ON ACTIONS OF INFINITESIMAL GROUP SCHEMES

GAETANA RESTUCCIA AND HANS-JÜRGEN SCHNEIDER

1. Introduction

Let k be a field of characteristic p > 0 and H a finite-dimensional commutative Hopf algebra over k with underlying algebra

H = k[X_{1}, …, X_{n}]/(X_{1}^{p^{s_{1}}}, …, X_{n}^{p^{sn}}), n ≥ 1, s_{1} ≥ ⋯ ≥ s_{n} ≥ 1. (1.1)

By the structure theorem of infinitesimal, that is finite and connected group schemes (see [W, 14.4]), any finite-dimensional, commutative and local Hopf algebra over a perfect field has this form.

Let A be a commutative algebra, and δ : A → A ⊗ H, a right Hcomodule algebra structure on A.

In other words, we are considering an action

X × G → X

of the infinitesimal group scheme represented by H on the affine scheme represented by A.

For any a ∈ A, we can write

δ(a) = a ⊗ 1 + ∑_{1≤i≤n} D_{i}(a) ⊗ x_{i} + terms of higher order in the x_{i},

where the x_{i} are the residue classes of the X_{i} in H, and where D_{i} : A → A, 1 ≤ i ≤ n, are k-linear derivations of A.

Let R = A^{coH} be the subalgebra of A of coinvariant elements under the coaction δ. The extension R ⊂ A is called an H-Galois extension (see [M]) (and the projection X → Y represented by the inclusion R ⊂ A is called a principal homogeneous space for G) if the canonical map

A ⊗_{R} A → A ⊗ H, a ⊗ b ↦ aδ(b),

is bijective.

In Theorem 4.1 we prove a Jacobi criterion for H-Galois extensions. For local algebras A this criterion says that the following statements are equivalent:

1991 Mathematics Subject Classification. Primary: 17B37; Secondary: 16W30.

Key words and phrases. Hopf Galois extensions, infinitesimal group schemes.

Version of May 10, 2001.

This work was partially supported by the Graduiertenkolleg of the Math. Institut (Universität München)

.

2 GAETANA RESTUCCIA AND HANS-JÜRGEN SCHNEIDER

• R ⊂ A is an H-Galois extension.

• There are elements y_{1}, …, y_{n} ∈ A such that the matrix (D_{i}(y_{j}))

is invertible over A. By renumbering the y_{i} we can assume that for all 1 ≤ m ≤ n the matrices (D_{i}(y_{j}))_{1≤i,j≤m} are invertible. Assuming this stronger condition we show as an application of our criterion for H-Galois extensions that the elements

y_{1}^{α_{1}} ⋯ y_{n}^{α_{n}}, 0 ≤ α_{i} < p^{s_{i}} − 1, 1 ≤ i ≤ n, (1.2)

form an R-basis of A (Theorem 4.3).

Under the additional assumption that all the powers y_{i}^{p^{s_{i}}} are Hcoinvariant we obtain rather precise results about the structure of the ring extension R ⊂ A for regular local rings A (Theorem 5.1). For example if A is local regular and complete, then the ring extension R ⊂ A is isomorphic to the extension

F [[Y_{1}^{p^{s_{1}}}, ….Y_{n}^{p^{sn}}, Z_{n+1}, …, Z_{d}]] ⊂ F [[Y_{1}, ….Y_{n}, Z_{n+1}, …, Z_{d}]]

of formal power series rings over a coefficient field k ⊂ F ⊂ R of R and A.

Theorem 5.1 was known before in some special cases. The Lie algebra case, that is when s_{1} = ⋯ = s_{n} = 1, was shown in [RM] in analogy to a theorem of Lipman [L, Theorem 2] on invariants of derivations in characteristic zero. The case n = 1 of (1.1) but for cocommutative Hopf algebras H over separably closed fields was treated in [RT]. In this paper H was cocommutative in order to apply the theory of formal groups in an essential way. Then the results in [RT] were partially generalized in [RU] to the case when n is arbitrary and H is cocommutative.

The condition about the invertibility of the matrix (D_{i}(y_{j}))_{1≤i,j≤n} appears in all these papers. We explain this condition as a criterion for H-Galois extensions, and then we can use the properties of H-Galois extensions to achieve inductive arguments for arbitrary Hopf algebras satisfying (1.1).

In Section 3 we consider coactions of the additive group where the x_{i} are primitive elements of the Hopf algebra H. In this Section the algebra A is an arbitrary not necessarily commutative algebra. For a general Hopf algebra H satisfying (1.1) we do not know whether our results can be generalized to coactions on non-commutative algebras.

2. Preliminaries on Hopf Galois extensions and free actions

Let H be a Hopf algebra over the field k with comultiplication Δ : H → H ⊗ H, counit ε : H → k and antipode S : H → H. We will use the Sweedler notation Δ(x) = x_{(1)} ⊗ x_{(2)} for x ∈ H. If V is a right H-comodule with structure map δ : V → V ⊗ H, we will use the notation δ(v) = v_{(0)} ⊗ v_{(1)} for all v ∈ V. The augmentation ideal of H will be denoted by H^{+} = ker(ε).

Let us first recall some notions and results from Hopf algebra theory [M].

Definition 2.1. Let A be an algebra, and

δ : A → A ⊗ H

an algebra map and a right H-comodule structure on A. Then (A, δ) is called a right H-comodule algebra, and

A^{coH} = {a ∈ A | δ(a) = a ⊗ 1}

is the subalgebra of H-coinvariant elements.

Let (A, δ) be a right H-comodule algebra, and R := A^{coH}. The extension R ⊂ A is called an H-Galois extension and δ is called H-Galois, if the canonical map

can : A ⊗_{R} A → A ⊗ H, x ⊗ y ↦ xδ(y) = xy_{(0)} ⊗ y_{(1)},

is bijective.

Let now H be a finite-dimensional Hopf algebra.

By a fundamental theorem of Larson and Sweedler [M, 2.1.3] the space of left integrals of H is one-dimensional. A left integral Λ of H is an element of H such that xΛ = ε(x)Λ for all x ∈ H.

Theorem 2.2. [Kreimer and Takeuchi] Let Λ be a non-zero integral of H, and δ : A → A ⊗ H a right H-comodule algebra with R = A^{coH}. Then the following are equivalent:

(1) R ⊂ A is an H-Galois extension.

(2) The canonical map can : A ⊗_{R} A → A ⊗ H is surjective.

(3) 1 ⊗ Λ ∈ A ⊗ H is in the image of the canonical map. In this case, A is a finitely generated and projective R-left and R-right module. In particular, A is faithfully flat over R if R is commutative.

Proof. The equivalence of (1) and (2), and the fact that A is finitely generated and projective over R on both sides is a result of Kreimer and Takeuchi (see [M, 8.3.1]). Since can : A ⊗_{R} A → A ⊗ H is left A-linear and right H-colinear with the obvious structure maps, and Λ generates H as a left H^{∗}-module or as a right H-comodule, (2) and (3) are equivalent. Finally, if R is commutative, then A is faithfully flat over R, since A_{𝔪} is non-zero and free over R_{𝔪} for any maximal ideal 𝔪 of R.

The next result is useful for induction arguments on the order of H. A Hopf subalgebra K ⊂ A is called normal if for all x ∈ H, and y ∈ K, x_{(1)}yS(x_{(2)}) ∈ K (or equivalently S(x_{(1)})yx_{(2)} ∈ K). Note that any

4 GAETANA RESTUCCIA AND HANS-JÜRGEN SCHNEIDER

Hopf subalgebra is trivially normal if H is commutative. If K ⊂ H is normal, then \overline{H} = H/HK^{+} is a quotient Hopf algebra of H.

We will say that an H-Galois extension R ⊂ A is faithfully flat if A is faithfully flat over R as a left (or equivalently right) module over R.

Theorem 2.3. [MS, 6.1, 6.2] Let K ⊂ H be a normal Hopf subalgebra, and R ⊂ A a faithfully flat H- Galois extension with structure map δ : A → A ⊗ H. Let π : H → \overline{H} = H/HK^{+} be the quotient map, and \overline{δ} = (id ⊗ π)δ : A → A ⊗ \overline{H} the induced \overline{H}-module algebra structure. Define B = A^{co\overline{H}}.

Then R ⊂ B is a faithfully flat K-Galois extension where the coaction isthe restriction of δ, and B ⊂ A is a faithfully flat \overline{H}-Galois extension with coaction \overline{δ}.

One ingredient of the proof of Theorem 4.2 is the basic Theorem of Nichols and Zoeller ([M, 3.1.5]) which says that H is free over any Hopf subalgebra.

We will apply Theorem 2.3 in the case when H is commutative, and k has characteristic p > 0. Then

H^{[1]} := k[H^{p}],

the subalgebra generated by all x^{p}, x ∈ H, is a Hopf subalgebra, and H/(x^{p} | x ∈ H^{+}) is the quotient Hopf algebra.

Remark 2.4. Let H be commutative, and A a commutative H-comodule algebra with structure map δ : A → A ⊗ H. In the language of affine schemes and affine group schemes (see [DG], [W]), H defines a functor G from commutative k-algebras to groups, A defines a functor X from commutative k-algebras to sets represented by H resp. A, and δ describes an action

μ : X × G → X,

that is for all commutative algebras T an action of the group G(T) on the set X(T). We will write μ(x, g) = xg for all x ∈ X(T), g ∈ G(T).

Recall that for all commutative algebras T, the group structure in G(T) = Alg(H, T) is defined by (f · g)(h) = f(h_{(1)})g(h_{(2)}), and the action is given by (xg)(a) = x(a_{(0)})g(a_{(1)}), for all f, g ∈ G(T), x ∈ X(T) = Alg(A, T), t ∈ T.

Then can : A ⊗_{R} A → A ⊗ H is surjective if and only if the action of G on X is (fixed point) free in the sense that for all commutative algebras T, and all g ∈ G(T), x ∈ X(T) the equality xg = x is only possible for g = 1.

The subgroup G_{(1)} of G represented by the quotient Hopf algebra H/(x^{p} | x ∈ H^{+}) is called the first Frobenius kernel of G.

Theorem 2.2 in the commutative case is the fundamental result on free actions of finite affine group schemes on affine schemes which goes back to Grothendieck.

Let us now fix some notations and assumptions.

In this paper we assume that k is a field of characteristic p > 0, and H will always be a commutative Hopf algebra with underlying algebra

H = k[X_{1}, …, X_{n}]/(X_{1}^{p^{s_{1}}}, …, X_{n}^{p^{sn}}), n ≥ 1, s_{1} ≥ ⋯ ≥ s_{n} ≥ 1. (2.1)

By the structure theorem of infinitesimal, that is finite and connected group schemes (see [W, 14.4]), any finite-dimensional, commutative and local Hopf algebra over a perfect field has this form.

Let 𝔸 be the set of all multi-indices α = (α_{1}, …, α_{n}) with 0 ≤ α_{i} < p^{s_{i}}, 1 ≤ i ≤ n.

For β = (β_{1}, …, β_{n}), γ = (γ_{1}, …, γ_{n}) ∈ ℕ^{n} we define

β + γ = (β_{1} + γ_{1}, …, β_{n} + γ_{n}), and |β| = β_{1} + ⋯ + β_{n}.

For all i, we denote the residue class of X_{i} in H by x_{i}. Then

x^{α} := x_{1}^{α_{1}} … x_{n}^{α_{n}}, α ∈ 𝔸,

is a k-basis of H. Note that ε(x_{i}) = 0 for all i, where ε : H → k is the counit of H, since H is local with maximal ideal (x_{1}, …, x_{n}). Let Λ := x_{1}^{p^{s_{1}}−1} … x_{n}^{p^{sn}−1}. Then Λ ≠ 0 and Λx_{i} = 0 for all 1 ≤ i ≤ n. Hence Λ is a non-zero integral in H.

Let A be an algebra, and δ : A → A⊗H, a right H-comodule algebra structure on A. We will always write

δ(a) = ∑_{α∈𝔸}D_{α}(a) ⊗ x^{α}, for all a ∈ A.

Thus for all α ∈ 𝔸 and a, b ∈ A,

D_{α}(ab) = ∑_{\begin{matrix}β+γ=α \\ β,γ∈𝔸 \end{matrix}}D_{β}(a)D_{γ}(b), and D_{(0,...,0)} = id.

For all i, let δ_{i} = (δ_{ij})_{1≤j≤n} ∈ 𝔸, where δ_{ij} = 1, if j = i, and δ_{ij} = 0, otherwise. We define D_{i} = D_{δ_{i}}, 1 ≤ i ≤ n. Thus the linear maps D_{i} : A → A are derivations of the algebra A, and for all a ∈ A we have

δ(a) = a ⊗ 1 + ∑_{1≤i≤n} D_{i}(a) ⊗ x_{i} + ∑_{\begin{matrix}α∈𝔸 \\ |α|≥2 \end{matrix}} D_{α}(a) ⊗ x_{α}. (2.2)

Let H^{∗} be the dual Hopf algebra of H and 𝔤 = P (H^{∗}) the p-Lie algebra of the primitive elements in H^{∗}. Thus 𝔤 is the Lie algebra of the group scheme represented by H. Recall that the primitive elements in H^{∗} are the ε-derivations d : H → k, that is the linear functions satisfying d(xy) = ε(x)d(y)+d(x)ε(y) for all x, y ∈ H. For all i, let d_{i} : H → k be the ε-derivation with d_{i}(x_{j}) = δ_{ij}, 1 ≤ i, j ≤ n (Kronecker δ). Then d_{1}, …, d_{n} is a basis of 𝔤. The coaction δ : A → A ⊗ H defines a left H^{∗}-module algebra structure on A, and the action of d_{i} is given by the derivation D_{i}. Thus

𝔤 → Der(A, A), d_{i} ↦ D_{i}, 1 ≤ i ≤ n,

is a map of p-Lie algebras. In particular,

[D_{i}, D_{j}], D_{i}^{p} ∈ ∑_{1≤l≤n} kD_{l}, 1 ≤ i, j ≤ n. (2.3)

In the case when s_{1} = ⋯ = s_{n} = 1, H^{∗} is the restricted enveloping algebra of 𝔤, and the action is determined by the action of 𝔤. Hence in this case A^{coH} = A^{{D_{1},...,D_{n}}}, where

A^{{D_{1},...,D_{n}}} := {a ∈ A | D_{i}(a) = 0 for all 1 ≤ i ≤ n}. (2.4)

3. The additive group

The Hopf algebra H_{a} with underlying algebra

H_{a} = k[X_{1}, …, X_{n}]/(X_{1}^{p^{s_{1}}}, …, X_{n}^{p^{sn}}), n ≥ 1, s_{1} ≥ ⋯ ≥ s_{n} ≥ 1, (3.1)

and comultiplication given by

Δ(x_{i}) = x_{i} ⊗ 1 + 1 ⊗ x_{i}, 1 ≤ i ≤ n, (3.2)

will be called the Hopf algebra of the additive group (where n and s_{1}, …, s_{n} are fixed).

Note that for any commutative algebra T, G_{a}(T) := Alg(H_{a}, T) ≃ {(t_{1}, …, t_{n}) ∈ k^{n} | t^{p^{s_{i}}} = 0, 1 ≤ i ≤ n}, where the group structure is componentwise addition.

We consider coactions of H_{a} on arbitrary, not necessarily commutative algebras A. However in all the later results about H-comodule algebras A for our general Hopf algebra H satisfying (2.1) we assume that A is commutative.

Theorem 3.1. Let H_{a} be the Hopf algebra defined by (3.1), (3.2), and A a right H-comodule algebra with structure map δ : A → A ⊗ H_{a}. Define the derivations D_{1}, …, D_{n} by (2.2) and let R = A^{coH_{a}}.

Then the following are equivalent:

(1) R ⊂ A is a faithfully flat H_{a}-Galois extension.

(2) There are y_{1}, …, y_{n} ∈ A with δ(y_{i}) = y_{i} ⊗ 1 + 1 ⊗ x_{i}, for all 1 ≤ i ≤ n.

Suppose (2) holds. Define y^{α} = y_{1}^{α_{1}} ⋯ y_{n}^{α_{n}}, α = (α_{1}, …, α_{n}) ∈ 𝔸. Then

R ⊗ H_{a} → A, r ⊗ x^{α} ↦ ry^{α}, r ∈ R, α ∈ 𝔸,

is a left R-linear and right H_{a}-colinear isomorphism. In particular, the elements

y^{α}, α ∈ 𝔸,

form an R-basis of A as a left R-module.

Proof. (1) ⇒ (2): By (1), A is an injective H_{a}-comodule (see for example [S, Theorem I]). Hence the right H_{a}-colinear map k → A, 1 ↦ 1, can be extended to an H_{a}-colinear map γ : H_{a} → A. Then for all i,

δ(γ(x_{i})) = (γ ⊗ id)Δ(x_{i}) = γ(x_{i}) ⊗ 1 + 1 ⊗ x_{i},

since γ is H_{a}-colinear and γ(1) = 1, and the claim follows with y_{i} := γ(x_{i}) for all 1 ≤ i ≤ n.

Suppose (2) holds. Then we define a k-linear map γ : H_{a} → A by γ(x^{α}) := y^{α} for all α ∈ 𝔸. Since Δ and δ are algebra maps, and for all i,

Δ(x_{i}) = x_{i} ⊗ 1 + 1 ⊗ x_{i}, δ(y_{i}) = y_{i} ⊗ 1 + 1 ⊗ x_{i},

γ is right H_{a}-colinear.

Since H_{a} as an algebra is generated by the group-like element 1 and the primitive elements x_{i}, k is the only simple subcoalgebra of H_{a} ([M, 5.5.1]). Therefore the map γ is convolution invertible (cf. [M, 5.2.10]). Thus the H_{a}-extension R ⊂ A is H_{a}-cleft, hence H_{a}-Galois, and

R ⊗ H_{a} → A, r ⊗ x^{α} ↦ ry^{α}, r ∈ R, α ∈ 𝔸,

is bijective (see [M, 8.2.4, 7.2.3]).

Example 3.2. We consider the Lie algebra case of the additive group

H_{a} = k[X_{1}, …, X_{n}]/(X_{1}^{p}, …, X_{n}^{p}).

In this case coactions of H_{a} are given by derivations D_{1}, …, D_{n} ∈ Der(A, A) with

D_{i}D_{j} = D_{j}D_{i}, and D_{i}^{p} = 0, 1 ≤ i, j ≤ n,

where

D_{α} = \frac{D_{1}^{α_{1}}}{α_{1}!} ⋯ \frac{D_{n}^{α_{n}}}{α_{n}!} , α = (α_{1}, …, α_{n}), 0 ≤ α_{i} < p, 1 ≤ i ≤ n.

Proof. This follows easily from the fact that e_{α} = \frac{x^{α_{1}}}{α_{1}!} ⋯ \frac{x^{αn}}{α_{n}!} , α ∈ 𝔸, is a k-basis of H_{a} with

Δ(e_{α}) = ∑_{\begin{matrix}β+γ=α \\ β,γ∈𝔸 \end{matrix}}e_{β} ⊗ e_{γ}.

Corollary 3.3. Assume s_{1} = ⋯ = s_{n} = 1. Let A be an algebra, and δ : A → A ⊗ H_{a} a coaction. Define the derivations D_{1}, …, D_{n} by (2.2) and let R = A^{coH}.

Then the following are equivalent:

(1) R ⊂ A is a faithfully flat H_{a}-Galois extension.

(2) There are y_{1}, …, y_{n} ∈ A with D_{i}(y_{j}) = δ_{ij} for all 1 ≤ i, j ≤ n. Suppose (2) holds. Then

R ⊗ H_{a} → A, r ⊗ x^{α} ↦ ry^{α}, r ∈ R, α ∈ 𝔸,

is a left R-linear and right H_{a}-colinear isomorphism.

Proof. This follows from Theorem 3.1 and Example 3.2

8 GAETANA RESTUCCIA AND HANS-JÜRGEN SCHNEIDER

The next Lemma shows that under certain conditions the coinvariant elements in the Lie algebra case are the coinvariant elements under some action of the additive group. The idea of this Lemma appears several times in the literature (see [L] resp. [RM] when the characteristic of the field is 0 resp. p > 0.) We include the proof for completeness.

Lemma 3.4. Let A be a commutative algebra, H a Hopf algebra satisfying (2.1) with s_{1} = ⋯ = s_{n} = 1, and δ a right H-comodule algebra structure on A with derivations D_{1}, …, D_{n} defined by (2.2).

Assume y_{1}, …, y_{n} ∈ A such that the matrix (D_{i}(y_{j}))_{1≤i,j≤n} is invertible.

Then there are derivations ∂_{1}, …, ∂_{n} ∈ Der(A, A) such that

[∂_{i}, ∂_{j}] = 0, ∂_{i}^{p} = 0, ∂_{i}(y_{j}) = δ_{ij}, 1 ≤ i, j ≤ n, and A^{coH} = A^{{∂_{1},...,∂_{n}}}.

In particular, the elements

y_{1}^{α_{1}} ⋯ y_{n}^{α_{n}}, 0 ≤ α_{i} < p, 1 ≤ i ≤ n,

form a basis of A over R.

Proof. Let (a_{ij}) be the inverse matrix of (D_{i}(y_{j})). Define

∂_{i} = ∑_{1≤j≤n} a_{ij}D_{j}, 1 ≤ i ≤ n.

Then ∂_{i}(y_{l}) = δ_{il} for all 1 ≤ i, l ≤ n, and ∑_{1≤i≤n} AD_{i} = ∑_{1≤i≤n} A∂_{i}. In particular, using (2.4),

A^{{∂_{1},...,∂_{n}}} = A^{{D_{1},...,D_{n}}} = A^{coH}.

Since A is commutative it is easy to see by (2.3) that

[∂_{i}, ∂_{j}] ∈ ∑_{1≤i≤n} AD_{i} = ∑_{1≤i≤n} A∂_{i}.

By Hochschild’s formula [Ma, 25.5], for any a ∈ A and any derivation D ∈ Der(A, A), (aD)^{p} is an A-linear combination of D^{p} and D. Hence for all i by (2.3),

∂_{i}^{p} ∈ ∑_{1≤i≤n} AD_{i} = ∑_{1≤i≤n} A∂_{i}.

Since an A-linear combination of ∂_{1}, …, ∂_{n} is zero if its evaluation on all y_{j}, 1 ≤ j ≤ n, vanishes, we see that [∂_{i}, ∂_{j}] = 0, ∂_{i}^{p} = 0, 1 ≤ i, j ≤ n.

Finally, the statement about the R-basis of A follows from Corollary 3.3 and Example 3.2.

4. A Jacobi-criterion for free actions

Theorem 4.1. Let H be a Hopf algebra satisfying (2.1), and A a commutative right H-comodule algebra with structure map δ : A → A ⊗ H. Define the derivations D_{1}, …, D_{n} by (2.2) and let R = A^{coH}.

Then the following are equivalent:

(1) R ⊂ A is an H-Galois extension.

(2) There are N ∈ ℕ and y_{1}, …, y_{N} ∈ A such that the matrix (D_{i}(y_{j})) is right invertible, that is there are a_{jl} ∈ A, 1 ≤ j ≤ N, 1 ≤ l ≤ n, with ∑_{1≤j≤N} D_{i}(y_{j})a_{jl} = δ_{il}, for all 1 ≤ i, l ≤ n.

(3) If A is local: There are y_{1}, …, y_{n} ∈ A such that for all 1 ≤ m ≤ n, the m × m matrix (D_{i}(y_{j}))_{1≤i,j≤m} over A is invertible.

Proof. (1) ⇒ (2): By (1), 1 ⊗ x_{l} is in the image of the canonical map A ⊗_{R} A → A ⊗ H for all 1 ≤ l ≤ n. Hence there are elements y_{j} ∈ A, 1 ≤ j ≤ N, for some N ∈ ℕ such that there are a_{jl} ∈ A, 1 ≤ j ≤ N, 1 ≤ l ≤ n, with

∑_{\begin{matrix}1≤j≤N \\ α∈𝔸 \end{matrix}} a_{jl}D_{α}(y_{j}) ⊗ x^{α} = 1 ⊗ x_{l}, 1 ≤ l ≤ n.

In particular, for all 1 ≤ i, l ≤ n, ∑_{1≤j≤N} a_{jl}D_{i}(y_{j}) = δ_{il}.

(2) ⇒ (1): By Theorem 2.2 it is enough to show that the integral Λ := x_{1}^{p^{s_{1}}−1} … x_{n}^{p^{sn}−1} is in the image of the canonical map can : A ⊗_{R} A → A ⊗ H.

For all 1 ≤ l ≤ n, using (2.2) and (2) we compute:

can(∑_{j} a_{jl} ⊗ y_{j}) = ∑_{j} a_{jl}y_{j} ⊗ 1 + ∑_{j,i} a_{jl}D_{i}(y_{j}) ⊗ x_{i}

+ ∑_{j,|α|≥2} a_{jl}D_{α}(y_{j}) ⊗ x^{α}

= ∑_{j} a_{jl}y_{j} ⊗ 1 + 1 ⊗ x_{l} + ∑_{j,|α|≥2} a_{jl}D_{α}(y_{j}) ⊗ x^{α}.

Defining z_{l} = ∑_{j} a_{jl} ⊗ y_{j} − ∑_{j} a_{jl}y_{j} ⊗ 1, and a_{l}^{α} = ∑_{j} a_{jl}D_{α}(y_{j}) for all 1 ≤ l ≤ n and α ∈ 𝔸, |α| ≥ 2, we obtain

can(z_{l}) = 1 ⊗ x_{l} + ∑_{|α|≥2} a_{l}^{α} ⊗ x^{α}, 1 ≤ l ≤ n.

We claim that

can(z_{1}^{p^{s_{1}}−1} ⋯ z_{n}^{p^{sn}−1}) = 1 ⊗ Λ.

Since the canonical map is an algebra map, we have

can(z_{1}^{p^{s_{1}}−1} ⋯ z_{n}^{p^{sn}−1}) = can(z_{1})^{p^{s_{1}}−1} ⋯ can(z_{n})^{p^{sn}−1}

= ∏_{1≤l≤n} (1 ⊗ x_{l} + ∑_{|α|≥2} a_{l}^{α} ⊗ x^{α})^{p^{s_{l}}−1}.

We see that this product is a sum of terms of the form

∏_{1≤l≤n}(1 ⊗ x_{l})^{c_{l}}( ∑_{|α|≥2} a_{l}^{α} ⊗ x^{α})^{d_{l}}, c_{l} + d_{l} = p^{s_{l}} − 1, for all 1 ≤ l ≤ n.

The general term of this sum can be written as

a ⊗ x^{γ}, a ∈ A, γ ∈ ℕ^{n}, |γ| ≥ ∑_{1≤l≤n}(c_{l} + 2d_{l}) = ∑_{1≤l≤n}(p^{s_{l}} − 1) + ∑_{1≤l≤n} d_{l}.

10 GAETANA RESTUCCIA AND HANS-JÜRGEN SCHNEIDER

Only the terms a⊗x^{γ} with γ = (γ_{1}, …, γ_{n}) and γ_{l} ≤ p^{s_{l}} −1, 1 ≤ l ≤ n, can be non-zero. Therefore we only have to consider a ⊗ x^{γ} with |γ| ≤ ∑_{1≤l≤n}(p^{s_{l}} − 1), that is with d_{l} = 0 for all l. This proves our claim.

(2) ⇒ (3) follows from the next Lemma.

Lemma 4.2. Let A be a commutative local ring, 1 ≤ n, N ∈ ℕ and d_{ij} ∈ A, 1 ≤ i ≤ n, 1 ≤ j ≤ N. Assume that there are a_{jl} ∈ A, 1 ≤ l ≤ n, 1 ≤ j ≤ N with ∑_{1≤j≤N} d_{ij}a_{jl} = δ_{il}, for all 1 ≤ i, l ≤ n.

Then there are 1 ≤ j_{1} < ⋯ < j_{n} ≤ N and a permutation σ ∈ S_{n} such that for all 1 ≤ m ≤ n, the m × m matrix (d_{ij_{σ(l)}})_{1≤i,l≤m} is invertible.

Proof. Since a square matrix over A is invertible if and only if its determinant is ≢ 0 mod 𝔪, 𝔪 the maximal ideal of A, we may assume that A is a field.

Since the rows of (d_{ij}) are linearly independent, there is a sequence 1 ≤ j_{1} < ⋯ < j_{n} ≤ N such that (d_{ij_{l}})_{1≤i,l≤n} is invertible.

Hence it suffices to show that for any invertible square matrix B = (b_{ij})_{1≤i,j≤n} over a field there is a permutation σ ∈ S_{n} such that each submatrix (b_{ij})_{1≤i,j≤m}, 1 ≤ m ≤ n, is invertible. We prove this result by induction on n. Since the first n − 1 rows of B are linearly independent, there are column indices 1 ≤ j_{1} < ⋯ < j_{n−1} ≤ n such that (b_{ij_{l}})_{1≤i,l≤n−1} is invertible. By induction there is a permutation τ ∈ S_{n−1} such that all submatrices (b_{ij_{τ(l)}})_{1≤i,l≤m}, 1 ≤ m ≤ n−1, are invertible. Let j_{n} be the column index not contained in {j_{1}, …, j_{n−1}}. Then (j_{τ(1)}, …, j_{τ(n−1)}, j_{n}) is the permutation of the columns we want.

Theorem 4.3. Let H, δ : A → A ⊗ H and R be as in Theorem 4.1 and assume y_{1}, …, y_{n} ∈ A such that for all 1 ≤ m ≤ n, (D_{i}(y_{j}))_{1≤i,j≤m} is invertible.

Then the elements

y^{α} := y_{1}^{α_{1}} ⋯ y_{n}^{α_{n}}, α = (α_{1} … α_{n}) ∈ 𝔸,

form an R-basis of A.

Proof. We proceed by induction on the biggest exponent s_{1}.

If s_{1} = 1, we are in the Lie algebra case, and the claim follows by Lemma 3.4. For the induction step suppose

s_{1} ≥ s_{2} ⋯ ≥ s_{m} > 1, s_{m+1} = ⋯ = s_{1}, for some 1 ≤ m ≤ n.

Recall from Section 2 that H^{[1]} = k[H^{p}] is a Hopf subalgebra of H with quotient Hopf algebra \overline{H}_{(1)} = H/(x^{p} | x ∈ H^{+}). Then

k[X_{1}, …, X_{m}]/(X_{1}^{p^{s_{1}−1}}, …, X_{m}^{p^{sm−1}}) → H^{[1]}, \overline{X_{i}} ↦ x_{i}^{p}, 1 ≤ i ≤ m,

and

k[X_{1}, …, X_{n}]/(X_{1}^{p}, …, X_{n}^{p}) → \overline{H}_{(1)}, \overline{X_{i}} ↦ \overline{x_{i}}, 1 ≤ i ≤ n,

are isomorphisms.

Let B = A^{co\overline{H}_{(1)}}. By Theorem 4.1, R ⊂ A is an H-Galois extension. Hence by Theorem 2.3, R ⊂ B is an H^{[1]}-Galois and B ⊂ A an \overline{H}_{(1)}- Galois extension. The coaction of B is the restriction of δ. Therefore we obtain from the formula (2.2) for the coaction for all b ∈ B,

δ(b) = b ⊗ 1 + ∑_{pα∈𝔸}D_{pα}(b) ⊗ x^{pα}.

Here we use the notation pα = (pα_{1}, …, pα_{n}) for all α = (α_{1}, …, α_{n}).

On the other hand, by taking p-th powers it follows from (2.2) that for all a ∈ A,

δ(a^{p}) = a^{p} ⊗ 1 + ∑_{1≤i≤n} D_{i}(a)^{p} ⊗ x_{i}^{p} + ∑_{\begin{matrix}α∈𝔸 \\ |α|≥2 \end{matrix}} D_{α}(a)^{p} ⊗ x_{pα}.

In particular, D_{i}(a)^{p} = D_{pδ_{i}}(a^{p}).

By assumption, the matrix (D_{i}(y_{j}))_{1≤i,j≤t} is invertible for all 1 ≤ t ≤ m. Hence also (D_{i}(y_{j})^{p})_{1≤i,j≤t} = (D_{pδ_{i}}(y_{j}^{p}))_{1≤i,j≤t} is invertible for all 1 ≤ t ≤ m. Therefore it follows by induction that

y_{1}^{pα_{1}} ⋯ y_{m}^{pα_{m}}, 0 ≤ α_{i} < p^{s_{i}−1}, 1 ≤ i ≤ m,

is an R-basis of B.

Moreover we know from the Lie algebra case that

y_{1}^{β_{1}} ⋯ y_{n}^{β_{n}}, 0 ≤ β_{l} < p, 1 ≤ l ≤ n,

is a B-basis of A.

Hence

y_{1}^{pα_{1}} ⋯ y_{m}^{pα_{m}}y_{1}^{β_{1}} ⋯ y_{n}^{β_{n}},
0 ≤ α_{i} < p^{s_{i}−1}, 1 ≤ i ≤ m, 0 ≤ β_{l} < p, 1 ≤ l ≤ n,

is an R-basis of A, and we are done.

Note that any H-Galois extension of commutative algebras R ⊂ A, H a finite-dimensional Hopf algebra, is finite by Theorem 2.2. In particular, if A is local, then also R is local.

Corollary 4.4. Let H, δ : A → A ⊗ H and R be as in Theorem 4.1, and assume that A is local with maximal ideal 𝔪_{A}. Then the following are equivalent:

(1) R ⊂ A is an H-Galois extension, and R + 𝔪_{A} = A, that is A and R have the same residue field.

(2) There are y_{1}, …, y_{n} ∈ 𝔪_{A} such that the matrix (D_{i}(y_{j}))_{1≤i,j≤n} is invertible.

Proof. (1) ⇒ (2): By Theorem 4.1 there are y_{1}, …, y_{n} ∈ A such that (D_{i}(y_{j}))_{1≤i,j≤n} is invertible. By assumption we can write y_{i} = r_{i} + ỹ_{i}, where r_{i} ∈ R and ỹ_{i} ∈ 𝔪_{A} for all 1 ≤ i ≤ n. Then D_{i}(r_{j}) = 0 for all 1 ≤ i, j ≤ n since R = A^{coH}. Thus (D_{i}(ỹ_{j}))_{1≤i,j≤n} = (D_{i}(y_{j}))_{1≤i,j≤n} is invertible.

12 GAETANA RESTUCCIA AND HANS-JÜRGEN SCHNEIDER

(2) ⇒ (1): By Theorem 4.1, R ⊂ A is H-Galois, and by Lemma 4.2 we can choose the y_{i} in 𝔪_{A} such that condition (3) in Theorem 4.1 is satisfied. Then the equality R+𝔪_{A} = A follows from Theorem 4.3.

Corollary 4.5. Let H, δ : A → A ⊗ H and R be as in Theorem 4.1 and assume y_{1}, …, y_{n} ∈ A such that for all 1 ≤ m ≤ n, (D_{i}(y_{j}))_{1≤i,j≤m} is invertible.

(1) If s_{1} = ⋯ = s_{n} =: s, then y_{i}^{p^{s}} ∈ R for all 1 ≤ i ≤ n.

(2) If y_{i}^{p^{s_{i}}} ∈ R for all 1 ≤ i ≤ n, then the algebra map

R[T_{1}, …, T_{n}]/(T_{1}^{p^{s_{1}}} − y_{1}^{p^{s_{1}}}, …, T_{n}^{p^{sn}} − y_{n}^{p^{sn}}) → A, \overline{T_{i}} ↦ y_{i}, 1 ≤ i ≤ n,

where R[T_{1}, …, T_{n}] is the polynomial ring in n variables, and the canonical map

R/(y_{1}^{p^{s_{1}}}, …, y_{n}^{p^{sn}}) → A/(y_{1}, …, y_{n})

are bijective.

Proof. (1) follows by raising (2.2) to the p^{s}-th power, and (2) follows from 4.3.

The examples below show that Theorem 4.3 does not hold if we just assume that the matrix (D_{i}(y_{j}))_{1≤i,j≤n} is invertible; also in the situation of Corollary 4.5 it is not always true that y_{i}^{p^{s_{i}}} ∈ R for all 1 ≤ i ≤ n.

Example 4.6. We consider the Hopf algebra

H = H_{a} = k[X_{1}, X_{2}]/(X_{1}^{p^{2}}, X_{2}^{p}) = k[x_{1}, x_{2}]

of the additive group with n = 2 and s_{1} = 2, s_{2} = 1, and the polynomial algebra in two variables A = k[T_{1}, T_{2}].

1) The algebra map δ : A → A ⊗ H, defined by

δ(T_{1}) = T_{1} ⊗ 1 + 1 ⊗ x_{2}, δ(T_{2}) = T_{2} ⊗ 1 + 1 ⊗ x_{1} + 1 ⊗ x_{2},

is an H-comodule algebra structure on A. Let y_{i} := T_{i}, 1 ≤ i ≤ 2. Then the matrix (D_{i}(y_{j}))_{1≤i,j≤2} is invertible, but the elements

y_{1}^{i_{1}}y_{2}^{i_{2}}, 0 ≤ i_{1} < p^{2}, 0 ≤ i_{2} < p,

do not form a basis of A over R = A^{coH}.

2) The algebra map δ : A → A ⊗ H, defined by

δ(T_{1}) = T_{1} ⊗ 1 + 1 ⊗ x_{1}, δ(T_{2}) = T_{2} ⊗ 1 + 1 ⊗ x_{1} + 1 ⊗ x_{2},

is an H-comodule algebra structure on A. Let y_{i} := T_{i}, 1 ≤ i ≤ 2. Then the matrices (D_{i}(y_{j}))_{1≤i,j≤m} are invertible for 1 ≤ m ≤ 2, and y_{i}^{p} ∉ R = A^{coH}, 1 ≤ i ≤ 2.

3) In both examples 1) and 2), δ induces an H-comodule algebra structure on the localization A_{𝔪}, 𝔪 := (T_{1}, T_{2}).

Proof. 1) It is easy to check that δ is coassociativ. By definition, D_{1}(T_{1}) = 0, D_{1}(T_{2}) = 1, D_{2}(T_{1}) = 1, D_{2}(T_{2}) = 1. Hence the elements

T_{2}^{i_{1}}T_{1}^{i_{2}}, 0 ≤ i_{1} < p^{2}, 0 ≤ i_{2} < p,

are an R-basis of A by Theorem 4.3. In particular, A is free over R of rank p^{3}. Therefore the elements

T_{1}^{i_{1}}T_{2}^{i_{2}}, 0 ≤ i_{1} < p^{2}, 0 ≤ i_{2} < p,

do not generate the R-module A, since T_{1}^{p} ∈ R, although the 2 × 2 matrix (D_{i}(T_{j}))_{1≤i,j≤2} is invertible.

2) is easy to check. To prove 3) note that for any commutative Hcomodule algebra structure δ : A → A ⊗ H, H commutative and local with (𝔪_{H})^{p^{s}} = 0, and for all a ∈ A, δ(a) ∈ a ⊗ 1 + A ⊗ 𝔪_{H}, hence (δ(a))^{p^{s}} = a^{p^{s}} ⊗ 1. Thus δ induces an H-comodule algebra structure on each localization of A.

5. An application

We finally apply the previous results to the case when A is regular local or regular local and complete.

Theorem 5.1. Let H be a Hopf algebra satisfying (2.1), and A a commutative right H-comodule algebra with structure map δ : A → A ⊗ H. Define the derivations D_{1}, …, D_{n} by (2.2) and let R = A^{coH}.

Assume that A is a regular local ring of dimension d with maximal ideal 𝔪_{A} and there are y_{1}, …, y_{n} ∈ 𝔪_{A} such that for all 1 ≤ m ≤ n, (D_{i}(y_{j}))_{1≤i,j≤n} is invertible.

Then R is regular local.

(1) Assume moreover that s_{1} = ⋯ = s_{n}, or more generally that for all 1 ≤ i ≤ n, y^{p^{s_{i}}} ∈ R.

Then there are z_{n+1}, …, z_{d} ∈ R such that

y_{1}, …, y_{n}, z_{n+1}, …, z_{d} is a regular system of parameters of A, and

y_{1}^{p^{s_{1}}}, …, y_{n}^{p^{sn}}, z_{n+1}, …, z_{d} is a regular system of parameters of R.

(2) Assume in addition to (1) that the regular local ring (A, 𝔪_{A}) is complete, and k ⊂ A/𝔪_{A} is a separable field extension. Then there is a subfield k ⊂ F ⊂ R of R such that

F ⟶^{≃} R/𝔪_{R} ⟶^{≃} A/𝔪_{A},

and an algebra isomorphism

F [[Y_{1}, ….Y_{n}, Z_{n+1}, …, Z_{d}]] ⟶^{≃} A,

inducing an algebra isomorphism

F [[Y_{1}^{p^{s_{1}}}, ….Y_{n}^{p^{sn}}, Z_{n+1}, …, Z_{d}]] ⟶^{≃} R.

Proof. By Theorem 4.1, R ⊂ A is H-Galois, in particular A is a finitely generated free R-module. Hence R is local, noetherian and regular (see [Ma, 23.7]).

Assume (1). Following the argument in [RM, Theorem 5], we first note that the residue classes of y_{1}, …, y_{n} in 𝔪_{A}/𝔪_{A}^{2} are linearly independent over A/𝔪_{A}. Suppose ∑_{1≤j≤n} a_{j}y_{j} ∈ 𝔪_{A}^{2} for some a_{j} ∈ A. By applying the derivations ∂_{i} of the proof of Lemma 3.4 we see that a_{i} ∈ 𝔪_{A} for all i.

Thus by 4.5 R/(y_{1}^{p^{s_{1}}}, …, y_{n}^{p^{sn}}) ⟶^{≃} A/(y_{1}, …, y_{n}) is regular, and we

can extend the y_{i}^{p^{s_{i}}} by z_{n+1}, …, z_{d} to a regular system of parameters of R. Then y_{1}, …, y_{n}, z_{n+1}, …, z_{d} is a regular system of parameters of A since A and R have the same residue field by 4.4.

To prove (2), we note that R is also complete local since A is finitely generated and free over A (The 𝔪_{R}-adic topology of A is the 𝔪_{A}-adic topology, since A/𝔪_{R}A is artinian. Hence R ⊂ R̂ ⊂ A, and A is free over R and R̂ of the same rank. Thus R = R̂ by Nakayama).

Since R/𝔪_{R} ⟶^{≃} A/𝔪_{A} is separable over k there is a coefficient field k ⊂ F ⊂ R by [Ma, 28.3]. The claim follows from (1) and [Ma, 29.7].

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Università di Messina, Dipartimento di Matematica,, Contrada Papardo, salita Sperone 31,, 98166 Sant’Agata (ME), Italy

  E-mail address: grest@dipmat.unime.it

Mathematisches Institut, Universität München, Theresienstraße 39, D-80333 München, Germany

E-mail address: hanssch@rz.mathematik.uni-muenchen.de