = ℂ[X, Y]/⟨XY ⟩ and L the total quotient ring of A. Then A has no nontrivial idempotent. But L has a nontrivial idempotent X/(X + Y). Excuse me for making such a mistake and I hope the reader of old versions was free from the wrong argument.

3.6. Translation into affine group schemes

For an inclusion of D-module algebras K ⊂ A, let Aut_{D, K-alg}(A) denote the group of D-linear K-algebra automorphisms of A. Let Aut_{D,K-alg}(A) denote the associated group functor over k = K^{D}; it associates to each commutative k-algebra T the automorphism group Aut_{D, K ⊗_{k} T -alg}(A⊗_{k} T), where T is considered as a D-module algebra by the trivial action dt = ε(d)t (d ∈ D, t ∈ T). As in [27, Appendix], we have the following:

Theorem 3.6.1. Let (L/K, A, H) be a PV extension and G(L/K) = SpecH the PV group scheme. Then the linear representation ϕ : G(L/K) → GL(A) arising from the H-comodule structure θ : A → A ⊗_{k} H gives an isomorphism G(L/K) ⟶^{∼} Aut_{D,K-alg}(A) of affine k-group schemes. In particular, G(L/K)(k) ≃ Aut_{D, K-alg}(A) = Aut_{D, K-alg}(L).

Proof. Let T be a commutative k-algebra. For α ∈ G(L/K)(T) = Alg_{k}(H, T), ϕ_{T} (α) is given by

ϕ_{T} (α) : A ⊗_{k} T ⟶^{∼} A ⊗_{k} T, a ⊗ t ↦ ∑_{(a)} a_{(0)} ⊗ α(a_{(1)})t.

We easily see ϕ_{T} (α) ∈ Aut_{D, K ⊗_{k} T -alg}(A ⊗_{k} T). We will construct the inverse ψ : Aut_{D,K-alg}(A) → G(L/K). For an element β ∈ Aut_{D, K ⊗_{k} T -alg}(A ⊗_{k} T), let _{A}β denote the left A-linear extension of β|_{A} : A → A ⊗_{k} T. Namely, _{A}β : A ⊗_{K} A → A ⊗_{k} T, a ⊗ b ↦ aβ(b ⊗ 1). Consider the D-linear A-algebra map

A ⊗_{k} H ⟶^{μ} A ⊗_{K} A ⟶^{_{A}β} A ⊗_{k} T.

We see this maps the constants H into T. Then we have a k-algebra map ψ_{T} (β) := (_{A}β ∘μ)^{D} = _{A}β|_{H} ∈ Alg_{k}(H, T) = G(L/K)(T) so that (id_{A} ⊗ψ_{T} (β))∘θ = β|_{A}. This gives a homomorphism ψ : Aut_{D,K-alg}(A) → G(L/K). Indeed, for β, γ ∈ Aut_{D, K ⊗_{k} T -alg}(A⊗_{k} T),

(γ ∘ β)(a ⊗ 1) = ∑_{(a)} γ(a_{(0)} ⊗ 1)ψ_{T} (β)(a_{(1)}) = ∑_{(a)} a_{(0)} ⊗ ψ_{T} (γ)(a_{(1)})ψ_{T} (β)(a_{(2)}) (a ∈ A).

One easily sees ϕ_{T} ∘ ψ_{T} = id. For α ∈ G(L/K)(T), we see _{A}ϕ_{T} (α) = (ε ⊗ α) ∘ (id ⊗θ) where ε : A ⊗_{K} A → A, the counit, and hence _{A}ϕ_{T} (α)|_{H} = (ε_{H} ⊗ α) ∘ Δ_{H} = α. This implies ψ_{T} ∘ ϕ_{T} = id. □

Theorem 3.5.4 and Proposition 3.5.7, 3.5.8 can be translated as follows.

Theorem 3.6.2. Let L/K be a PV extension of AS D-module algebras.

(i) If M is an intermediate AS D-module algebra of L/K, then L/M is also a PV extension and the PV group scheme G(L/M) is identified with a closed subgroup scheme of G(L/K). Then intermediate AS D-module algebras of L/K and closed subgroup schemes of G(L/K) correspond bijectively by M ↦ G(L/M).

(ii) Under the correspondence above, M/K is a PV extension iff G(L/K) ▷ G(L/M). If this is the case, we have an isomorphism G(M/K) ≃ G(L/K)/G(L/M) of affine group schemes.

3.7. Copying and interlacing

In this section, we investigate how PV extensions change through the functor Φ and Ψ described in Section 3.2.

First we easily see the following:

Lemma 3.7.1. Let G_{1} ⊂ G be a subgroup of finite index. Write Φ = Φ_{G_{1}}. Let K_{1} ⊂ L_{1} be an inclusion of AS D(G_{1})-module algebras. Then (L_{1}/K_{1}, A_{1}, H) is a PV extension iff (Φ(L_{1})/Φ(K_{1}), Φ(A_{1}), H) is a PV extension of AS D-module algebras.

Proof. This follows by Proposition 3.2.5 and Lemma 3.2.6. □

Remark 3.7.2. Let K ⊂ L be an inclusion of AS D-module algebras. Choose 𝔭 ∈ Ω(K), and let P_{1}, …, P_{r} be all those elements in Ω(L) that lie over 𝔭. Define K_{1} = K/𝔭, L_{1} = L/𝔭L = ∏_{i=1}^{r} L/P_{i}. Then we have an inclusion K_{1} ⊂ L_{1} of AS D(G_{𝔭})-module algebras such that the induced inclusion Φ_{G_{𝔭}}(K_{1}) ⊂ Φ_{G_{𝔭}}(L_{1}) is identified with K ⊂ L. We can thus reduce to the case where K is a field, especially to discuss PV extensions by Lemma 3.7.1.

Example 3.7.3. Let G_{1} ⊂ G be a normal subgroup of finite index. Let K be a D-module field. Regarding this as a D(G_{1})-module algebra, define L = Φ_{G_{1}}(K). We then have the inclusion

K ↪ L = ⨁_{g∈G/G_{1}} g ⊗ K, x ↦ ∑_{g} g ⊗ g^{−1}x

of AS D-module algebras. If K^{D(G_{1})} = K^{D}, then K^{D} = L^{D} (=: k) by Lemma 3.2.6 (i). Moreover, (L/K, L, H) is a PV extension with H = k(G/G_{1})^{∗}, the dual of the group algebra k(G/G_{1}). In fact, we see that the elements

e_{g} := ∑_{h∈G/G_{1}}(h ⊗ 1) ⊗_{K} (hg ⊗ 1) (g ∈ G/G_{1})

in L ⊗_{K} L are D-invariant, and behave as the dual basis in H of the group elements g (∈ G/G_{1}) in k(G/G_{1}). Namely, Δ(e_{g}) = ∑_{h} e_{gh^{−1}} ⊗ e_{h}, ε(e_{g}) = δ_{1,g}, S(e_{g}) = e_{g^{−1}}. The H-comodule structure θ : L → L ⊗_{k} H is given by

θ(h ⊗ x) = ∑_{g}(hg^{−1} ⊗ gx) ⊗_{k} e_{g},

as is seen from following computation in L ⊗_{K} L:

1⊗_{K} (h⊗x) = ∑_{f}(f ⊗f^{−1}hx)⊗_{K} (h⊗1) = ∑_{g}(hg^{−1}⊗gx)⊗_{K} (h⊗1) = ∑_{g}(hg^{−1}⊗gx)·e_{g}.

Proposition 3.7.4. Let (L/K, A, H) be a PV extension of AS D-module algebras. Choose arbitrarily P ∈ Ω(L), and write Φ = Φ_{G_{P}}. Let 𝔭 = P ∩ K (∈ Ω(K)). Define

K_{1} = K/𝔭, A_{1} = A/P ∩ A, L_{1} = L/P.

Then, (i) A ≃ Φ(A_{1}). (ii) Φ(K_{1}) is identified with the K-subalgebra K̂ of L which is spanned over K by the

primitive idempotents in L.

(iii) (L_{1}/K_{1}, A_{1}, H̄ = H/I) is a PV extension of D(G_{P})-module fields, where I = H ∩ Ker(L ⊗_{K} L → L ⊗K̂ L); cf. [20, Corollary 1.16].

(iv) The subalgebra of H

B = {h ∈ H | Δ(h) ≡ h ⊗ 1 mod H ⊗_{k} I} (= H^{co H̄})

is a separable k-algebra. We have a right H̄-colinear B-algebra isomorphism H ≃ B⊗_{k} H̄.

(v) If G_{P} is normal in G_{𝔭}, then B ⊂ H is a Hopf subalgebra which is isomorphic to k(G_{𝔭}/G_{P})^{∗}, and we have an extension

k(G_{𝔭}/G_{P})^{∗} ↣ H ↠ H̄

of Hopf algebras; cf. [20, Corollary 1.17].

Proof. (i) This follows from Proposition 3.5.9.

(ii) This is easy to see.

(iii) By Proposition 3.5.7 (i), we have a PV extension

(L/K̂, A, H̄) = (Φ(L_{1})/Φ(K_{1}), Φ(A_{1}), H̄).

part (iii) now follows by Lemma 3.7.1.

For the remaining (iv), (v), we may suppose that K is a field and so 𝔭 = 0, G_{𝔭} = G by Remark 3.7.2.

(iv) By Proposition 3.5.7 (ii), we see A ⊗_{k} B ≃ A ⊗_{K} K̂. Since this is a separable A-algebra, B is a separable k-algebra by [7, Ch. II, Proposition 1.8].

Recall that A has the natural, right H̄-comodule k-algebra structure A ⟶^{1⊗−} A ⊗K̂ A ≃ A ⊗_{k} H̄; in fact, A is also a left H̄-comodule k-algebra. We see that the map

(3.9)  σ : Φ(A_{1} ⊗_{K} A_{1}) = A ⊗K̂ A → A ⊗_{K} A

given by g ⊗(a⊗_{K} b) ↦ (g ⊗a)⊗_{K} (g ⊗b) (g ∈ G/G_{P}) is a D-linear, two-sided H̄-colinear k-algebra splitting of A ⊗_{K} A → A ⊗K̂ A. The induced σ^{D} : H̄ → H is a two-sided H̄-colinear k-algebra splitting of H → H̄. It follows by [16, Theorem 7.2.2] that

(3.10)  B ⊗_{k} H̄ → H, b ⊗ x ↦ bσ^{D}(x)

gives a right H̄-colinear B-algebra isomorphism.

(v) If G_{P} is normal in G, then (K̂/K, K̂, k(G/G_{P})^{∗}) is a PV extension by Example 3.7.3. Hence the assertion follows from Proposition 3.5.8. □

Theorem 3.7.5. Let K ⊂ L be an inclusion of AS D-module algebras. Choose arbitrarily P ∈ Ω(L), and let 𝔭 = P ∩ K (∈ Ω(K)). Then L/K is a PV extension if

(a) G_{P} is normal in G_{𝔭}, and

(b) the inclusion K_{1} := K/𝔭 ⊂ L_{1} := L/P of D(G_{P})-module fields is a PV extension. The converse holds true if the field K^{D} (= L^{D}) of D-invariants is separably closed.

Proof. This follows by slightly modifying the last proof, as follows. We may suppose that K is a field.

Suppose that (L_{1}/K_{1}, A_{1}, H̄) is a PV extension. Define A = Φ(A_{1}) with Φ = Φ_{G_{P}}. Recall from Proposition 3.7.4 that if L/K is PV, the principal algebra must be A. As was seen in the last proof, A ⊗_{K} A is a right H̄-comodule k-algebra and the map σ given in (3.9) induces an H̄-colinear k-algebra map σ^{D} : H̄ → (A ⊗_{K} A)^{D}. Again by [16, Theorem 7.2.2], we have a D-linear and H̄-colinear isomorphism

A ⊗_{K} Φ(K) ⊗_{k} H̄ ≃ A ⊗_{K} A

of algebras over A ⊗_{K} Φ(K); see (3.10). It follows that L/K is a PV extension iff the natural injection

(3.11)  A ⊗_{k} (A ⊗_{K} Φ(K))^{D} → A ⊗_{K} Φ(K)

is surjective. If G_{P} is normal in G, then this is surjective since by Example 3.7.3, A ⊗_{k} (Φ(K) ⊗_{K} Φ(K))^{D} → A ⊗_{K} Φ(K) is already surjective.

To prove the converse, we may suppose (b) by Proposition 3.7.4 (iii), and that the map given in (3.11) is an isomorphism by the argument above. It follows that

dim_{k}(A ⊗_{K} Φ(K))^{D} = [G : G_{P}].

If k is a separably closed field, then B = (A ⊗_{K} Φ(K))^{D} is isomorphic to k × ⋯ × k, the product of [G : G_{P}] copies of k. The isomorphism (3.11) induces a D(G_{P})-linear isomorphism

L_{1} ⊗_{k} B ⟶^{∼} L_{1} ⊗_{K} Φ(K) ⟶^{∼} Φ(L_{1}) = L.

Thus all primitive idempotents in L are fixed by the action of G_{P} and hence G_{P} = G_{Ω(L)}. This implies that G_{P} is normal in G. □

As will be seen from the following, the second half does not necessarily hold true unless k is separably closed.

Example 3.7.6. Let G = D_{3} = {1, σ, σ^{2}, τ, στ, σ^{2}τ} be the dihedral group of order 6 (σ^{3} = 1, τ^{2} = 1, στ = τσ^{2}), G_{1} = \{1, τ\} ⊂ G, and D = ℚG. Put k = ℚ, K = ℚ(\sqrt{−1}), and identify G_{1} with Gal(K/k). Then G acts on K so that σa = a for all a ∈ K. Take the system of representatives {1, σ, σ^{2}} for G/G_{1} and let L = Φ_{G_{1}}(K) = 1⊗K+σ⊗K+σ^{2}⊗K. Then L/K is a PV extension of AS D-module algebras with the Hopf algebra

H = ℚ[z_{1}, z_{2}, z_{3}]/⟨z_{1} + z_{2} − 1, z_{2}^{2} − z_{2}, z_{3}^{2} + z_{2}, z_{2}z_{3} − z_{3}⟩

where ε(z_{i}) = δ_{1i} (i = 1, 2, 3) and

Δ(z_{1}) = z_{1} ⊗ z_{1} + \frac{1}{2}z_{2} ⊗ z_{2} + \frac{1}{2}z_{3} ⊗ z_{3},
Δ(z_{2}) = z_{1} ⊗ z_{2} + z_{2} ⊗ (z_{1} + \frac{1}{2}z_{2}) − \frac{1}{2}z_{3} ⊗ z_{3},
Δ(z_{3}) = z_{1} ⊗ z_{3} − \frac{1}{2}z_{2} ⊗ z_{3} + z_{3} ⊗ (z_{1} − \frac{1}{2}z_{2}).

Indeed, a Hopf algebra isomorphism H ⟶^{∼} (L ⊗_{K} L)^{D} is given by

z_{1} ↦ (1 ⊗ 1) ⊗_{K} (1 ⊗ 1) + (σ ⊗ 1) ⊗_{K} (σ ⊗ 1) + (σ^{2} ⊗ 1) ⊗_{K} (σ^{2} ⊗ 1),
z_{2} ↦ (1 ⊗ 1) ⊗_{K} (σ ⊗ 1 + σ^{2} ⊗ 1) + (σ ⊗ 1) ⊗_{K} (1 ⊗ 1 + σ^{2} ⊗ 1)
+ (σ^{2} ⊗ 1) ⊗_{K} (1 ⊗ 1 + σ ⊗ 1),
z_{3} ↦ −\sqrt{−1}(1 ⊗ 1) ⊗_{K} (σ ⊗ 1 − σ^{2} ⊗ 1) + \sqrt{−1}(σ ⊗ 1) ⊗_{K} (1 ⊗ 1 − σ^{2} ⊗ 1)
− \sqrt{−1}(σ^{2} ⊗ 1) ⊗_{K} (1 ⊗ 1 − σ ⊗ 1).

The PV group scheme G(L/K) = SpecH is a twisted form of ℤ/3ℤ (see [29, (6.4)]):

ℚ(\sqrt{−1}) ⊗_{ℚ} H ⟶^{∼} ℚ(\sqrt{−1}) × ℚ(\sqrt{−1}) × ℚ(\sqrt{−1}) ≃ ℚ(\sqrt{−1})(ℤ/3ℤ)^{∗}
z_{1} ↦ (1, 0, 0)
\frac{1}{2}(z_{2} + \sqrt{−1}z_{3}) ↦ (0, 1, 0)
\frac{1}{2}(z_{2} − \sqrt{−1}z_{3}) ↦ (0, 0, 1).

3.8. Splitting algebras

Let K be an AS D-module algebra and V a K#D-module. The rank rk_{K}(V) of the free K-module V will be called the K-rank; see Corollary 3.3.4.

Definition 3.8.1. Let K ⊂ L be an inclusion of AS D-module algebras and V a K#Dmodule. We say that V splits in L/K, or L/K is a splitting algebra for V iff there is an L#D-linear injection L ⊗_{K} V ↪ L^{Λ} into some power L^{Λ} of L. K⟨V ⟩ denotes the smallest AS D-module subalgebra of L including K and f(V) for all f ∈ Hom_{K#D}(V, L). If L = K⟨V ⟩ and V splits in L/K, we say L/K is a minimal splitting algebra for V.

Similarly to [27, Proposition 3.1], we have the following:

Proposition 3.8.2. Let K ⊂ L be an inclusion of AS D-module algebras and V a K#Dmodule.

(i) If V splits in L/K, every K#D-submodule of V splits in L/K.

(ii) If V splits in L/K, it splits in K⟨V ⟩/K.

(iii) V splits in L/K iff the canonical L-module map

(3.12)  L ⊗_{L^{D}} Hom_{K#D}(V, L) → Hom_{K}(V, L),

has a dense image; in other words, the map

(3.13)  L ⊗_{K} V → Hom_{L^{D}}(Hom_{K#D}(V, L), L), a ⊗ v ↦ [f ↦ af(v)]

is injective.

Proof. (i) Since all K#D-modules are free K-modules, this follows immediately by the definition.

(ii) If V splits in L/K, then the image of L ⊗_{K} V → L^{Λ} is in K⟨V ⟩^{Λ} by the definition. Thus V splits in K⟨V ⟩/K.

(iii) Recall that Hom_{K}(V, L) has an L#D-module structure by D-conjugation (3.5) and the map (3.12) is necessarily injective by Corollary 3.1.4.

(“If” part.) Since L^{D} is a field, Hom_{K#D}(V, L) is a free L^{D}-module. By taking a dual basis, we can identify Hom_{L^{D}}(Hom_{K#D}(V, L), L) with some power L^{Λ} of L. Then the injective L-module map L ⊗_{K} V → Hom_{L^{D}}(Hom_{K#D}(V, L), L) ⟶^{∼} L^{Λ} can be considered as an injective L#D-module map.

(“Only if” part.) By the definition, there is an L#D-linear injection φ : L ⊗_{K} V → L^{Λ} for some power L^{Λ}. Let {f_{i}}_{i∈Λ} ⊂ Hom_{K#D}(V, L) be the family of K#D-module maps induced by φ(1 ⊗ v) = (f_{i}(v))_{i∈Λ} (v ∈ V). Take an arbitrary element w = ∑_{j} a_{j} ⊗ v_{j} ∈ L ⊗_{K} V. If the image of w by the map (3.13) is 0, then we have ∑_{j} a_{j}f_{i}(v_{j}) = 0 for all i ∈ Λ and so φ(w) = 0 (⇔ w = 0). Thus the map (3.13) is injective. □

Proposition 3.8.3. Let K ⊂ L be an inclusion of AS D-module algebras and V a K#Dmodule with finite K-rank rk_{K}(V) = r < ∞. Then

(3.14)  dim_{L^{D}} Hom_{K#D}(V, L) ≤ r

and the following are equivalent:

(a) V splits in L/K;

(b) L ⊗_{L^{D}} Hom_{K#D}(V, L) ⟶^{∼} Hom_{K}(V, L);

(c) dim_{L^{D}} Hom_{K#D}(V, L) = r;

(d) There is an isomorphism L ⊗_{K} V ⟶^{∼} L^{r} as L#D-modules;

(e) L ⊗_{L^{D}} (L ⊗_{K} V)^{D} ⟶^{∼} L ⊗_{K} V;

(f) dim_{L^{D}}(L ⊗_{K} V)^{D} = r;

(g) There is an injective L#D-module map L ⊗_{K} V → L^{n} for some integer n.

Proof. The inequality (3.14) follows since L ⊗_{L^{D}} Hom_{K#D}(V, L) → Hom_{K}(V, L) is an injective L#D-module map whose cokernel is a free L-module with finite rank. Then ((a) ⇔ (b) ⇔ (c) ⇔ (d)) follows from (the proof of) Proposition 3.8.2 (iii). ((d) ⇔ (e) ⇔ (f)) is easily seen. ((d) ⇒ (g) ⇒ (a)) is trivial. □

Lemma 3.8.4. Let K ⊂ L be an inclusion of AS D-module algebras, V a K#D-module, and W a K#D-submodule of V with finite K-rank. If V splits in L/K, then the restriction res. : Hom_{K#D}(V, L) → Hom_{K#D}(W, L) is surjective.

Proof. Consider the following L#D-module map:

φ : L ⊗_{L^{D}} Hom_{K#D}(V, L) → Hom_{K}(W, L), a ⊗ f ↦ af|_{W} .

Notice that Im φ is a direct summand of Hom_{K}(W, L) as an L-module. The transposed L-linear map of φ is given by

L ⊗_{K} W ↪ L ⊗_{K} V → Hom_{L^{D}}(Hom_{K#D}(V, L), L),

which is injective by Proposition 3.8.2 (iii). Thus φ is surjective. Since the functor (−)^{D} is exact, we have that φ^{D} = res. is surjective. □

We see that the functor Φ preserves splitting algebras:

Lemma 3.8.5. Let G_{1} ⊂ G, K_{1} ⊂ L_{1} be as in Lemma 3.7.1. Write Φ = Φ_{G_{1}}. Then, L_{1}/K_{1} is a (minimal) splitting algebra for a K_{1}#D(G_{1})-module V_{1}, iff Φ(L_{1})/Φ(K_{1}) is a (minimal) splitting algebra for the Φ(K_{1})#D-module Φ(V_{1}).

Proof. This easily follows from Proposition 3.2.4 if one notices that

Φ(K_{1}⟨V_{1}⟩) = Φ(K_{1})⟨Φ(V_{1})⟩

to see the equivalence on minimality. □

Let K ⊂ L be an inclusion of AS D-module algebras. For finitely many elements u_{1}, …, u_{m} in L, let K⟨u_{1}, …, u_{m}⟩ denote the smallest AS D-module subalgebra in L including K and u_{1}, …, u_{m}.

Definition 3.8.6. L/K is said to be finitely generated iff L is of the form K⟨u_{1}, …, u_{m}⟩. This is equivalent to that L_{1}/K_{1} is finitely generated, where K_{1} = K/P ∩ K, L_{1} = L/P for an arbitrarily chosen P ∈ Ω(L).

Theorem 3.8.7. Let K ⊂ L be as above. Suppose K^{D} = L^{D}. Then the following are equivalent:

(a) L/K is a finitely generated PV extension;

(b) L/K is a minimal splitting algebra for a cyclic K#D-module of finite K-rank;

(c) L/K is a minimal splitting algebra for a K#D-module of finite K-rank;

(d) L = K⟨x_{ij}⟩, where X = (x_{ij})_{i,j} is a GL_{n}-primitive in Kolchin’s sense [13]: X ∈ GL_{n}(L), and for every d ∈ D, (dX)X^{−1} ∈ M_{n}(K) with dX = (dx_{ij})_{i,j}.

Proof. We write k = K^{D} (= L^{D}).

(a) ⇒ (b). By Lemmas 3.7.1 and 3.8.5, we may assume that K is a field. Suppose that (L/K, A, H) is a finitely generated PV extension. By Proposition 3.7.4, we have a finitely generated PV extension (L_{1}/K, A_{1}, H̄) of module fields over C := D(G_{P}) with P ∈ Ω(L), such that L = Φ(L_{1}), A = Φ(A_{1}).

There exist those finitely many elements u_{1}, …, u_{m} in A which span an H-subcomodule over k, and satisfy L = K⟨u_{1}, …, u_{m}⟩. (In fact, take some x_{1}, …, x_{l} ∈ A which satisfy L = K⟨x_{1}, …, x_{l}⟩. Then there exists a finite dimensional H-subcomodule U ⊂ A such that x_{1}, …, x_{l} ∈ U (see [29, (3.3)]). Choose u_{1}, …, u_{m} which span U over k.) Set an element u = (u_{1}, …, u_{m}) in A^{m}, and let V = (K#D)u, the cyclic K#D-submodule generated by u. Since L ⊗_{K} A ≃ L ⊗_{k} H, we see that L/K is a minimal splitting algebra for A^{m}, and hence for V.

It remains to prove that the K-dimension dim_{K}(V) is finite. It suffices to prove that the natural image V (P), say, of V under the projection A^{m} → A_{1}^{m} has a finite Kdimension, since V is naturally embedded into ∏_{P ∈Ω(L)} V (P). Let g_{1}, …, g_{s} be a system of representatives of the right cosets G_{P} \G. Then we have

V = ∑_{i=1}^{s}(K#C)g_{i}u.

Fix an i ∈ {1, …, s}, and let w = (w_{1}, …, w_{m}) ∈ A_{1}^{m} denote the natural image of g_{i}u. It suffices to prove that W := (K#C)w has a finite K-dimension. By re-numbering we have a k-basis, w_{1}, …, w_{r} (r ≤ m), of the k-subspace in A_{1} spanned by w_{1}, …, w_{m}. There is a rank r matrix T with entries in k, such that w = w′T with w′ = (w_{1}, …, w_{r}). It suffices to prove that W′ := (K#C)w′ has a finite K-dimension, since W′ ≃ W under the right multiplication by T.

Notice that for any g ∈ G, gu_{1}, …, gu_{m} span an H-subcomodule in A since the comodule structure map θ : A → A ⊗_{k} H is D-linear. It then follows that w_{1}, …, w_{r} form a k-basis of an H̄-subcomodule in A_{1}. By applying Proposition 3.1.6 for w_{1}, …, w_{r} ∈ L_{1}, there exist r elements h_{1}, …, h_{r} ∈ C such that (h_{i}(w_{j}))_{i,j} is an invertible matrix. We claim that (cw′)(h_{i}(w_{j}))_{i,j}^{−1} ∈ K^{r} for all c ∈ C. If it follows, then W′ = Kh_{1}(w′)+⋯+Kh_{r}(w′) and hence we have dim_{K} W′ < ∞, as desired.

Let θ_{1} : A_{1} → A_{1} ⊗_{k} H̄ be the comodule structure map associated to the PV extension L_{1}/K. Write

θ_{1}(w_{j}) = ∑_{s=1}^{r} w_{s} ⊗_{k} z_{sj} (z_{sj} ∈ H, j = 1, …, r).

By applying μ-isomorphism A_{1} ⊗_{k} H ⟶^{∼} A_{1} ⊗_{K} A_{1} in each side, we have

(3.15)  1 ⊗_{K} w_{j} = ∑_{s=1}^{r}(w_{s} ⊗_{K} 1)z_{sj} in A_{1} ⊗_{K} A_{1}.

Hence

1 ⊗_{K} h_{i}(w_{j}) = ∑_{s=1}^{r}(h_{i}(w_{s}) ⊗_{K} 1)z_{sj} (i, j = 1, …, r),

i.e. 1 ⊗_{K} (h_{i}(w_{j}))_{i,j} = ((h_{i}(w_{j}))_{i,j} ⊗_{K} 1)Z with Z = (z_{ij})_{i,j}. Since (h_{i}(w_{j}))_{i,j} is invertible, we have

Z = ((h_{i}(w_{j}))_{i,j}^{−1} ⊗_{K} 1)(1 ⊗_{K} (h_{i}(w_{j}))_{i,j}) ∈ GL_{r}(L_{1} ⊗_{K} L_{1}).

On the other hand, recalling (3.15) we have

1 ⊗_{K} (cw′) = ((cw′) ⊗_{K} 1)Z

for all c ∈ C. Thus, by multiplying 1 ⊗_{K} (h_{i}(w_{j}))_{i,j}^{−1} = Z^{−1}((h_{i}(w_{j}))_{i,j}^{−1} ⊗_{K} 1) from the right,

1 ⊗_{K} (cw′)(h_{i}(w_{j}))_{i,j}^{−1} = ((cw′) ⊗_{K} 1)ZZ^{−1}((h_{i}(w_{j}))_{i,j}^{−1} ⊗_{K} 1) = (cw′)(h_{i}(w_{j}))_{i,j}^{−1} ⊗_{K} 1

for all c ∈ C. This implies the claim above.

(b) ⇒ (c). This is trivial.

(c) ⇒ (d). Suppose that L/K is a minimal splitting algebra for V with finite K-free basis v_{1}, …, v_{n}. By Proposition 3.8.3 (c), we have a k-basis f_{1}, …, f_{n} in Hom_{K#D}(V, L). Define

(3.16)  X = (x_{ij})_{i,j} = (f_{j}(v_{i}))_{i,j}, v = ^{t}(v_{1}, …, v_{n}).

Then we have X ∈ GL_{n}(L) since there is an L-module isomorphism

L^{n} ≃ L ⊗_{K} V ⟶^{∼} Hom_{k}(Hom_{K#D}(V, L), L) ≃ L^{n}

which is precisely the multiplication of X (see the proof of Proposition 3.8.2, 3.8.3). If we write dv_{i} = ∑_{s=1}^{n} c_{is}(d)v_{s} (c_{is}(d) ∈ K) for d ∈ D, then we have dx_{ij} = f_{j}(dv_{i}) = ∑_{s=1}^{n} c_{is}(d)f_{j}(v_{s}) = ∑_{s=1}^{n} c_{is}(d)x_{sj}. This implies that X is GL_{n}-primitive such that

(3.17)  (dX)X^{−1}v = dv (d ∈ D),

i.e. (dX)X^{−1} = (c_{ij}(d))_{i,j} ∈ M_{n}(K). By the definition, we have L = K⟨V ⟩ = K⟨x_{ij}⟩. (d) ⇒ (a). This is shown by modifying [27, Example 2.5c] as follows. Put Y = (y_{ij})_{i,j} = X^{−1} and A = K[x_{ij}, y_{ij}]. First we shall show that A is a D-module

subalgebra of L. Define ϕ ∈ Hom_{R}(D, M_{n}(K)) by ϕ_{d} = d(X)X^{−1} = d(X)Y (d ∈ D). Since ϕ_{g}^{−1} = Xg(Y) = g(g^{−1}(X)X^{−1}) ∈ GL_{n}(K) for all g ∈ G, ϕ is convolution-invertible in Hom_{R}(D, M_{n}(K)) by [23, Corollary 9.2.4]. We see that the ψ ∈ Hom_{R}(D, M_{n}(L)) given by ψ_{d} = Xd(Y), is the inverse of ϕ, and so ψ ∈ Hom_{R}(D, M_{n}(K)). This implies that A is a D-module subalgebra of L. Since Q(A), the total quotient ring of A, is an AS D-module subalgebra of L containing K and x_{11}, …, x_{nn} (recall Corollary 3.3.8), we have Q(A) = K⟨x_{ij}⟩ = L.

Put

Z = (Y ⊗_{K} 1)(1 ⊗_{K} X), Z^{−1} = (1 ⊗_{K} Y)(X ⊗_{K} 1) ∈ GL_{n}(A ⊗_{K} A).

For all d ∈ D,

d(Z) = ∑_{(d)}(d_{(1)}(Y) ⊗_{K} 1)(1 ⊗_{K} d_{(2)}(X))
= ∑_{(d)}(Y ψ_{d_{(1)}} ⊗_{K} 1)(1 ⊗_{K} ϕ_{d_{(2)}}X)
= ∑_{(d)}(Y ⊗_{K} 1)(ψ_{d_{(1)}} ⊗_{K} 1)(1 ⊗_{K} ϕ_{d_{(2)}})(1 ⊗_{K} X)
= ∑_{(d)}(Y ⊗_{K} 1)(ψ_{d_{(1)}} ⊗_{K} 1)(ϕ_{d_{(2)}} ⊗_{K} 1)(1 ⊗_{K} X)
= ∑_{(d)}(Y ⊗_{K} 1)(ψ_{d_{(1)}}ϕ_{d_{(2)}} ⊗_{K} 1)(1 ⊗_{K} X)
= ε(d)Z.

Thus Z has entries in H := (A ⊗_{K} A)^{D}. Similarly we have that the entries in Z^{−1} are also in H and hence Z ∈ GL_{n}(H). Then,

1 ⊗_{K} X = (X ⊗_{K} 1)Z, 1 ⊗_{K} Y = Z^{−1}(Y ⊗_{K} 1) ∈ GL_{n}(A · H).

This implies A ⊗_{K} A = A · H. Therefore (L/K, A, H) is a PV extension. □

Remark 3.8.8. Keep the notation just as above.

(i) Write Z = (z_{ij}), Z^{−1} = (w_{ij}). Then A ⊗_{K} A = A[z_{ij}, det(z_{ij})^{−1}] and H = k[z_{ij}, det(z_{ij})^{−1}] = k[z_{ij}, w_{ij}]. Taking μ, θ as in Proposition 3.5.2, we see

\begin{aligned}θ(X) & = & μ^{−1}(1 ⊗_{K} X) = μ^{−1}((X ⊗_{K} 1)Z) = μ^{−1} (∑_{s=1}^{n}(x_{is} ⊗_{K} 1)z_{sj})_{i,j} \\ & = & (∑_{s=1}^{n}x_{is} ⊗_{k} z_{sj})_{i,j} = (X ⊗_{k} 1)(1 ⊗_{k} Z). \end{aligned}

This is often written like

(3.18)  θ(X) = X ⊗_{k} Z.

It follows that the Hopf algebra structure of H is given by

Δ(Z) = Z ⊗_{k} Z, ε(Z) = I, S(Z) = Z^{−1},

here I denotes the identity matrix; see [29, (3.2), Corollary]. We have a Hopf algebra surjection,

k[GL_{n}] = k[T_{ij}, det(T_{ij})^{−1}] ↠ H, T_{ij} ↦ z_{ij},

which gives a closed embedding G(L/K) → GL_{n} of affine k-group schemes.

(ii) Suppose that D = R[τ, τ^{−1}], the group algebra of the free abelian group of rank 1, and K is a field; K is then an inversive difference field. A difference system τy = By with B ∈ GL_{n}(K) arises uniquely from a K#D-module of K-dimension n, together with its K-basis. We see from (3.17) that the X in (3.16) is a fundamental matrix [20, Definition 1.4] for the difference system arising from the V and the v above, and so that A is the Picard-Vessiot ring [20, Definition 1.5] for the system. It will follow from Theorems 3.8.7, 3.8.11 that if k (= K^{D}) is algebraically closed, a Picard-Vessiot ring for any difference system as above uniquely exists, and is given by such an A as above.

Corollary 3.8.9. Let (L/K, A, H) be a PV extension of AS D-module algebras. The following are equivalent:

(a) L/K is finitely generated;

(b) L is the total quotient ring of a finitely generated K-subalgebra in L;

(c) A is finitely generated as a K-algebra;

(d) H is finitely generated as a k-algebra.

Proof. ((a) ⇒ (c) ⇒ (b) ⇒ (a)) and ((a) ⇒ (d)) follow by the proof of Theorem 3.8.7. If H is a finitely generated k-algebra, then we have an ascending chain condition for Hopf ideals of H. Hence we have an ascending chain condition for intermediate AS D-module algebras of L/K, which implies (a). □

Corollary 3.8.10. Let K ⊂ L be an inclusion of AS D-module algebras such that K^{D} = L^{D} =: k. Then L/K is a PV extension iff it is a minimal splitting algebra for such a K#D-module V that is a directed union, V = ⋃_{λ} V_{λ}, of K#D-submodules V_{λ} of finite K-rank.

Proof. This follows in the same way as [27, Corollary 3.5]. We include the proof for convenience.

(“Only if” part.) Let (L/K, A, H) be a PV extension. Then H is a directed union of Hopf subalgebras which are finitely generated k-algebras (see [29, (3.3)]). It follows by Proposition 3.5.8 and Corollary 3.8.9 that L is a directed union, say L = ⋃_{λ} L_{λ}, of AS D-module subalgebras which are finitely generated PV extensions over K. By Theorem 3.8.7, each L_{λ}/K is a minimal splitting algebra for a K#D-module V_{λ} of finite K-rank. Then L/K is a minimal splitting algebra for the direct sum V = ⨁_{λ} V_{λ}.

(“If” part.) Suppose that L/K is a minimal splitting algebra for a K#D-module V = ⋃_{λ} V_{λ}, a directed union of K#D-modules V_{λ} of finite K-rank. Since every V_{λ} splits in L/K, each L_{λ} := K⟨V_{λ}⟩ is a minimal splitting algebra for V_{λ} and is a finitely generated PV extension over K by Theorem 3.8.7. By Lemma 3.8.4, the union ⋃_{λ} L_{λ} is a directed union of AS D-module subalgebras of L. Thus ⋃_{λ} L_{λ} is an AS D-module subalgebra of L by Lemma 3.3.7. For every f ∈ Hom_{K#D}(V, L), we have f(V) = ⋃_{λ} f(V_{λ}) ⊂ ⋃_{λ} L_{λ}. Hence L = K⟨V ⟩ = ⋃_{λ} L_{λ}. Let A_{λ} (resp. H_{λ}) be the principal algebra (resp. the Hopf algebra) for L_{λ}/K. Then one sees that (L/K, ⋃_{λ} A_{λ}, ⋃_{λ} H_{λ}) is a PV extension. □

Theorem 3.8.11. Let K be an AS D-module algebra such that the field K^{D} of Dinvariants is algebraically closed. Let V be a K#D-module of finite K-rank. Then there exists an AS D-module algebra L including K such that K^{D} = L^{D}, and L/K is a (necessarily finitely generated) minimal splitting algebra for V. Such an algebra is unique up to D-linear isomorphism of K-algebras.

To prove this, we need the following:

Lemma 3.8.12. Let K be an AS D-module algebra. Let A be a simple D-module algebra, and let L = Q(A) be the total quotient ring of A; by Lemma 3.3.6, L is uniquely a Dmodule algebra. If A is finitely generated as a K-algebra, then L^{D}/K^{D} is an algebraic extension of fields.

Proof. We follow Levelt [14, Appendix] for this proof. If x ∈ L^{D}, then (A : x) = {a ∈ A | ax ∈ A} is a D-stable ideal. Since this contains a non-zero divisor, we have that (A : x) = A, and so A^{D} = L^{D}.

If A is finitely generated, then it is noetherian. By Proposition 3.3.3, we may suppose that K is a field (and A is a domain). If P ⊂ A is a maximal ideal, then the field A^{D} is included in the field A/P, which is algebraic over K. Therefore if x ∈ A^{D}, it is algebraic over K. Let φ(T) = T^{n} + c_{1}T^{n−1} + ⋯ + c_{n} denote the minimal polynomial of x over K. Since for any d ∈ D, ε(d)T^{n} + (dc_{1})T^{n−1} + ⋯ + dc_{n} has x as a root, each c_{i} ∈ K^{D} by the minimality of φ(T). Thus x is algebraic over K^{D}. □

Proof of Theorem 3.8.11. Existence; this is proved by modifying the proof of [27, Theorem 4.5], as follows. Let v_{1}, …, v_{r} be a K-basis for V. For d ∈ D, write

dv_{i} = ∑_{s=1}^{r} c_{is}(d)v_{s}

with c_{is}(d) ∈ K. Define a D-module algebra structure on K[X_{ij}], the polynomial Kalgebra in r^{2} indeterminates, by

d(X_{ij}) = ∑_{s=1}^{r} c_{is}(d)X_{sj} (d ∈ D).

For each g ∈ G, we see that (c_{ij}(g)) is an invertible matrix: (c_{ij}(g))^{−1} = (gc_{ij}(g^{−1})). Thus det(c_{ij}(g)) is invertible in K for each g ∈ G, and the D-module algebra structure of K[X_{ij}] is uniquely extended to F = K[X_{ij}, det(X_{ij})^{−1}] by Lemma 3.3.6. Let I be a maximal D-stable ideal of F, and put A = F/I. Since K is simple, I ∩ K = 0. Hence A is a noetherian simple D-module algebra including K. Let L be the total quotient ring of A; this is an AS D-module algebra by Proposition 3.3.3 and Lemma 3.3.6. By Lemma 3.8.12, we have L^{D} = K^{D}. Let x_{ij} denote the image of X_{ij} in A, and define K-linear maps f_{j} : V → L (j = 1, …, r) by f_{j}(v_{i}) = x_{ij}. Then these maps are in Hom_{K#D}(V, L), and are linearly independent over L^{D}, since (x_{ij})_{i,j} ∈ GL_{r}(L). Therefore, L/K is a minimal splitting algebra for V by Lemma 3.8.3 (c).

Uniqueness; also this proof is essentially the same as the proof given in [27, Theorem 4.6]. Let L_{1}/K and L_{2}/K are two minimal splitting algebra for V such that L_{1}^{D} = L_{2}^{D} = K^{D} = k. By Theorem 3.8.7, L_{1}/K and L_{2}/K are finitely generated PV extensions. Let A_{i} be the principal algebra for L_{i}/K (i = 1, 2) respectively. Put A = A_{1} ⊗_{K} A_{2} and let I be a maximal D-stable ideal of A. We see A_{1}, A_{2} are noetherian simple Dmodule algebras which are finitely generated K-algebras by the proof of Theorem 3.8.7 (d) ⇒ (a), and by Proposition 3.5.9 (i). Thus A_{i} ∩ I = 0 (i = 1, 2) and hence A_{1}, A_{2} are identified with D-module subalgebras of A/I. Let L be the total quotient ring of A/I. By Lemma 3.3.6 and Proposition 3.3.3, L is an AS D-module algebra since A/I is a noetherian simple D-module algebra. Furthermore, since A/I is a finitely generated K-algebra, we have L^{D} = K^{D} = k by Lemma 3.8.12. Let χ_{i} : L_{i} ↪ L (i = 1, 2) denote the induced inclusions of D-module algebras over K. The injective k-linear maps Hom_{K#D}(V, L_{i}) → Hom_{K#D}(V, L) (i = 1, 2) are precisely isomorphisms by Proposition 3.8.3. Therefore f(V) ⊂ χ_{1}(L_{1})∩χ_{2}(L_{2}) for all f ∈ Hom_{K#D}(V, L). Since L_{i} are generated over K by the image of all f ∈ Hom_{K#D}(V, L_{i}), we have χ_{1}(L_{1}) = χ_{2}(L_{2}) = K⟨V ⟩ in L. Thus we have χ_{2}^{−1} ∘ χ_{1} : L_{1} ⟶^{∼} L_{2}, a D-module algebra isomorphism over K. □

Let K be an AS D-module algebra. We have the K^{D}-abelian symmetric tensor category (_{K#D}M, ⊗_{K}, K). Let V be an object in _{K#D}M of finite K-rank. Then the K-linear dual V^{∗} := Hom_{K}(V, K) is a dual object under the D-conjugation; see (3.5). Thus the tensor full subcategory _{K#D}M_{fin} consisting of the finite K-rank objects is rigid. Let {{V }} denote the abelian, rigid tensor full subcategory of _{K#D}M generated by V, that is, the smallest full subcategory containing V that is closed under subquotients, finite direct sums, tensor products and duals. Thus an object in {{V }} is precisely a subquotient of some finite direct sum W_{1} ⊕ ⋯ ⊕ W_{r}, where each W_{i} is the tensor product of some copies of V, V^{∗}; see [21, Theorem 2.33] also for comparing with the following.

Theorem 3.8.13. Let (L/K, A, H) be a finitely generated PV extension of AS D-module algebras. By Theorem 3.8.7, we have such a K#D-module V of finite K-rank for which L/K is a minimal splitting algebra.

(i) Let W ∈ {{V }}. Regard the A ⊗_{K} W as a right H-comodule with the structure induced by A. Then (A ⊗_{K} W)^{D} is an H-subcomodule with k-dimension rk_{K}(W).

(ii) W ↦ (A ⊗_{K} W)^{D} gives a k-linear equivalence

{{V }} ≈ M_{fin}^{H}

of symmetric tensor categories, where M_{fin}^{H} = (M_{fin}^{H}, ⊗_{k}, k) denotes the rigid symmetric tensor category of finite-dimensional right H-comodules; notice that this is isomorphic to the category Rep_{G(L/K)} of the same kind, consisting of finite-dimensional linear representations of the PV group scheme G(L/K) = SpecH.

Proof. Put D_{k} = D⊗_{R}k, a cocommutative Hopf algebra over k, and consider D_{k} as a right H-comodule algebra with the trivial structure map d ↦ d ⊗ 1. Regard naturally A as an algebra in the symmetric tensor category (_{D_{k}}M^{H}, ⊗_{k}, k) of right (H, D_{k}^{op})-Hopf modules (see [16, §8.5]); its objects are D_{k}-modules N which has a D_{k}-linear, right H-comodule structure ρ_{N} : N → N ⊗_{k} H. We then have the symmetric tensor category _{A}(_{D_{k}}M^{H}) of A-modules in _{D_{k}}M^{H}, which is denoted by (_{A#D}M^{H}, ⊗_{A}, A); this is k-abelian. Define k-linear functors

M^{H} ⟶^{Θ_{1}} ; ⟵_{Ξ_{1}}_{A#D}M^{H} ⟶^{Θ_{2}} ; ⟵_{Ξ_{2}}_{K#D}M

by

Θ_{1}(U) = A ⊗_{k} U; H coacts codiagonally,
Ξ_{1}(N) = N^{D},
Θ_{2}(N) = N^{co H} (= {n ∈ N | ρ_{N}(n) = n ⊗_{k} 1}),
Ξ_{2}(W) = A ⊗_{K} W; H coacts on A.

We see that Θ_{1} and Ξ_{2} are symmetric tensor functors with the obvious tensor structures. Moreover by [22], Θ_{2} and Ξ_{2} are quasi-inverses of each other, since A/K is H-Galois by Proposition 3.5.2 (iii). Since A^{D} = k, Ξ_{1} ∘ Θ_{1} is isomorphic to the identity functor. Suppose N ∈ _{A#D}M^{H}. Since A is simple by Corollary 3.5.9 (i), we see from Corollary 3.1.4 that the morphism in _{A#D}M^{H}

μ_{N} : Θ_{1} ∘ Ξ_{1}(N) = A ⊗_{k} N^{D} → N

is an injection. Let N denote the full subcategory of _{A#D}M^{H} consisting of those N for which μ_{N} is an isomorphism. Since each Θ_{1}(U) is in N, Θ_{1} gives an equivalence

M^{H} ≈ N .

Necessarily, N is closed under tensor products, and this is an equivalence of symmetric tensor categories.

Since A ⊗_{K} V ≃ A^{n} (n = rk_{K}(V)) in _{A#D}M, Ξ_{2}(V) = A ⊗_{K} V ∈ N. We see that Θ_{1} is exact, and N is closed under subquotients. Therefore for (ii), it suffices to prove that

Ṽ := Ξ_{1} ∘ Ξ_{2}(V) = (A ⊗_{K} V)^{D}

generates M_{fin}^{H}. Let v_{1}, …, v_{n} be a K-free basis of V, and define X, v as in (3.16). We see from (3.17) that the entries in ṽ := X^{−1} ⊗_{K} v (∈ (A ⊗_{K} V)^{n}) are D-invariant, and hence form a k-basis in Ṽ. By (3.18), the H-comodule structure ρṼ : Ṽ → Ṽ ⊗_{k} H on Ṽ is given by

ρṼ (^{t}ṽ) = ^{t}ṽ ⊗_{k} ^{t}Z^{−1},

where ^{t} denotes the transpose of matrices. This means that the coefficient k-space of Ṽ is the subcoalgebra in H spanned by the entries w_{ij} in ^{t}Z^{−1}. Since w_{ij} together with the entries S(w_{ij}) in Z generate the k-algebra H (see the proof of Theorem 3.8.7 (d) ⇒ (a)), Ṽ generates M_{fin}^{H}; see [29, (3.5)]. This proves part (ii).

If W ∈ {{V }}, then Ξ_{2}(W) ∈ N, and so

dim_{k}(A ⊗_{K} W)^{D} = rk_{A}(A ⊗_{K} W) = rk_{K}(W).

This proves part (i). □

3.9. Liouville extensions

Finally we define the notion of Liouville extensions and show the solvability theorem. As is described in Introduction, we should define Liouville group schemes and study how strong the definition is.

3.9.1. Liouville group schemes.

Definition 3.9.1. Let G be an algebraic affine group scheme over a field k.

(1) We say G is (k-)Liouville (cf. [13, p. 374]) iff there exists a normal chain of closed subgroup schemes

(3.19)  G = G_{0} ▷ G_{1} ▷ ⋯ ▷ G_{r} = {1}

such that each G_{i−1}/G_{i} (i = 1, …, r) is at least one of the following types: finite etale, a closed subgroup scheme of G_{a}, or a closed subgroup scheme of G_{m}. In this case, we call (3.19) a Liouville normal chain (LNC).

(2) In (3.19), if each G_{i−1}/G_{i} is merely a closed subgroup scheme of G_{a} or a closed subgroup scheme of G_{m}, then we call it a restricted Liouville normal chain (RLNC).

We use the following abbreviation of some types on group schemes which arise as factor group schemes in an LNC: we say G is of G_{a}-type (resp., G_{m}-type) iff it is a closed subgroup scheme of G_{a} (resp., G_{m}), and a group scheme of RL-type (resp., L-type) means that it is of G_{a}-type or G_{m}-type (resp., RL-type or finite etale).

Lemma 3.9.2. (1) If G is Liouville (resp., has an RLNC), then every closed subgroup scheme of G is Liouville (resp., has an RLNC). Especially G is Liouville iff the connected component G^{∘} is Liouville.

(2) Let H be a normal closed subgroup scheme of G. Then G is Liouville (resp., has an RLNC) iff both H and G/H are Liouville (resp., have an RLNC).

(3) If G is connected Liouville, then G is solvable.

Proof. First we take an LNC (resp., an RLNC): G = G_{0} ▷ G_{1} ▷ ⋯ ▷ G_{r} = {1} in each proof of (1), “only if” part of (2), and (3).

(1) Let H be a closed subgroup scheme of G and put H_{i} := H∩G_{i} (i = 0, …, r). Then we have H_{0} = H and H_{i} = H_{i−1} ∩G_{i} = Ker(H_{i−1} → G_{i−1}/G_{i}) for i = 1, …, r. It follows that H_{i−1} ▷ H_{i} and H_{i−1}/H_{i} is a closed subgroup scheme of G_{i−1}/G_{i} for i = 1, …, r. Therefore H = H_{0} ▷ H_{1} ▷ ⋯ ▷ H_{r} = {1} is an LNC (resp., an RLNC).

(2) (“Only if” part.) H is Liouville (resp., has an RLNC) by (1). Put F_{i} := G_{i}/H ∩ G_{i} (i = 0, …, r). Each k[F_{i−1}/F_{i}] is identified with a Hopf subalgebra of k[G_{i−1}/G_{i}], since the quotient morphism G_{i−1} → F_{i−1} → F_{i−1}/F_{i} goes through G_{i−1} → G_{i−1}/G_{i}; consider the following commutative diagram:

{1} ⟶ G_{i} ⟶ G_{i−1} ⟶ G_{i−1}/G_{i} ⟶ {1} (exact)
↓^{quotient} ↓^{quotient} ↓^{∃}^{quotient}
{1} ⟶ F_{i} ⟶ F_{i−1} ⟶ F_{i−1}/F_{i} ⟶ {1} (exact).

Thus each F_{i−1}/F_{i} is of L-type (resp., RL-type) for i = 1, …, r. Therefore G/H = F_{0} ▷ F_{1} ▷ ⋯ ▷ F_{r} = {1} is an LNC (resp., an RLNC).

(”If” part.) Let G/H = F_{0} ▷ F_{1} ▷ ⋯ ▷ F_{r} = {1} be an LNC (resp., an RLNC) and (0) = I_{0} ⊂ I_{1} ⊂ ⋯ ⊂ I_{r} the corresponding sequence of Hopf ideals of k[G/H]. Each I_{i}/I_{i−1} is the normal Hopf ideal of k[F_{i−1}] = k[G/H]/I_{i−1} corresponding to the Hopf subalgebra k[F_{i−1}/F_{i}], and thus I_{i}/I_{i−1} = k[F_{i−1}] · k[F_{i−1}/F_{i}]^{+}. If we put I′_{i} := k[G] · I_{i} (i = 0, …, r), then each I′_{i} becomes a Hopf ideal of k[G]. Let G_{i} be the closed subgroup scheme of G which corresponds to I′_{i}. Since k[G] is a faithfully flat k[G/H]-module, we have an inclusion

k[F_{i−1}] = k[G/H]/I_{i−1} ↪ k[G] ⊗_{k[G/H]} (k[G/H]/I_{i−1}) ≃ k[G]/I′_{i−1} = k[G_{i−1}]

for each i = 1, …, r+1. Considering I_{i}/I_{i−1} = k[F_{i−1}]·k[F_{i−1}/F_{i}]^{+} through this inclusion, we have that I′_{i}/I′_{i−1} = k[G_{i−1}] · k[F_{i−1}/F_{i}]^{+}. Hence we have a normal chain G = G_{0} ▷ G_{1} ▷ ⋯ ▷ G_{r} = H such that G_{i−1}/G_{i} ≃ F_{i−1}/F_{i} (i = 1, …, r). Therefore G is Liouville (resp., has an RLNC).

(3) We use induction on the least length r of LNC. The case r = 0 is clear. Let r > 0. Since G is connected, G/G_{1} is also connected. Then G/G_{1} is of RL-type and hence abelian. Therefore 𝒟 G (see [29, (10.1)]) is a connected closed subgroup scheme of G_{1}. By (1) and its proof, 𝒟 G is connected Liouville and has an LNC with length ≤ r − 1. Then 𝒟 G is solvable by inductive assumption, concluding the proof. □

The converse of (3) above does not hold in general:

Example 3.9.3. (1) A nontrivial anisotropic torus T is connected solvable but not Liouville since both Hom(T, G_{m}) and Hom(T, G_{a}) are trivial.

(2) Let k be the prime field of ch(k) = 2 and H = k[x]/⟨x^{4} + x^{2} + x⟩ with x primitive. Then H is a commutative Hopf algebra and G = SpecH is abelian, finite etale, and unipotent. The Cartier dual G^{∗} is finite connected of multiplicative type and hence solvable. Since H^{∗} does not have any nontrivial grouplike, Hom(G^{∗}, G_{m}) is trivial. Therefore G^{∗} is not Liouville.

Proposition 3.9.4. Let G be a connected algebraic affine group scheme over a field k. Then G is Liouville iff G has an RLNC.

Proof. The “if” part is clear. For the “only if” part, we use induction on the least length r of LNC G = G_{0} ▷ G_{1} ▷ ⋯ ▷ G_{r} = {1}. The case r = 0 is clear. Let r > 0 and assume G_{1}^{∘} has an RLNC. By the argument in the proof of Lemma 3.9.2 (3), we have G ▷ G_{1}^{∘} and G/G_{1}^{∘} is abelian. Thus the proof can be reduced to the case that G is connected abelian by Lemma 3.9.2 (2).

Let G be connected abelian and put H = k[G]. Let H_{u} (= H^{1}) be the irreducible component of H which contains 1 and H_{s} = H/HH_{u}^{+}. Then we have the exact sequence

(3.20)  H_{u} ↣ H ↠ H_{s}.

Let k̄ denote the algebraic closure of k. It is known that H_{u} ⊗_{k} k̄ is also the irreducible component of H ⊗_{k} k̄ containing 1. (Let ∧_{k} (resp. ∧k̄) denotes the wedge product of k-subspaces (resp. k̄-subspaces) of H (resp. H ⊗_{k} k̄) in the sense of [23, Ch. IX]. Then (H ⊗_{k} k̄)^{1} = ⋃_{n=0}^{∞} ∧n ; k̄k̄ = ⋃_{n=0}^{∞}(∧_{k}k) ⊗_{k} k̄ = H_{u} ⊗_{k} k̄.) The exact sequence (3.20) splits over k̄ (the Jordan decomposition of Gk̄ [29, (9.5)]), and G_{s} := SpecH_{s} is connected of multiplicative type since (G_{s})k̄ is connected diagonalizable. Put G_{u} := SpecH_{u} (= G/G_{s}); this is unipotent. We see G_{u} has an RLNC whose all factor group schemes are of G_{a}-type (see [29, Ch. 16, Ex. 5]). Then it suffices to show that G_{s} has an RLNC. Let T be a maximal torus of G_{s}. T includes no nontrivial anisotropic subtorus since it is Liouville. Hence, by [29, (7.4)], we see T is a split torus and has an RLNC. Put H = G_{s}/T; this is finite connected, Liouville, and of multiplicative type. Let H ▷ H_{1} ▷ ⋯ ▷ H_{r} = {1} be an LNC. We see H/H_{1} is of G_{m}-type. Since H is finite connected, k[H] is a local algebra of finite dimension. Then its quotient k[H_{1}] is also a local algebra of finite dimension and hence H_{1} is connected. By inductive assumption, H_{1} has an RLNC. Therefore H also has an RLNC, concluding the proof. □

Proposition 3.9.5. Let k be an algebraically closed field and G an algebraic affine group scheme over k. Then G is Liouville iff G^{∘} is solvable.

Proof. In fact we have proved the “only if” part in Lemma 3.9.2 over an arbitrary field. For the “if” part, we use induction on the least m such that 𝒟 ^{m}G^{∘} = {1}. The case m = 0 is clear. Let m > 0 and assume that 𝒟 G^{∘} has an RLNC. By Lemma 3.9.2 (2), it suffices to show that G^{∘}/𝒟 G^{∘} has an RLNC. Thus the proof can be reduced to the case that G is (connected) abelian.

Let G be abelian and take the Jordan decomposition G = G_{s} ×G_{u}. G_{u} has an RLNC. Since k is algebraically closed, G_{s} is diagonalizable and hence has an RLNC. Therefore G has an RLNC. □

As is seen in the following example, we see that the triangulability is certainly stronger than the condition to have an RLNC, even if k is algebraically closed.

Example 3.9.6 ([29, Ch. 10, Ex. 3]). Let k be a field with ch(k) = 2 and G the closed subgroup scheme of SL_{2} over k defined by the Hopf algebra

k[G] = k[X_{11}, X_{12}, X_{21}, X_{22}]/⟨X_{11}X_{22} + X_{12}X_{21} + 1, X_{11}^{2} + 1, X_{22}^{2} + 1, X_{12}^{2}, X_{21}^{2}⟩. Put I = ⟨X_{11}X_{22} + X_{12}X_{21} + 1, X_{11}^{2} + 1, X_{22}^{2} + 1, X_{12}^{2}, X_{21}^{2}⟩ ⊂ k[X_{11}, X_{12}, X_{21}, X_{22}]. Since \sqrt{I} = ⟨X_{11} + 1, X_{22} + 1, X_{12}, X_{21}⟩, we have k[X_{11}, X_{12}, X_{21}, X_{22}]/\sqrt{I} = k and hence G is finite connected. Let H_{1} = k[G]/k[G]X_{21} and G_{1} = SpecH_{1}. Then we have exact sequences

\begin{matrix}k[α_{2}] = k[X]/⟨X^{2}⟩ & ↣ & k[G] & ↠ & H_{1} \\ X & ↦ & X_{21}X_{22} \end{matrix}

and

\begin{matrix}k[α_{2}] & ↣ & H_{1} & ↠ & H_{1}/H_{1}X_{12} ≃ k[μ_{2}]. \\ X & ↦ & X_{11}X_{12} \end{matrix}

Thus G ▷ G_{1} ▷ μ_{2} ▷ {1} is an RLNC. Since

kX_{11} ⊕ kX_{12} ⊕ kX_{21} ⊕ kX_{22} ⊂ k[G]

is a 4-dimensional simple subcoalgebra, k[G] is not pointed and hence G is not triangulable.

It is known that G is unipotent iff G has an RLNC whose all factor group schemes are of G_{a}-type. We say that G is G_{m}-composite iff G has an RLNC whose all factor group schemes are of G_{m}-type. When k is algebraically closed and G comes from the affine algebraic group G(k) (in the sense of [29, (4.5)]), G is G_{m}-composite iff G(k) is solvable and “quasicompact” in Kolchin’s terminology, which implies that each element of G(k) is semisimple [11, §6, Theorem 2]. In general it is difficult to characterize the condition to be G_{m}-composite. As is seen above, not all group schemes of multiplicative type are G_{m}-composite. On the other hand, non-diagonalizable group schemes can be G_{m}-composite; see the following example.

Example 3.9.7. (1) Let k be the prime field with ch(k) = p > 0 and take the commutative Hopf algebra H = k[x, y]/⟨x^{p} − x, y^{p} − x − y⟩ with x, y primitive. One sees G = SpecH is abelian, finite etale, and unipotent. Hence the Cartier dual G^{∗} is of multiplicative type and connected. We have the RLNC of G:

k[x]/⟨x^{p} − x⟩ ↣ H ↠ k[y]/⟨y^{p} − y⟩.

By dualizing this we see that G^{∗} is G_{m}-composite:

k[μ_{p}] ↞ H^{∗} ↢ k[μ_{p}].

The grouplikes of H^{∗} is given by

Coalg_{k}(k, H^{∗}) ≃ Alg_{k}(H, k) = {(a, b) ∈ k^{2} | a^{p} − a = 0, b^{p} − a − b = 0}. Thus we have G^{∗} is not diagonalizable since p^{2} = dim_{k} H^{∗} ≠ p = #Alg_{k}(H, k).

(2) [12, §2, Remark 2]. Let k be an infinite field of arbitrary characteristic. Let G be the closed subgroup scheme of GL_{2} over k defined by the Hopf algebra k[G] = k[X_{11}, X_{12}, X_{21}, X_{22}, 1/(X_{11}X_{22} − X_{12}X_{21})]/⟨X_{11}X_{12}, X_{12}X_{22}, X_{11}X_{21}, X_{21}X_{22}⟩. Since G(k) can not be simultaneously diagonalized, G is not diagonalizable (see [29, (4.6)]). We see π_{0}G = ℤ/2ℤ. Indeed, π_{0}(k[G]) = ke_{0} ⊕ ke_{1} where

e_{0} = \frac{X_{11}X_{22}}{X_{11}X_{22} − X_{12}X_{21}} , e_{1} = −\frac{X_{12}X_{21}}{X_{11}X_{22} − X_{12}X_{21}} .

Let H_{1} = k[G]/k[G]X_{12}X_{21}, H_{2} = H_{1}/H_{1}(X_{11}X_{22} − 1), and G_{i} = SpecH_{i} (i = 1, 2). Then we have exact sequences

k(ℤ/2ℤ)^{∗} ↣ k[G] ↠ H_{1}

and

\begin{matrix}k[G_{m}] = k[X, X^{−1}] & ↣ & H_{1} & ↠ & H_{2} ≃ k[G_{m}]. \\ X & ↦ & X_{11}X_{22} \end{matrix}

Thus G ▷ G_{1} ▷ G_{2} ▷ {1} is an LNC. If ch(k) ≠ 2, then ℤ/2ℤ ≃ μ_{2} and hence G is G_{m}-composite.

3.9.2. Finite etale extensions. In what follows, we always assume that L/K is an extension of AS D-module algebras such that L^{D} = K^{D} =: k.

Definition 3.9.8. We say that L/K is a finite etale extension iff L is a separable Kalgebra in the sense of [7], i.e. L is a projective L ⊗_{K} L-module.

For a commutative K-algebra A, let π_{0}(A) denote the union of all separable Ksubalgebras. If we take a maximal ideal P of L and put L′ = L/P and K′ = K/P ∩ K, then the following are equivalent:

• L/K is a finite etale extension.

• L is a finitely generated K-algebra and π_{0}(L) = L.

• L′/K′ is a finite separable field extension. Take a maximal ideal 𝔭 of K. We say a finite etale extension L/K is copied (resp., anticopied) iff L′ = K′ (resp., 𝔭L is a maximal ideal of L); this condition is independent of the choice of P (resp., 𝔭).

Lemma 3.9.9. Let (L/K, A, H) be a finitely generated PV extension.

(1) (π_{0}(A)/K, π_{0}(A), π_{0}(H)) is also a PV extension and hence π_{0}(A) is the intermediate AS D-module algebra which corresponds to G(L/K)^{∘}.

(2) π_{0}(A) = π_{0}(L).

Especially L/K is a finite etale extension iff G(L/K) is finite etale.

Proof. (1) The μ-isomorphism A ⊗_{k} H ⟶^{∼} A ⊗_{K} A restricts to an algebra isomorphism π_{0}(A) ⊗_{k} π_{0}(H) ⟶^{∼} π_{0}(A) ⊗_{K} π_{0}(A). Hence θ^{−1}(A ⊗_{k} π_{0}(H)) = π_{0}(A); this implies that (π_{0}(A)/K, π_{0}(A), π_{0}(H)) is a PV extension by Proposition 3.5.8.

(2) By Proposition 3.7.4, we may assume that K is a field and A is an integral domain. Put M = π_{0}(A). It suffices to show that M is separably closed in L. Write H_{0} = H/Hπ_{0}(H)^{+}. Then we have an algebra isomorphism L ⊗_{k} H_{0} ⟶^{∼} L ⊗_{M} A since (L/M, A, H_{0}) is a PV extension by Proposition 3.5.7. By [29, (6.6)], (L ⊗_{k} H_{0})/nilradical is an integral domain. It follows that (L ⊗_{M} L)/nilradical is also an integral domain since it is a localization of (L ⊗_{M} A)/nilradical by a multiplicative subset not containing zero. Thus L ⊗_{M} L has no nontrivial idempotent. This implies that M is separably closed in L. Indeed, let S be an intermediate field of L/M such that S/M is finite separable. If e ∈ S ⊗_{M} S (⊂ L⊗_{M} L) is a separability idempotent for S (see [7, p. 40]), then necessarily e = 1 ⊗ 1. Let J := Ker(mult : S ⊗_{M} S → S). Then 0 = Je = J. Hence a ⊗ 1 − 1 ⊗ a = 0 in S ⊗_{M} S for all a ∈ S. This implies S = M. □

Corollary 3.9.10. Let L/K be a finitely generated PV extension. Then G(L/K) is connected iff π_{0}(L) = K.

Let 𝒢 be a finite group and put H = (k𝒢)^{∗}. Consider D_{k} = D ⊗_{R} k as a right Hcomodule algebra with the trivial structure map d ↦ d ⊗ 1. We say L/K is a 𝒢-extension iff

(i) L is an algebra in the symmetric tensor category (_{D_{k}}M^{H}, ⊗_{k}, k), and (ii) L/K is a right H-Galois extension.

Here _{D_{k}}M^{H} denotes the category of right (H, D_{k}^{op})-Hopf modules as in the proof of Theorem 3.8.13. We easily see that L/K is a 𝒢-extension iff (L/K, L, (k𝒢)^{∗}) is a PV extension. If L/K is an anticopied 𝒢-extension, then L′/K′ is a Galois extension of fields in ordinary sense such that Gal(L′/K′) = 𝒢. Conversely, when L/K is a finite Galois extension of fields, L/K is a Gal(L/K)-extension iff every element of Gal(L/K) is D-linear.

3.9.3. G_{a}-primitive extensions and G_{m}-primitive extensions.

Definition 3.9.11. (1) An x ∈ L is called G_{a}-primitive over K iff d(x) ∈ K for all d ∈ D^{+}. In this case, we say that K⟨x⟩/K is a G_{a}-primitive extension.

(2) An x ∈ L is called G_{m}-primitive over K iff x is a non-zero divisor of L and d(x)x^{−1} ∈ K for all d ∈ D. In this case, we say that K⟨x⟩/K is a G_{m}-primitive extension.

As in [27, (2.5a), (2.5b)], we have the following lemmas:

Lemma 3.9.12. (1) Let K⟨x⟩/K be a G_{a}-primitive extension. Put A = K[x] and l = 1 ⊗_{K} x − x ⊗_{K} 1 ∈ (A ⊗_{K} A)^{D}. Then (K⟨x⟩/K, A, k[l]) is a PV extension with l primitive and the PV group scheme G(K⟨x⟩/K) of G_{a}-type.

(2) Let K⟨x⟩/K be a G_{m}-primitive extension. Put A = K[x, x^{−1}] and g = x^{−1} ⊗_{K} x ∈ (A ⊗_{K} A)^{D}. Then (K⟨x⟩/K, A, k[g, g^{−1}]) is a PV extension with g grouplike and the PV group scheme G(K⟨x⟩/K) of G_{m}-type.

Proof. (1) x is G_{a}-primitive iff there exists a φ ∈ Hom_{R}(D, K) such that d(x) = ε(d)x + φ(d) for all d ∈ D. Then

X = ( \begin{matrix}1 & 1 \\ x & x + 1 \end{matrix} ) ∈ GL_{2}(K⟨x⟩)

is GL_{2}-primitive over K. In fact,

dX = ( \begin{matrix}ε(d) & 0 \\ φ(d) & ε(d) \end{matrix} ) X (d ∈ D).

Recalling the proof of Theorem 3.8.7 (d) ⇒ (a), we see

Z = (X^{−1} ⊗_{K} 1)(1 ⊗_{K} X) = ( \begin{matrix}1 − l & −l \\ l & 1 + l \end{matrix} ) ,

which concludes the proof.

(2) This is equivalent to saying that x is GL_{1}-primitive over K. □

Lemma 3.9.13. (1) If l ∈ (L ⊗_{K} L)^{D} and if l is primitive in the L-coring L ⊗_{K} L, then there exists an x ∈ L such that l = 1 ⊗_{K} x − x ⊗_{K} 1 and x is G_{a}-primitive over K.

(2) If g ∈ (L ⊗_{K} L)^{D} and if g is grouplike in L ⊗_{K} L, then there exists a non-zero divisor x ∈ L such that g = x^{−1} ⊗_{K} x and x is G_{m}-primitive over K.

Proof. (1) Primitive elements in the L-coring L ⊗_{K} L are precisely 1-cocycles in the Amitsur complex:

0 → L ⟶^{δ_{0}} L ⊗_{K} L ⟶^{δ_{1}} L ⊗_{K} L ⊗_{K} L ⟶^{δ_{2}} ⋯,

δ_{0}(x) = 1 ⊗_{K} x − x ⊗_{K} 1,

δ_{1}(∑x_{i} ⊗_{K} y_{i}) = ∑1 ⊗_{K} x_{i} ⊗_{K} y_{i} − ∑x_{i} ⊗_{K} 1 ⊗_{K} y_{i} + ∑x_{i} ⊗_{K} y_{i} ⊗_{K} 1, …,

whose n-th cohomology is H^{n}(L/K, G_{a}). But H^{1}(L/K, G_{a}) = 0 since L/K is a faithfully flat extension (see [29, Ch. 17, Ex. 10]). Then l ∈ Ker δ_{1} = Im δ_{0} and hence there exists some x ∈ L such that l = 1 ⊗_{K} x − x ⊗_{K} 1. Since dl = ε(d)l for all d ∈ D, we have (dx) ⊗_{K} 1 = 1 ⊗_{K} (dx) for all d ∈ D^{+}. This implies dx ∈ K for all d ∈ D^{+}.

(2) Grouplike elements in L ⊗_{K} L are precisely 1-cocycles in the complex:

{1} → G_{m}(L) ⟶^{δ_{0}} G_{m}(L ⊗_{K} L) ⟶^{δ_{1}} G_{m}(L ⊗_{K} L ⊗_{K} L) ⟶^{δ_{2}} ⋯,

δ_{0}(x) = (1 ⊗_{K} x)(x ⊗_{K} 1)^{−1} = x^{−1} ⊗_{K} x,

δ_{1}(∑x_{i} ⊗_{K} y_{i}) = (∑1 ⊗_{K} x_{i} ⊗_{K} y_{i})(∑x_{i} ⊗_{K} 1 ⊗_{K} y_{i})^{−1}(∑x_{i} ⊗_{K} y_{i} ⊗_{K} 1), …, whose n-th cohomology is H^{n}(L/K, G_{m}). But H^{1}(L/K, G_{m}) = Pic(L/K) ⊂ Pic(K) = {1} since K is a finite product of fields. Then g ∈ Ker δ_{1} = Im δ_{0} and hence there exists some x ∈ G_{m}(L) such that g = x^{−1} ⊗_{K} x. Since dg = ε(d)g, we have

1 ⊗_{K} dx = d(1 ⊗_{K} x) = d((x ⊗_{K} 1)g) = d(x)x^{−1} ⊗_{K} x

for all d ∈ D. By multiplying 1 ⊗_{K} x^{−1}, we have 1 ⊗_{K} d(x)x^{−1} = d(x)x^{−1} ⊗_{K} 1 for all d ∈ D, which implies d(x)x^{−1} ∈ K for all d ∈ D. □

Proposition 3.9.14. L/K is a G_{a}-primitive (resp., G_{m}-primitive) extension iff L/K is a PV extension and G(L/K) is of G_{a}-type (resp., G_{m}-type).

Proof. (“Only if” part.) This has been proved in Lemma 3.9.12.

(“If” part.) Let k[l] (resp., k[g, g^{−1}]) be the Hopf algebra for L/K. By Lemma 3.9.13, there exists the corresponding x ∈ L. Then K⟨x⟩ is an intermediate AS D-module algebra of L/K such that K⟨x⟩/K is a PV extension. Since the Hopf algebras of K⟨x⟩/K and L/K coincide, we have L = K⟨x⟩. □

3.9.4. The solvability theorem.

Definition 3.9.15. Let F/K be a finitely generated extension of AS D-module algebras. We call F/K a Liouville extension iff F^{D} = K^{D} = k and there exists a sequence of AS D-module algebras

(3.21)  K = F_{0} ⊂ F_{1} ⊂ ⋯ ⊂ F_{r} = F

such that each F_{i}/F_{i−1} (i = 1, …, r) is at least one of the following types: G_{a}-primitive extension, G_{m}-primitive extension, or finite etale extension. In this case, the sequence (3.21) is called a Liouville chain. Moreover, F/K is called a Liouville extension of type (j) (j = 1, …, 10) iff F/K has a Liouville chain (3.21) such that each extension F_{i}/F_{i−1} (i = 1, …, r) is

(1) G_{a}-primitive, G_{m}-primitive, or finite etale,

(2) G_{a}-primitive or G_{m}-primitive,

(3) G_{m}-primitive or finite etale, (4) G_{a}-primitive or finite etale,

(5) G_{a}-primitive or a 𝒢-extension for a finite solvable group 𝒢,

(6) G_{m}-primitive,

(7) G_{a}-primitive,

(8) finite etale,

(9) a 𝒢-extension for a finite solvable group 𝒢,

(10) trivial (i.e. F_{i} = F_{i−1}), respectively. Here we are taking priority of the compatibility with [11, §24]. We observe an anticopied 𝒢-extension for a finite solvable group 𝒢 is identified with a Galois extension by radicals and is also a Liouville extension of type (6).

To show the solvability theorem, we need the following lemma (cf. [11, §21]).

Lemma 3.9.16. Let L/K be a finitely generated PV extension and F an AS D-module algebra including L such that F^{D} = K^{D} = k. Take one t ∈ F. Then L⟨t⟩/K⟨t⟩ is a finitely generated PV extension and G(L⟨t⟩/K⟨t⟩) ≃ G(L/K⟨t⟩ ∩ L).

Proof. By Theorem 3.8.7, there exists a GL_{n}-primitive X = (x_{ij}) ∈ GL_{n}(L) over K such that L = K⟨x_{ij}⟩. Since L⟨t⟩ = K⟨t, x_{ij}⟩, we have that L⟨t⟩/K⟨t⟩ is a finitely generated PV extension. Write M = K⟨t⟩ ∩ L, Z = (X^{−1} ⊗_{M} 1)(1 ⊗_{M} X) = (z_{ij}), and Z^{−1} = (w_{ij}). Then H = k[z_{ij}, w_{ij}] becomes the Hopf algebra for L/M. Similarly by writing Z′ = (X^{−1} ⊗_{K⟨t⟩} 1)(1 ⊗_{K⟨t⟩} X) = (z′_{ij}), and (Z′)^{−1} = (w′_{ij}), we obtain the Hopf algebra H′ = k[z′_{ij}, w′_{ij}] for L⟨t⟩/K⟨t⟩. It follows that there exists a surjective Hopf algebra map φ : H ↠ H′, z_{ij} ↦ z′_{ij}. This implies that G(L⟨t⟩/K⟨t⟩) is a closed subgroup scheme of G(L/M). Let I = Ker φ be the corresponding Hopf ideal.

φ is the restriction (to H) of the natural map φ̃ : L ⊗_{M} L → L⟨t⟩ ⊗_{K⟨t⟩} L⟨t⟩. Since the coideal I · (L ⊗_{M} L) of L ⊗_{M} L, which corresponds to I, is included in Ker φ̃, we have {a ∈ L | a⊗_{M}1−1⊗_{M}a ∈ I·(L⊗_{M}L)} ⊂ {a ∈ L | a⊗_{M}1−1⊗_{M}a ∈ Ker φ̃} = L∩K⟨t⟩ = M. This implies that the intermediate AS D-module algebra of L/M which corresponds to I is M. Thus I = 0. □

Theorem 3.9.17. Let L/K be a finitely generated PV extension. Then the following are equivalent:

(a) L/K is a Liouville extension.

(b) There exists a Liouville extension F/K such that L ⊂ F.

(c) G(L/K) is Liouville. When k is algebraically closed, these are equivalent to:

(d) G(L/K)^{∘} is solvable.

Proof. ((a) ⇒ (b)) This is clear.

((b) ⇒ (c)) Take a Liouville chain of F/K:

K = F_{0} ⊂ F_{1} ⊂ ⋯ ⊂ F_{r} = F.

We use induction on r. The case r = 0 is obvious. Let r > 0. Since there are finite t_{1}, …, t_{s} ∈ F such that F_{1} = K⟨t_{1}, …, t_{s}⟩, we have that L⟨t_{1}, …, t_{s}⟩/F_{1} is a finitely generated PV extension and G(L⟨t_{1}, …, t_{s}⟩/F_{1}) ≃ G(L/F_{1} ∩ L) by Lemma 3.9.16. By the inductive assumption, G(L/F_{1} ∩ L) is Liouville.

If F_{1}/K is a finite etale extension, then F_{1} ∩ L ⊂ π_{0}(L). Hence we have G(L/F_{1} ∩ L) ⊃ G(L/π_{0}(L)) = G(L/K)^{∘} (Lemma 3.9.9). Thus, G(L/F_{1} ∩ L)^{∘} = G(L/K)^{∘} and hence (c) holds by Lemma 3.9.2 (1).

If F_{1}/K is a G_{a}-primitive extension, then there exists a G_{a}-primitive x ∈ F_{1} such that F_{1} = K⟨x⟩. Write L_{1} = F_{1} ∩ L. One sees that L_{1}/K is also a G_{a}-primitive extension (see [27, (2.9a)]). Hence G(L/K) ▷ G(L/L_{1}) and G(L/K)/G(L/L_{1}) = G(L_{1}/K) is of G_{a}-type. Therefore (c) holds.

If F_{1}/K is a G_{m}-primitive extension, then there exists a G_{m}-primitive x ∈ F_{1} such that F_{1} = K⟨x⟩. Write L_{1} = F_{1} ∩ L. One sees that L_{1}/K is also a G_{m}-primitive extension (see [27, (2.9b)]). Then we obtain (c) in the same way to the above.

((c) ⇒ (a)) Let G(L/K) = G_{0} ▷ G_{1} ▷ ⋯ ▷ G_{r} = {1} be an LNC and L_{i} (i = 0, …, r) the intermediate AS D-module algebra which corresponds to G_{i}. Then by Lemma 3.9.9 and by Proposition 3.9.14, K = L_{0} ⊂ L_{1} ⊂ ⋯ ⊂ L_{r} = L is a Liouville chain. □

By Proposition 3.9.4, we have the following.

Corollary 3.9.18. Let L/K be a finitely generated PV extension. If L/K is a Liouville extension, then there exists a Liouville chain

K = L_{0} ⊂ π_{0}(L) = L_{1} ⊂ L_{2} ⊂ … ⊂ L_{r} = L

such that each L_{i}/L_{i−1} (i = 2, …, r) is G_{m}-primitive or G_{a}-primitive extension.

Corollary 3.9.19. Let L/K be a finitely generated PV extension. Then L/K is (included in) a Liouville extension of type (j) (j = 1, …, 10) iff (1) G(L/K) is Liouville,

(2) G(L/K) has an RLNC,

(3) G(L/K)^{∘} is G_{m}-composite,

(4) G(L/K)^{∘} is unipotent,

(5) π_{0}G(L/K) is finite constant and solvable, and G(L/K)^{∘} is unipotent,

(6) G(L/K) is G_{m}-composite,

(7) G(L/K) is unipotent,

(8) G(L/K) is finite etale,

(9) G(L/K) is finite constant and solvable,

(10) G(L/K) is trivial, respectively.

This corollary can become more explicit when K is a perfect field and k is algebraically closed. In such a case, if (L/K, A, H) is a finitely generated PV extension, then A ⊗_{K} A is reduced (see [29, Ch. 6, Ex. 2]), and so H is reduced. Thus G(L/K) corresponds to the affine algebraic group G(L/K)(k) = Aut_{D, K-alg}(L) in the sense of [29, (4.5)]. There we can do the following replacement on condition about G(L/K):

(1) G(L/K)^{∘} is solvable (⇔ G(L/K)^{∘} is triangulable),

(2) G(L/K) is solvable,

(3) G(L/K)^{∘} is diagonalizable,

(4) G(L/K)^{∘} is unipotent,

(5) G(L/K) is solvable and G(L/K)^{∘} is unipotent,

(6) G(L/K)(k) is solvable and quasicompact (in Kolchin’s sense),

(7) G(L/K) is unipotent,

(8) G(L/K) is finite constant,

(9) G(L/K) is finite constant and solvable,

(10) G(L/K) is trivial.

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