Invariants of Hopf Algebras

Gregor Kemper

Department of Mathematics and Statistics, Queen’s University,

Kingston, Ontario K7L 3N6, Canada

                 E-mail Gregor.Kemper@iwr.uni-heidelberg.de

July 10, 1999

Introduction

In the winter of 1999 I gave a series of lectures at Queen’s university about some recent results concerning the Cohen-Macaulay property of invariants of Hopf algebras. Tony Geramita asked me to write up my notes for the Queen’s Papers, and I happily took up his suggestion. Although this article focuses on the proof of one main theorem (Theorem 2.11 on page 12), it has some of the character of a survey article, since the results already appeared in my (German) habilitation thesis [11], and I am trying to explain concepts in more detail and give more background than in an original article.

This work originated in the study of the Cohen-Macaulay property of invariant rings of finite groups [12], which led to results about invariants of algebraic groups as well [10]. Actions of Hopf algebras are a natural generalization of group actions. Apart from group rings, Lie algebras (or, more precisely, their universal enveloping algebras) are an interesting incarnation of Hopf algebras, so everything that we say in this article will apply to invariants if Lie algebras as well. By a celebrated theorem of Hochster and Roberts [8], invariant rings of linearly reductive groups are always Cohen-Macaulay, i.e., they are free modules of finite rank over a subalgebra which is isomorphic to a polynomial ring. With the proper definition, this carries over to invariants of linearly reductive Hopf algebras (see Theorem 3.4 in this paper). For finite groups, Hochster and Roberts’s result means that the invariant ring is Cohen-Macaulay if the characteristic of the ground field does not divide the group order (see Hochster and Eagon [7]). On the other hand, it has been generally observed that invariant rings of finite groups tend not to be Cohen-Macaulay if the characteristic divides the group order. However, it is still an open question to characterize exactly those groups and representations for which the invariant ring is Cohen-Macaulay.

In this article we prove that if a Hopf algebra is geometrically reductive but not linearly reductive, then there exists a representation whose invariant ring is not Cohen-Macaulay. This constitutes a partial converse to the (generalized) theorem of Hochster and Roberts. This means, for example, that for each of the classical algebraic groups in positive characteristic there exists a representation such that the invariant ring is not Cohen-Macaulay. Moreover, every finite group of order divisible by the characteristic of the ground field has a representation with non-Cohen-Macaulay invariant ring. This gives a partial answer to the question raised above. For (universal enveloping algebras of) Lie algebras, the main theorem says the following: Any non-zero, finite dimensional Lie algebra 𝔤 in positive characteristic has a finite dimensional representation V such that the invariant ring (i.e., the set of polynomials in the symmetric algebra S(V) on which all elements of 𝔤 act as zero) is not Cohen-Macaulay.

In the first section of the paper we recall the concept of Hopf algebras and their invariants, and introduce cohomology. This is used as the basic tool in Section 2 to develop a criterion for regular sequences in the invariant ring. It turns out that in order to prove that the invariant ring is not Cohen-Macaulay, it suffices to find a non-zero element α ∈ H^{1}(Λ, S(V)) such that the annihilator Ann_{S(V)^{Λ}}(α) contains a sequence of length three which generates an ideal of height three in S(V) as well as in S(V^{Λ}). In Section 3 such a situation is manufactured under the assumption that we have a geometrically reductive but not linearly reductive Hopf algebra. This proves the main theorem. In order to apply the theorem, one must interpret the concept of linear and geometric reductivity for the various classes of objects which occur as Hopf algebras. This is done in the fourth section.

The work for this paper benefited greatly from conversations (or correspondences) with Hanspeter Kraft, Kay Magaard, Vladimir Popov, Alexander Premet, Jean-Pierre Serre, Moss Sweedler, and David Wehlau. I would like to express my thanks to all of them, and to those who participated in the winter lectures at Queen’s.

1 Hopf Algebras

Suppose that a group G acts by automorphisms on an algebra R over a field K. Then R is a module over the group ring KG, and for a, b ∈ R and σ ∈ G we have σ(ab) = σ(a)σ(b). Thus R is not an algebra over KG in the usual sense. Moreover, if U and V are KGmodules, then U ⊗_{K} V is also a KG-module, where

aσ(u ⊗ v) = a(σ(u) ⊗ σ(v))

for a ∈ K, σ ∈ G, u ∈ U, and v ∈ V. This is quite unusual, since in general the tensor product of two modules over an algebra A does not have the structure of an A-module. We make the convention here that throughout this article all tensor products are over K. Also note that Hom_{K}(U, V) (and in particular the dual V^{∗} = Hom_{K}(V, K)) is again a KG-module. These “peculiarities” of the algebra KG are captured by the fact that KG is a Hopf algebra with antipode. We recall the definition of a Hopf algebra and then present a few examples, including group rings.

Definition 1.1. (a) A Hopf algebra over a field K is an associative K-algebra Λ with unit and homomorphisms of algebras Δ: Λ → Λ ⊗ Λ (“comultiplication”) and ϵ: Λ → K (“counit”) such that the diagrams

[figure: Λ; Δ; Λ ⊗ Λ; Δ; id ⊗Λ; Λ ⊗ Λ; Δ⊗id; Λ ⊗ Λ ⊗ Λ]

and

[figure: Λ; ≅; Δ; ≅; Λ ⊗ K; id ⊗ϵ; Λ ⊗ Λ; ϵ⊗id; K ⊗ Λ]

commute.

(b) A Hopf algebra Λ is said to be commutative if it is commutative as a K-algebra, and cocommutative if the diagram

[figure: Λ; Δ; Δ; Λ ⊗ Λ; swap; Λ ⊗ Λ]

commutes, where the lower map is a ⊗ b ↦ b ⊗ a.

(c) An antipode of a Hopf algebra Λ is a K-linear map η: Λ → Λ such that the diagram

[figure: Λ ⊗ Λ; Δ; Λ; Δ; Λ ⊗ Λ; ϵ; η⊗id; K; id ⊗η; Λ ⊗ Λ; mult; Λ; mult; Λ ⊗ Λ]

commutes.

(d) A module over a Hopf algebra Λ is a (left-)module over Λ as a ring. K becomes a Λ-module via the counit ϵ. If U and V are Λ-modules, then U ⊗ V has a natural structure as a (Λ ⊗ Λ)-module (this is true for any K-algebra Λ), and becomes a Λ-module via the comultiplication Δ. More explicitly, if Δ(λ) = ∑_{i=1}^{r} μ_{i} ⊗ ν_{i} for a λ ∈ Λ, then for u ∈ U and v ∈ V we have

λ(u ⊗ v) = ∑_{i=1}^{r} μ_{i}u ⊗ ν_{i}v.

If V is a Λ-module, then the dual V^{∗} has a natural structure as a module over the opposite algebra Λ^{op}. It can be shown that an antipode η always is an anti-homomorphism of algebras (see Sweedler [22, Proposition 4.0.1 (1)]). Therefore if Λ has an antipode η, V^{∗} becomes a Λ-module via η. Now the natural isomorphism Hom_{K}(U, V) ≅ U^{∗}⊗V makes Hom_{K}(U, V) into a Λ-module. More explicitly, if Δ(λ) = ∑_{i=1}^{r} μ_{i} ⊗ ν_{i} for a λ ∈ Λ, then for f ∈ Hom_{K}(U, V) we have

λf: U → V, u ↦ ∑_{i=1}^{r} μ_{i}f(η(ν_{i})u).

(e) If V is a module over a Hopf algebra Λ, we write

V^{Λ} := {v ∈ V | λv = ϵ(λ)v ∀λ ∈ Λ}

and call this the space of invariants of Λ (see Montgomery [13, p. 13]).

(f) A Λ-algebra is an associative, commutative K-algebra R with unit which has a structure of a Λ-module such that 1_{R} ∈ R^{Λ} (the unit is an invariant), and the multiplication mult : R ⊗ R → R is a homomorphism of Λ-modules. More explicitly, we demand that λ(ab) = ∑_{i=1}^{r}(μ_{i}a)(ν_{i}b) for a, b ∈ R.

We look at a few examples of Hopf algebras, which show that Hopf algebras provide an umbrella for various structures that are quite familiar.

Example 1.2. (a) Let G be a group, K a field, and Λ := KG the group ring. Λ becomes a cocommutative Hopf algebra with Δ(σ) = σ ⊗ σ and ϵ(σ) = 1 for σ ∈ G. An antipode is given by η(σ) = σ^{−1}. We see that with these definitions tensor products and homomorphism spaces of Λ-modules are given structures as Λ-modules in the usual way. If V is a Λ-module, we sometimes write V^{G} for V^{Λ}, which is the fixed space under the G-action. Also notice that a Λ-algebra is nothing else than a Kalgebra with an action of G by automorphisms fixing K.

(b) Let 𝔤 be a Lie algebra over a field K and Λ := U(𝔤) the universal enveloping algebra. In order to make Λ into a Hopf algebra, it is enough to define Δ and ϵ on 𝔤, since Λ is generated by 𝔤 as a K-algebra. We set Δ(x) = x ⊗ 1 + 1 ⊗ x and ϵ(x) = 0 for x ∈ 𝔤. An antipode is given by η(x) = −x.

A module over Λ is also a module over the Lie algebra 𝔤 by restricting the action, and conversely a 𝔤-module is endowed with a Λ-action by the universal property of the universal enveloping algebra. In more sophisticated language, the module categories of 𝔤 and Λ are equivalent. A vector v ∈ V is invariant if and only if xv = 0 for all x ∈ 𝔤. We sometimes write V^{𝔤} for V^{Λ}. By the special form of the comultiplication, a Λ-algebra is a K-algebra with an action of 𝔤 by derivations.

(c) A commutative Hopf algebra Λ with antipode is nothing else than the coordinate ring of an affine group scheme G = Spec(Λ) (see, for example, Waterhouse [23, Theorem 1.4]). Λ is cocommutative if and only if G is abelian.

(d) The following example is closely related to the above: Let G be a finite group, K a field, and let Λ := Fun(G, K) be the algebra of functions from G into K with pointwise addition and multiplication. Define ϵ: Λ → K by ϵ(f) = f(1), and Δ: Λ → Λ ⊗ Λ ≅ Fun(G × G, K) by

Δ(f): G × G → K, (σ, τ) ↦ f(στ).

This makes Λ a commutative Hopf algebra with an antipode given by η(f): G → K, σ ↦ f(σ^{−1}). For σ ∈ G, denote by δ_{σ} ∈ Λ the function τ ↦ δ_{σ,τ} (the Kronecker delta).

Let V be a Λ-module. Since the δ_{σ} are orthogonal idempotents, V_{σ} := δ_{σ}V yields a decomposition

V = ⨁_{σ∈G} V_{σ}.

Thus V has the structure of a G-graded vector space. Conversely, any G-graded vector space V becomes a Λ-module by setting f · v = f(σ)v for f ∈ Λ and v ∈ V_{σ}. In other words, the module category of Λ is equivalent to the category of G-graded vector spaces. The invariant space of V is simply the graded component V_{1} associated to the unit element of G. A Λ-algebra is a G-graded algebra.

(e) Let Λ := P^{∗} be the Steenrod algebra over a finite field K = 𝔽_{q}. We refer to Smith [19, Section 11.1] for the definition. P^{∗} is generated by the Steenrod operations P^{k}. Λ becomes a cocommutative Hopf algebra by Δ(P^{k}) = ∑_{i=0}^{k} P^{i}⊗P^{k−i} and ϵ(P^{k}) = δ_{k,0}. However, Λ does not have an antipode. A Λ-algebra is exactly what Smith calls an algebra over the Steenrod algebra. ◁

Proposition 1.3. Let Λ be a Hopf algebra and R a Λ-algebra.

(a) For f ∈ R^{Λ}, multiplication with f provides a homomorphism R → R of Λ-modules.

(b) R^{Λ} is a subalgebra of R.

Proof. (a) Take f ∈ R^{Λ}, g ∈ R, and λ ∈ Λ with Δ(λ) = ∑_{i=1}^{r} μ_{i} ⊗ ν_{i}. Then

λ(fg) = ∑_{i=1}^{r}(μ_{i}f)(ν_{i}g) = ∑_{i=1}^{r}(ϵ(μ_{i})f)(ν_{i}g) = f · ∑_{i=1}^{r} ϵ(μ_{i})ν_{i}g = f · (λg),

where the second diagram in Definition 1.1(a) was used.

(b) We have 1_{R} ∈ R^{Λ}. Moreover, if f, g ∈ R^{Λ} and λ ∈ Λ, then by (a)

λ(fg) = f · (λg) = fϵ(λ)g = ϵ(λ)(fg),

hence fg ∈ R^{Λ}.

We will need the following lemma, whose proof was shown to me by Moss Sweedler.

Lemma 1.4. Let Λ be a Hopf algebra with antipode and U, V Λ-modules. Then

(Hom_{K}(U, V))^{Λ} = Hom_{Λ}(U, V).

Proof. Take φ ∈ Hom_{Λ}(U, V). Then for λ ∈ Λ with Δ(λ) = ∑_{i=1}^{r} μ_{i} ⊗ ν_{i} and u ∈ U we have

(λφ)(u) = ∑_{i=1}^{r} μ_{i}φ(η(ν_{i})u) = ∑_{i=1}^{r} μ_{i}η(ν_{i})φ(u) = ϵ(λ)φ(u),

where the diagram in Definition 1.1(c) was used. Hence φ ∈ Hom_{K}(U, V)^{Λ}.

The opposite inclusion is much harder. For λ ∈ Λ with Δ(λ) = ∑_{i=1}^{r} μ_{i} ⊗ ν_{i}, set

Δ(μ_{i}) = ∑_{j=1}^{s} α_{i,j} ⊗ β_{i,j} and Δ(ν_{i}) = ∑_{k=1}^{t} γ_{i,k} ⊗ δ_{i,k}.

By coassociativity (the first diagram in Definition 1.1(a)) we have

∑_{i,j} α_{i,j} ⊗ β_{i,j} ⊗ ν_{i} = (Δ ⊗ id)(Δ(λ)) = (id ⊗Δ)(Δ(λ)) = ∑_{i,k} μ_{i} ⊗ γ_{i,k} ⊗ δ_{i,k}.

For φ ∈ Hom_{K}(U, V) and u ∈ U, this identity gives rise to

∑_{i,j} α_{i,j}φ(η(β_{i,j})ν_{i}u) = ∑_{i,k} μ_{i}φ(η(γ_{i,k})δ_{i,k}u) = ∑_{i} μ_{i}φ(ϵ(ν_{i})u) = λφ(u).

If φ ∈ Hom_{K}(U, V)^{Λ}, then

∑_{j=1}^{s} α_{i,j}φ(η(β_{i,j})ν_{i}u) = (μ_{i}φ)(ν_{i}u) = ϵ(μ_{i})φ(ν_{i}u).

With the above, this yields

λφ(u) = ∑_{i=1}^{r} φ(ϵ(μ_{i})ν_{i}u) = φ(λu).

This shows that φ ∈ Hom_{Λ}(U, V).

We now introduce cohomology of Hopf algebras. Take a projective resolution

⋯ ⟶^{∂_{2}} P_{2} ⟶^{∂_{1}} P_{1} ⟶^{∂_{0}} P_{0} ⟶ K ⟶ 0

of K (as a Λ-module). For a Λ-module V, we obtain the complex

Hom_{Λ}(P_{0}, V) ⟶^{∂_{0}^{∗}} Hom_{Λ}(P_{1}, V) ⟶^{∂_{1}^{∗}} Hom_{Λ}(P_{2}, V) ⟶^{∂_{2}^{∗}} ⋯,

with ∂_{i}^{∗}(f) = f ∘ ∂_{i}. The i-th cohomology is defined as

H^{i}(Λ, V) := Ext_{Λ}^{i}(K, V) := ker(∂_{i}^{∗})/ im(∂_{i−1}^{∗}),

where we formally set ∂_{i}^{∗} = 0 for i < 0. The cohomology H^{i}(Λ, V) is independent of the choice of the projective resolution (see Benson [1, p. 29]). As an important tool we have the long exact sequence of cohomology (see Benson [1, Proposition 2.5.3(ii)]). For later use and to gain some experience in working with cohomology, we prove the following lemma.

Lemma 1.5. If Λ is a Hopf algebra and V a Λ-module, then

H^{0}(Λ, V) ≅ V^{Λ}.

Proof. By the second diagram in Definition 1.1(a), ϵ is not the zero-map, hence we can choose a projective resolution which starts as

⋯ ⟶ P_{1} ⟶^{∂_{0}} Λ ⟶^{ϵ} K ⟶ 0.

Hence H^{0}(Λ, V) = {φ ∈ Hom_{Λ}(Λ, V) | φ ∘ ∂_{0} = 0}. We have an isomorphism of vector spaces

Hom_{Λ}(Λ, V) ⟶^{∼} V, φ ↦ φ(1).

Assume that φ(1) ∈ V^{Λ}. Then for f ∈ P_{1} we have

φ(∂_{0}(f)) = ∂_{0}(f)φ(1) = ϵ(∂_{0}(f))φ(1) = 0

by the exactness of the resolution, hence φ ∈ H^{0}(Λ, V). Conversely, take φ ∈ H^{0}(Λ, V). Then ker(ϵ) = im(∂_{0}) ⊆ ker(φ). For λ ∈ Λ we have that λ − ϵ(λ) lies in the kernel of ϵ, hence

λφ(1) = (λ − ϵ(λ))φ(1) + ϵ(λ)φ(1) = φ(λ − ϵ(λ)) + ϵ(λ)φ(1) = ϵ(λ)φ(1).

Therefore φ(1) ∈ V^{Λ}. Thus the restriction of the above isomorphism to H^{0}(Λ, V) yields the desired isomorphism.

2 Regular sequences and Cohen-Macaulay rings

In this section, all rings and algebras (except for Hopf algebras) are associative, commutative, and with unit. If a_{1}, …, a_{k} ∈ R are elements of a ring, we write

(a_{1}, …, a_{k})R = {a_{1}f_{1} + ⋯ + a_{k}f_{k} | f_{1}, …, f_{k} ∈ R}

for the ideal generated by the a_{i}. We recall the definition of a regular sequence.

Definition 2.1. Let R be a ring. A sequence a_{1}, …, a_{k} ∈ R is called R-regular if the following two conditions are satisfied.

(a) (a_{1}, …, a_{k})R ≠ R, and

(b) for 1 ≤ i ≤ k, multiplication with a_{i} is injective on R/(a_{1}, …, a_{i−1})R.

The goal of this section is to study the following question: If R is an algebra over a Hopf algebra Λ and a_{1}, …, a_{k} ∈ R^{Λ} are invariants forming an R-regular sequence, under which condition is this sequence also R^{Λ}-regular? For reasons that will become clear later, we will be especially interested in regular sequences of length 3.

Let R be a ring and a_{1}, a_{2}, a_{3} ∈ R. We consider the Koszul complex K(a_{1}, a_{2}, a_{3}; R), which in this case is given by

0 ⟶ R ⟶^{∂_{2}} R^{3} ⟶^{∂_{1}} R^{3} ⟶^{∂_{0}} R,

where the R-homomorphisms ∂_{i} are as follows: If e_{1}, e_{2}, e_{3} is the standard basis for R^{3}, then ∂_{0}(e_{i}) = a_{i}. Furthermore, if we label the standard basis vectors of the other R^{3} by e_{i,j} (1 ≤ i < j ≤ 3), then ∂_{1}(e_{i,j}) = a_{i}e_{j} − a_{j}e_{i}. Finally, ∂_{2}(1) = a_{1}e_{2,3} − a_{2}e_{1,3} + a_{3}e_{1,2}. It is a general fact that the Koszul complex K(a_{1}, …, a_{k}; R) is exact if a_{1}, …, a_{k} is R-regular (see Eisenbud [4, Corollary 17.5]). Since we need a partial converse for this result, we will give an elementary proof for the case k = 3.

Lemma 2.2. Let R be a ring and a_{1}, a_{2}, a_{3} ∈ R.

(a) If a_{1}, a_{2}, a_{3} is R-regular, then K(a_{1}, a_{2}, a_{3}; R) is exact.

(b) If the part

R^{3} ⟶^{∂_{1}} R^{3} ⟶^{∂_{0}} R

of K(a_{1}, a_{2}, a_{3}; R) is exact, then multiplication with a_{3} is injective on R/(a_{1}, a_{2})R.

Proof. (a) Since a_{1} is not a zero-divisor, ∂_{2} in K(a_{1}, a_{2}, a_{3}; R) is injective. Now suppose that ∂_{1}(y_{1}e_{2,3} + y_{2}e_{1,3} + y_{3}e_{1,2}) = 0. This means that

(−y_{2}a_{3} − y_{3}a_{2})e_{1} + (−y_{1}a_{3} + y_{3}a_{1})e_{2} + (y_{1}a_{2} + y_{2}a_{1})e_{3} = 0.

From the regularity it follows that y_{1} = za_{1} with z ∈ R. Now (za_{2} + y_{2})a_{1} = 0 and (−za_{3} +y_{3})a_{1} = 0, hence y_{2} = −za_{2} and y_{3} = za_{3}. Therefore y_{1}e_{2,3} +y_{2}e_{1,3} +y_{3}e_{1,2} = ∂_{2}(z). This shows the exactness at the left R^{3}. To prove the exactness at the right R^{3}, suppose ∂_{0}(x_{1}e_{1}+x_{2}e_{2}+x_{3}e_{3}) = 0, i.e., x_{1}a_{1}+x_{2}a_{2}+x_{3}a_{3} = 0. By the regularity, this implies x_{3} ∈ (a_{1}, a_{2})R, so x_{3} = y_{2}a_{1} + y_{1}a_{2} with y_{i} ∈ R. We obtain the equation

(x_{1} + y_{2}a_{3})a_{1} + (x_{2} + y_{1}a_{3})a_{2} = 0,

from which x_{2} + y_{1}a_{3} = y_{3}a_{1} with y_{3} ∈ R follows by the regularity. Hence x_{2} = y_{3}a_{1} − y_{1}a_{3}. Finally, we obtain

(x_{1} + y_{2}a_{3} + y_{3}a_{2})a_{1} = 0,

which yields x_{1} = −y_{2}a_{3} − y_{3}a_{2}. By the definition of ∂_{1}, it now follows that

x_{1}e_{1} + x_{2}e_{2} + x_{3}e_{3} = ∂_{1}(y_{1}e_{2,3} + y_{2}e_{1,3} + y_{3}e_{1,2}),

which completes the proof of (a).

(b) Suppose that we have x_{3}a_{3} ∈ (a_{1}, a_{2})R with x_{3} ∈ R. Then x_{1}a_{1}+x_{2}a_{2}+x_{3}a_{3} = 0 with x_{i} ∈ R, hence ∂_{0}(x_{1}e_{1} + x_{2}e_{2} + x_{3}e_{3}) = 0. By the exactness, there exist y_{1}, y_{2}, y_{3} ∈ R such that

x_{1}e_{1} + x_{2}e_{2} + x_{3}e_{3} = ∂_{1}(y_{1}e_{2,3} + y_{2}e_{1,3} + y_{3}e_{1,2}).

Evaluating the e_{3}-coefficient of this equation yields x_{3} = y_{1}a_{2} + y_{2}a_{1}. Therefore x_{3} ∈ (a_{1}, a_{2})R, as claimed.

We return to the question posed after Definition 2.1 and give a cohomological criterion which decides whether an R-regular sequence a_{1}, a_{2}, a_{3} ∈ R^{Λ} is also R^{Λ}-regular. Notice that since multiplication with an invariant is a Λ-endomorphism on R by Proposition 1.3(a), it induces endomorphisms on the cohomology spaces H^{i}(Λ, R).

Theorem 2.3. Let Λ be a Hopf algebra, R a Λ-algebra, and a_{1}, a_{2}, a_{3} ∈ R^{Λ} a R-regular sequence consisting of invariants. Then a_{1}, a_{2}, a_{3} is R^{Λ}-regular if and only if the homomorphism

H^{1}(Λ, R) ⟶ H^{1}(Λ, R)^{3}  (2.1)

induced by multiplication with the a_{i} is injective.

Proof. We have 1 ∉ (a_{1}, a_{2}, a_{3})R and therefore also 1 ∉ (a_{1}, a_{2}, a_{3})R^{Λ}. Hence the condition (a) in Definition 2.1 is satisfied. Moreover, multiplication with a_{1} is injective on R and hence also on R^{Λ}. Now suppose that x_{1}a_{1} = x_{2}a_{2} with x_{1}, x_{2} ∈ R^{Λ}. Then x_{2} = ya_{1} with y ∈ R. We claim that y ∈ R^{Λ}. Indeed, for λ ∈ Λ we have

a_{1}(λ − ϵ(λ))y = (λ − ϵ(λ))(a_{1}y) = (λ − ϵ(λ))x_{2} = 0,

hence (λ − ϵ(λ))y = 0, as claimed. Therefore multiplication with a_{2} is injective on R^{Λ}/(a_{1})R^{Λ}. Thus we have to show that multiplication with a_{3} is injective on R^{Λ}/(a_{1}, a_{2})R^{Λ} if and only if (2.1) is injective. By Lemma 2.2 the first condition is equivalent to the exactness of the sequence

(R^{Λ})^{3} ⟶ (R^{Λ})^{3} ⟶ R^{Λ}  (2.2)

from K(a_{1}, a_{2}, a_{3}; R^{Λ}). Set M := ker(∂_{0}) ⊆ R^{3} with ∂_{0} from K(a_{1}, a_{2}, a_{3}; R). Since the ∂_{i} are Λ-homomorphisms, restriction to the invariants yields an exact sequence

0 ⟶ M^{Λ} ⟶ (R^{Λ})^{3} ⟶ R^{Λ}.

By Lemma 2.2(a), M = im(∂_{1}), so we obtain the commutative diagram

[figure: M^{Λ}; (R^{Λ})^{3}; (R^{Λ})^{3}; R^{Λ}.]

Thus (2.2) is exact if and only if (R^{Λ})^{3} → M^{Λ} is surjective. From the short exact sequence

0 ⟶ R ⟶^{∂_{2}} R^{3} ⟶ M ⟶ 0

we obtain, using Lemma 1.5,

0 ⟶ R^{Λ} ⟶ (R^{Λ})^{3} ⟶ M^{Λ} ⟶ H^{1}(Λ, R) ⟶ H^{1}(Λ, R^{3}).

Therefore (R^{Λ})^{3} → M^{Λ} is surjective if and only if the map H^{1}(Λ, R) → H^{1}(Λ, R^{3}) induced by ∂_{2} is injective. But from the definition of ∂_{2}, this is equivalent to the injectivity of the map (2.1). Summing up, we have proved that a_{1}, a_{2}, a_{3} is R^{Λ}-regular if and only if (2.1) is injective.

Remark 2.4. There is also a version of Theorem 2.3 giving a criterion for the R^{Λ}-exactness of longer sequences, under a certain condition. It can be found in Kemper [11, Satz 1.9] and states the following: Assume for a non-negative integer r that H^{i}(Λ, R) = 0 for all i with 0 < i < r. Then an R-regular sequence a_{1}, …, a_{r+2} ∈ R^{Λ} is R^{Λ}-regular if and only if the map

H^{r}(Λ, R) ⟶ H^{r}(Λ, R)^{r+2}

induced by multiplication with the a_{i} is injective.

Theorem 2.3 is the version for r = 1 (in which case the hypothesis is vacuous). ◁

The following example illustrates a situation where the map (2.1) is not injective. Example 2.5. Let {0} ≠ G ≤ K^{+} be a subgroup of the additive group of K. We write elements of G as σ_{c}, c ∈ K. G acts on the polynomial ring R := K[x_{1}, x_{2}, x_{3}, y_{1}, y_{2}, y_{3}] by

σ_{c}(x_{i}) = x_{i} and σ_{c}(y_{i}) = y_{i} + cx_{i}.

This makes R into an algebra over Λ := KG. The homomorphism

φ: G → K^{+}, σ_{c} ↦ c

defines a non-zero element of H^{1}(G, K). K occurs as the polynomials of degree 0 in R and is hence a direct summand. Therefore φ defines a non-zero element α ∈ H^{1}(G, R) via the embedding H^{1}(G, K) ↪ H^{1}(G, R). The sequence x_{1}, x_{2}, x_{3} ∈ R^{G} is clearly R-regular, but we have

x_{i}φ(σ_{c}) = cx_{i} = (σ_{c} − 1)y_{i},

which is a coboundary. Hence x_{i}α = 0, so the map (2.1) is not injective, and x_{1}, x_{2}, x_{3} is not R^{G}-regular. ◁

In fact it is very common (to say the least) that elements in H^{i}(Λ, R) are torsion elements. To illustrate this, consider the case Λ = KG with G a finite group, and let the Λ-algebra R be a domain. Then H^{i}(G, Quot(R)) = 0 for i > 0 by (the additive version of) Hilbert’s Theorem 90. Thus every element in H^{i}(G, R) is torsion. However, this consideration does not provide sufficient information to decide if a non-zero element in H^{1}(G, R) exists whose annihilator in R^{G} contains an R-regular sequence of length 3.

The main goal of this paper is to study the Cohen-Macaulay property of invariant rings, which we define now.

Definition 2.6. Let R be an algebra over a field K.

(a) A sequence a_{1}, …, a_{k} ∈ R is called a partial system of parameters if (a_{1}, …, a_{k})R ≠ R and ht ((a_{1}, …, a_{k})R) = k. Recall that the height of an ideal I is the minimal height of the prime ideals containing I. By Krull’s principal ideal theorem (see Eisenbud [4, Theorem 10.2]), we have ht ((a_{1}, …, a_{k})R) ≤ k for any a_{1}, …, a_{k} ∈ R satisfying (a_{1}, …, a_{k})R ≠ R.

(b) R is called Cohen-Macaulay if it is noetherian and every partial system of parameters is an R-regular sequence.

Remark 2.7. (a) It can be shown that any R-regular sequence is a partial system of parameters (this follows from Eisenbud [4, Theorem 3.1(b)]).

(b) If R has finite Krull dimension n, then n is the maximal length of a partial system of parameters, and a partial system of parameters of this length is called a system of parameters.

(c) In Definition 2.6 we gave an alternative definition of Cohen-Macaulayness. The “official” one is as follows: R is called Cohen-Macaulay if for every prime ideal P ∈ Spec(R) there exists an R_{P} -regular sequence in P_{P} (the maximal ideal of the local ring R_{P}) of length equal to the Krull dimension dim(R_{P}).

(d) If R is a graded algebra and K = R_{0} is the homogeneous part of degree 0, then R is Cohen-Macaulay if and only if for some (and then for every) homogeneous system of parameters f_{1}, …, f_{n}, R is a free module over the subalgebra K[f_{1}, …, f_{n}]. We also remark that for any homogeneous system of parameters R is a finitely generated module over K[f_{1}, …, f_{n}] (independently of the Cohen-Macaulay property). This characterization shows what a nice property Cohen-Macaulayness is. In the sequel we shall focus on graded algebras. ◁

Example 2.8. (a) A polynomial ring K[x_{1}, …, x_{n}] is Cohen-Macaulay. This can be seen, for example, by using Remark 2.7(d).

(b) If |G| < ∞ in Example 2.5, then R^{G} is not Cohen-Macaulay. Indeed, R is integral over R^{G} is this case, and we see from Eisenbud [4, Proposition 9.2] that

ht ((x_{1}, x_{2}, x_{3})R^{G}) = ht ((x_{1}, x_{2}, x_{3})R) = 3.

Hence x_{1}, x_{2}, x_{3} is a partial system of parameters for R^{G}, but not R^{G}-regular. ◁

If V is a module over a cocommutative Hopf algebra Λ, then the symmetric algebra S(V) is a Λ-module. (We need cocommutativity so that the ideal factored out from the tensor algebra to form S(V) is a Λ-submodule.) In the case where Λ is the group ring of a linearly reductive group, the celebrated theorem of Hochster and Roberts says the following.

Theorem 2.9 (Hochster and Roberts [8]). Let G be a linearly reductive algebraic group over an algebraically closed field K and V a G-module (i.e., a KG-module such that the action G → GL(V) is a morphism of algebraic groups). Then S(V)^{G} is Cohen-Macaulay.

The main goal of this paper is to study, in the more general setting of Hopf algebras, to what extent a converse of Theorem 2.9 holds. In order to do so, we must generalize the concept of a linearly reductive group. Here we face the difficulty that linear reductivity is restricted to G-modules, which form a subcategory of all KG-modules. We address this difficulty by allowing subcategories of the module category with certain properties.

Definition 2.10. Let Λ be a cocommutative Hopf algebra with antipode.

(a) A full subcategory C of the module category MOD(Λ) of Λ is called admissible if the following properties hold:

(i) All objects of C have finite K-dimension.

(ii) K is an object of C.

(iii) If V is an object of C, then so are all symmetric powers S^{i}(V) and all submodules

U ≤ V. (iv) If U and V are objects of C, then so are U ⊕ V and Hom_{K}(U, V).

(b) If C is an admissible subcategory of MOD(Λ), then Λ is called C-linearly reductive if every short exact sequence of modules from C splits.

(c) If C is an admissible subcategory of MOD(Λ), then Λ is called C-geometrically reductive if for every module V from C and for every 0 ≠ v ∈ V^{Λ} there exist an r > 0 and an f ∈ S^{r}(V^{∗})^{Λ} such that f(v) ≠ 0.

Examples of admissible subcategories of MOD(Λ) are the category of all finite dimensional Λ-modules, or the category of all G-modules if Λ = KG with G an algebraic group over an algebraically closed field K. In the latter case, Λ is C-linearly or C-geometrically reductive if and only if G is linearly or geometrically reductive, respectively. We will see in Proposition 3.1 that linear reductivity implies geometric reductivity. The main goal of this paper is to prove the following theorem.

Theorem 2.11 (Main Theorem). Let Λ be a cocommutative Hopf algebra with antipode and C an admissible subcategory of MOD(Λ) such that Λ is C-geometrically reductive. Then the following two statements are equivalent:

(a) Λ is C-linearly reductive.

(b) For every object V of C, the invariant ring S(V)^{Λ} is Cohen-Macaulay.

A corollary of Theorem 2.11 is that if G is a finite group and K a field whose characteristic divides |G|, then there exists a finite dimensional KG-module V such that S(V)^{G} is not Cohen-Macaulay. To interpret Theorem 2.11 for other examples of cocommutative Hopf algebras with antipodes requires an analysis of the different notions of reductivity for these Hopf algebras.

The proof of the implication “(a) ⇒ (b)” is a straight forward generalization of Hochster and Roberts’s arguments (or, more precisely, the proof due to F. Knop given in Bruns and Herzog [3, Section 6.5]). The proof of the converse implication “(b) ⇒ (a)” breaks down into a number of steps, as follows:

(1) If Λ is not C-linearly reductive, we find an object U of C such that H^{1}(Λ, U) ≠ 0.

(2) For α ∈ H^{1}(Λ, U), we construct a module W from C such that W has a non-zero invariant 0 ≠ w ∈ W^{Λ} with w ⊗ α = 0 in H^{1}(Λ, W ⊗ U). In other words, we hire a killer for α.

(3) Then we build the Λ-module

V := W ⊕ W ⊕ W ⊕ U.

Using Theorem 2.3, we conclude that the three copies w_{1}, w_{2}, w_{3} of w in V form a sequence in S(V)^{Λ} which is not S(V)^{Λ}-regular.

(4) Using the hypothesis that Λ is C-geometrically reductive, we show that w_{1}, w_{2}, w_{3} is a partial system of parameters for S(V)^{Λ}. Thus S(V)^{Λ} is not Cohen-Macaulay.

3 The proof of the Main Theorem

In this section Λ is always a cocommutative Hopf algebra with antipode, and C is an admissible subcategory of MOD(Λ). We will first prove the implication “(a) ⇒ (b)” and then embark on the proof of the converse. First of all, however, we show that linear reductivity implies geometric reductivity.

Proposition 3.1. If Λ is C-linearly reductive, it is also C-geometrically reductive.

Proof. Let V be a Λ-module from C and 0 ≠ v ∈ V^{Λ}. Using the rule ϵ∘η = ϵ (Sweedler [22, Proposition 4.0.1(3)]), it is routine to check that the map φ: V^{∗} → K, f ↦ f(v) is a Λ-homomorphism. It follows that if U ≤ V^{∗} is an irreducible submodule with φ(U) ≠ 0, then U ≅ K. By the linear reductivity, V^{∗} is decomposable into irreducible Λ-submodules, and therefore φ(V^{∗}) = φ ((V^{∗})^{Λ}). But v ≠ 0 implies φ ≠ 0, hence there exists f ∈ (V^{∗})^{Λ} with f(v) = φ(f) ≠ 0.

We now prove that for linearly reductive Hopf algebras there exists a so-called Reynolds operator.

Proposition 3.2. Assume that Λ is C-linearly reductive and that V is a Λ-module from C. Then with R := S(V) there exists a Λ-homomorphism π: R → R^{Λ} with π|_{R^{Λ}} = id. Any such π is also a homomorphism of R^{Λ}-modules. It is called a Reynolds operator.

Proof. For each i the embedding S^{i}(V)^{Λ} ↪ S^{i}(V) splits by the linear reductivity. This yields Λ-homomorphisms π_{i} which we put together to obtain π. Now fix an a ∈ R^{Λ} and consider the Λ-endomorphism

φ: R → R, f ↦ π(af).

If U ≤ R is an irreducible submodule with φ(U) ≠ 0, then U ≅ K. Take f ∈ S^{i}(V) and set f′ := f − π(f). Since S^{i}(V) is decomposable into a direct sum of irreducible submodules, we conclude that φ(f′) = 0. Therefore

π(af) = φ(π(f) + f′) = φ(π(f)) = π(aπ(f)) = aπ(f).

Now the second claim follows.

Lemma 3.3. In the situation of Proposition 3.2, let I ⊆ R^{Λ} be an ideal and let IR be the ideal generated by I in R. Then

IR ∩ R^{Λ} = I.

Proof. The inclusion I ⊆ IR∩R^{Λ} is clear. Conversely, take f = a_{1}b_{1} +⋯+a_{r}b_{r} ∈ IR∩R^{Λ} with a_{i} ∈ R and b_{i} ∈ I. Applying the Reynolds operator, we obtain

f = π(f) = π(a_{1})b_{1} + ⋯ + π(a_{r})b_{r} ∈ I.

We can now use Theorem 6.5.2 from Bruns and Herzog [3] to generalize Theorem 2.9.

Theorem 3.4. Suppose that Λ is C-linearly reductive and V is a Λ-module from C. Then S(V)^{Λ} is Cohen-Macaulay.

Proof. We first remark that S(V)^{Λ} is a graded K-algebra. Hence Hilbert’s classical proof of the finite generation of invariant rings (see, for example, the proof of Theorem 2.1.3 in Sturmfels [21]) is applicable. Thus S(V)^{Λ} is finitely generated and therefore noetherian. Now the claim follows from Bruns and Herzog [3, Theorem 6.5.2], where the essential hypothesis is provided by Lemma 3.3.

We are now going to take the first step in proving the implication “(b) ⇒ (a)” from Theorem 2.11. Fix a Λ-module V from C. A short exact sequence

0 ⟶ V ⟶ W ⟶ K ⟶ 0  (3.1)

with W an object of C gives rise to the exact sequence

0 ⟶ V^{Λ} ⟶ W^{Λ} ⟶ K ⟶^{δ} H^{1}(Λ, V).

We see that (3.1) splits if and only if δ(1) = 0. Denote by H_{C}^{1}(Λ, V) the subset of H^{1}(Λ, V) consisting of all δ(1) which are obtained from exact sequences (3.1) with W objects of C.

Proposition 3.5. Λ is C-linear reductive if and only if H_{C}^{1}(Λ, V) = 0 for all Λ-modules V from C.

Proof. If Λ is C-linear reductive, every short exact sequence of the type (3.1) splits, hence H_{C}^{1}(Λ, V) = 0.

Conversely, assume that H_{C}^{1}(Λ, V) = 0, so that every sequence (3.1) splits. We first claim that an epimorphism U → V of Λ-modules from C restricts to an epimorphism U^{Λ} → V^{Λ}. Indeed, take any invariant 0 ≠ v ∈ V^{Λ}, and let U′ ⊆ U be the preimage of Kv. Since Kv is a submodule of V, U′ is also a submodule of U. By the hypothesis, the epimorphism U′ → Kv splits. Thus v has a preimage in (U′)^{Λ} ⊆ U^{Λ}, which proves the claim.

Now let

0 ⟶ V ⟶ W ⟶ U ⟶ 0  (3.2)

be any short exact sequence with objects of C. This leads to an exact sequence

0 ⟶ Hom_{K}(U, V) ⟶ Hom_{K}(W, V) ⟶ Hom_{K}(V, V) ⟶ 0.

We have id_{V} ∈ Hom_{Λ}(V, V) = Hom_{K}(V, V)^{Λ} by Lemma 1.4. By the above, id_{V} has a preimage φ ∈ Hom_{K}(W, V)^{Λ} = Hom_{Λ}(W, V), again using Lemma 1.4. This φ yields the desired splitting of the sequence (3.2).

I owe the idea of the current version of the preceding proof to Hanspeter Kraft. We now address the second step in the proof, the construction of a module W with an invariant vector 0 ≠ w ∈ W^{Λ} which kills a given α ∈ H^{1}(Λ, V). The homomorphism V → W ⊗ V, v ↦ w ⊗ v induces a map H^{1}(Λ, V) → H^{1}(Λ, W ⊗ V). We denote the image of α under this map by w ⊗ α.

Proposition 3.6. Let V be a Λ-module from C and α ∈ H_{C}^{1}(Λ, V). Then there exists a Λ-module W from C and a non-zero invariant 0 ≠ w ∈ W^{Λ} such that w ⊗ α = 0.

Proof. We start by considering any Λ-module W with a non-zero invariant 0 ≠ w ∈ W^{Λ} and derive conditions on W which are equivalent to w ⊗ α = 0. In order to find a short exact sequence to which w ⊗ α is associated, we have to look at the connection between short exact sequences and first cohomology in more detail. Indeed, α arises from an exact sequence

0 ⟶ V ⟶ Ṽ ⟶^{π} K ⟶ 0

with Ṽ a module from C. In explicit terms, this means that α is given by the homomorphism ψ_{1} in the commutative diagram

[figure: ⋯; P_{1}; ∂_{1}; P_{0}; ∂_{0}; K; 0; ψ_{1}; ψ_{0}; ||; 0; V; Ṽ; π; K; 0,]

where the upper row is a projective resolution and the homomorphisms ψ_{i} owe their existence to projectivity. We have a Λ-homomorphism φ: K → W, c ↦ cw. Taking the pullback

M := (W ⊗ Ṽ) ×_{W} K := {(a, c) ∈ (Ṽ ⊗ W) ⊕ K | (id_{W} ⊗π)(a) = φ(c)},

we obtain a commutative diagram

[figure: 0; W ⊗ V; M; K; 0; ||; φ; 0; W ⊗ V; W ⊗ Ṽ; id_{W} ⊗π; W; 0]

with exact rows. We claim that w ⊗ α is the element of H^{1}(Λ, W ⊗ V) associated to the upper row. Indeed, it is easily checked that the homomorphisms

\begin{aligned}ψ′_{0} & : P_{0} → M, x ↦ (w ⊗ ψ_{0}(x), ∂_{0}(x)) and \\ ψ′_{1} & : P_{1} → W ⊗ V, x ↦ w ⊗ ψ_{0}(∂_{1}(x)) = w ⊗ ψ_{1}(x) \end{aligned}

make the diagram

[figure: ⋯; P_{1}; ∂_{1}; P_{0}; ∂_{0}; K; 0; ψ′_{1}; ψ′_{0}; ||; 0; W ⊗ V; M; K; 0]

commute. So we have seen that the sequence 0 → W ⊗ V → M → K → 0 splits if and only if w ⊗ α = 0. By the definition of M, the sequence splits if and only if there exists a Λ-homomorphism K → W ⊗ Ṽ such that

[figure: K; φ; W ⊗ Ṽ; id_{W} ⊗π; W]

commutes. The natural isomorphism W ⊗ Ṽ ⟶^{∼} Hom_{K}(Ṽ^{∗}, W) and the homomorphism Hom_{K}(Ṽ^{∗}, W) → W, f ↦ f(π) make the diagram

[figure: W ⊗ Ṽ; id_{W} ⊗π; W; ≅; Hom_{K}(Ṽ^{∗}, W)]

commute. Using this, we conclude that w ⊗ α = 0 if and only if there exists a Λhomomorphism K → W such that

[figure: K; φ; Hom_{K}(Ṽ^{∗}, W); W]

commutes. This is equivalent to the existence of f ∈ Hom_{K}(Ṽ^{∗}, W)^{Λ} = Hom_{Λ}(Ṽ^{∗}, W) (by Lemma 1.4) such that f(π) = w. Thus W has a 0 ≠ w ∈ W^{Λ} with w ⊗ α = 0 if and only if there exists a Λ-homomorphism f: Ṽ^{∗} → W with f(π) ≠ 0. Hence we can choose W = Ṽ^{∗} (which is an object of C) and f = id.

Suppose that Λ is not C-linearly reductive. Putting things together as suggested at the end of Section 2, we obtain a Λ-module V from C and an S(V)-regular sequence w_{1}, w_{2}, w_{3} ∈ S(V)^{Λ} such that every w_{i} annihilates a non-zero α ∈ H^{1}(Λ, S(V)), which exists by Proposition 3.5. Therefore w_{1}, w_{2}, w_{3} is not S(V)^{Λ}-regular by Theorem 2.3. The proof of Theorem 2.11 is complete if we can show that w_{1}, w_{2}, w_{3} is a partial system of parameters for S(V)^{Λ}. For this we will need the hypothesis that Λ be C-geometrically reductive.

Lemma 3.7. Let φ: A → B be an epimorphism of Λ-algebras which need not have a unit. Assume that every a ∈ A lies in a Λ-submodule of A which is an object of C, and that Λ is C-geometrically reductive. Then for every b ∈ B^{Λ} there exists an a ∈ A^{Λ} and an r > 0 such that φ(a) = b^{r}.

Proof. We follow the proof given by Mumford et al. [14, Lemma A.1.2] of the analogous result for geometrically reductive groups. We can assume that b ≠ 0. Choose an a′ ∈ A with φ(a′) = b, and a submodule E ⊆ A from C containing a′. Since b is an invariant, we obtain for λ ∈ Λ:

λa′ ∈ ϵ(λ)a′ + (E ∩ ker(φ)).

Hence V := Ka′ ⊕ (E ∩ ker(φ)) ⊆ E is also a submodule from C. There exists an f ∈ V^{∗} with f(a′) = 1 and f|_{E∩ker(φ)} = 0. For λ ∈ Λ with Δ(λ) = ∑_{i} μ_{i} ⊗ ν_{i}, α ∈ K and v ∈ E ∩ ker(φ) we have

(λf)(αa′ + v) = ∑_{i} μ_{i} · f (η(ν_{i}) · (αa′ + v)) =

= ∑_{i} ϵ(μ_{i}) · f (ϵ(ν_{i})αa′) = ϵ(λ)α = (ϵ(λ) · f)(αa′ + v),

so f lies in (V^{∗})^{Λ}. By the geometric reductivity there exist r > 0 and ã ∈ S^{r}(V^{∗∗})^{Λ} with ã(f) = 1. Using (3), (4), and (6) from Sweedler [22, Proposition 4.0.1], we verify that the canonical embedding V → V^{∗∗} is a Λ-homomorphism. Since V is finite dimensional, we can therefore assume ã ∈ S^{r}(V). The inclusion V ⊆ A induces a Λ-homomorphism ψ: S^{r}(V) → A. The image of ψ is the direct sum of K · (a′)^{r} with a K-vector space all of whose elements are multiples of elements from E ∩ ker(φ). Thus we obtain

φ(ψ(ã)) = b^{r},

so a := ψ(ã) satisfies φ(a) = b^{r}.

From this lemma we derive a “geometric” version of Lemma 3.3, which holds under the hypothesis of geometric reductivity. For the case of a geometrically reductive group, this result is well known (see Newstead [17, Lemma 3.4.2]).

Proposition 3.8. Assume that Λ is C-geometrically reductive. Let V be a Λ-module from C, R = S(V) the symmetric algebra, and I ⊆ R^{Λ} an ideal. Then

\sqrt{IR} ∩ R^{Λ} = \sqrt{I}.

Proof. The inclusion \sqrt{I} ⊆ \sqrt{IR}∩R^{Λ} is clear. For the converse inclusion it suffices to show

IR ∩ R^{Λ} ⊆ \sqrt{I}.

Take f = ∑_{i=1}^{m} a_{i}f_{i} ∈ IR ∩ R^{Λ} with a_{i} ∈ R, f_{i} ∈ I. Consider the algebra A := R[t_{1}, …, t_{m}]_{+} of all polynomials in indeterminates t_{1}, …, t_{m} and coefficients in R with 0 as constant coefficient. A becomes a Λ-algebra (without unit) with trivial action on the t_{i}. Every element from A lies in a submodule from C. We have a homomorphism φ: A → R of Λ-algebras given by at_{i} ↦ af_{i} for a ∈ R. If B is the image of this homomorphism, then f ∈ B^{Λ}. Moreover,

φ(A^{Λ}) = φ(R^{Λ}[t_{1}, …, t_{m}]_{+}) ⊆ I.

Now by Lemma 3.7 there exists r > 0 such that f^{r} ∈ I. This completes the proof.

Remark 3.9. Proposition 3.8 has important geometrical consequences. With the notation of Proposition 3.8, consider the map

π_{Λ}: X := Spec(R) → X//Λ := Spec(R^{Λ})

obtained from intersecting a prime ideal from R with R^{Λ}. This is the categorical quotient of the Λ-action on R. Now Proposition 3.8 implies that π_{Λ} is onto if Λ is C-geometrically reductive. ◁

We deduce the following lemma from Proposition 3.8, which will be used in the proof of Theorem 2.11.

Lemma 3.10. Suppose that Λ is C-geometrically reductive and V is a Λ-module from C. Then if v_{1}, …, v_{k} ∈ V^{Λ} ⊂ S(V)^{Λ} are linearly independent over K, they form a partial system of parameters for S(V)^{Λ}.

Proof. We write R = S(V). Clearly I := (v_{1}, …, v_{k})R^{Λ} ≠ R^{Λ}. We have to show that ht(I) ≥ k. The ideals P_{i} := (v_{1}, …, v_{i})R ∩ R^{Λ} (0 ≤ i ≤ k) are prime ideals in R^{Λ} forming a strictly increasing sequence. Let P ⊆ R^{Λ} be any prime ideal containing I. Then by Proposition 3.8 we obtain

P_{k} = \sqrt{P_{k}} = \sqrt{(v_{1}, …, v_{k})R} ∩ R^{Λ} = \sqrt{IR} ∩ R^{Λ} = \sqrt{I} ⊆ \sqrt{P} = P.

Thus ht(P) ≥ ht(P_{k}) ≥ k, which completes the proof.

Proof of Theorem 2.11. The implication “(a) ⇒ (b)” was shown in Theorem 3.4. Suppose that Λ is not C-linearly reductive. By Theorem 3.5 there exists an object U of C such that H_{C}^{1}(Λ, U) ≠ 0. Choose 0 ≠ α ∈ H_{C}^{1}(Λ, U). By Proposition 3.6 there exists an object W of C with an invariant 0 ≠ w ∈ W^{Λ} such that w ⊗ α = 0. Form the Λ-module

V := U ⊕ W ⊕ W ⊕ W

and consider R = S(V). By Lemma 3.10, the three copies w_{1}, w_{2}, w_{3} of w in V form a partial system of parameters for R^{Λ}. They also form an R-regular sequence. Since U is a direct summand of R, we can view α as a non-zero element of H^{1}(Λ, R). Applying the map H^{1}(Λ, R ⊗ R) → H^{1}(Λ, R) induced by the multiplication, we find that w_{i}α = 0 for all i. Thus w_{1}, w_{2}, w_{3} is not R^{Λ}-regular by Theorem 2.3. Therefore by Definition 2.6(b) R is not Cohen-Macaulay. This completes the proof.

Remark 3.11. The way in which the hypothesis of geometric reductivity comes into the proof of Theorem 2.11 may seem odd, and possibly unnecessary. However, the example of the additive group G_{a} (of ℂ, say) shows that this hypothesis is indispensable. Indeed, for every G_{a}-module V we have that S(V)^{G_{a}} is Cohen-Macaulay (see Kemper [11, Anmerkung 2.16] or [10, Remark 8] for details). G_{a} is the prototype of a non-geometrically reductive group. ◁

4 Interpretations of the Main Theorem

In Example 1.2 we saw a few examples of cocommutative Hopf algebras with antipodes. In order to reap some fruits from Theorem 2.11, we have to analyze the reductivity properties for the various instances of Hopf algebras. Thus we can hope to obtain interesting theorems on invariants of groups and of Lie algebras, and possibly other types of objects. Before considering the various cases of Hopf algebras, we proof one general result.

Proposition 4.1. Let Λ be a finite dimensional Hopf algebra over a field K, and let C be an admissible subcategory of MOD(Λ). Then Λ is C-geometrically reductive.

Proof. Let V be an object of C and take 0 ≠ v ∈ V^{Λ}. Then there exists an f ∈ V^{∗} with f(v) = 1. By Ferrer Santos [5, Theorem 4.2.1], the finite dimensionality of Λ implies that R is integral over R^{Λ}. Thus there exist a_{1}, …, a_{r} ∈ S(V^{∗})^{Λ} such that

f^{r} + a_{1}f^{r−1} + ⋯ + a_{r−1}f + a_{r} = 0.

Without loss we can assume that a_{i} ∈ S^{i}(V^{∗})^{Λ}. Evaluating the above equation at v yields

1 + a_{1}(v) + ⋯ + a_{r−1}(v) + a_{r}(v) = 0,

hence at least one of the a_{i}(v) must be non-zero. This means that Λ is C-geometrically reductive.

4.1 Algebraic groups

We consider the case where G is a linear algebraic group over an algebraically closed field K and C is the subcategory of MOD(KG) consisting of the G-modules (i.e., the finite dimensional KG-modules V such that the action G → GL(V) is given by a morphism of algebraic groups). Theorem 2.11 now yields:

Theorem 4.2 ([10, Theorem 7]). A geometrically reductive algebraic group G is linearly reductive if and only if for all G-modules V the invariant ring S(V)^{G} is Cohen-Macaulay.

Remark 4.3. Using the language of schemes, we can also consider a linear algebraic group G over a non-algebraically closed field K. This poses a few difficulties, since the category of G-modules is not a subcategory of the Λ-modules, with Λ the group ring of the group G(K) of K-rational points of G. These difficulties can be circumvented by taking a detour over the algebraic closure of K (see [11, Section 2.2]). The result is that Theorem 4.2 also holds over non-algebraically closed fields. ◁

The following corollary is implicit in [12, Corollary 2.4].

Corollary 4.4. Let G be a finite group and K a field. Then S(V)^{G} is Cohen-Macaulay for all finite dimensional KG-modules V if and only if char(K) ∤ |G|.

Proof. Let C be the category of all finite dimensional modules over Λ := KG. By Proposition 4.1, Λ is C-geometrically reductive. Moreover, Λ is C-linearly reductive if and only if char(K) ∤ |G|. Now the corollary follows from Theorem 2.11.

We consider the two notions of reductivity for linear algebraic groups. Concerning geometric reductivity, it was shown by Haboush [6] that G is geometrically reductive if and only if it is (group theoretically) reductive, i.e., the unipotent radical of G is trivial. Therefore the classical groups are examples of geometrically reductive groups. If the characteristic of K is 0, then G is reductive if and only if it is linearly reductive (see Springer [20, V., Satz 1.1]). Therefore Theorem 4.2 is void in characteristic 0. The situation is remarkably different in positive characteristic. Here the linearly reductive groups were classified by Nagata [15]. The result is that G is linearly reductive if and only if the number of connected components of G is not divisible by char(K), and the 1-component G^{0} is a torus. Thus the two concepts of reductivity fall very much apart in positive characteristic, and Theorem 4.2 is far from void here. For example, we get the result that for the classical groups in positive characteristic there are modules whose invariant rings are not Cohen-Macaulay.

Theorem 4.2 ties in nicely with results of Popov [18] and Nagata [16] to give the following characterization of linearly reductive groups by their invariants.

Corollary 4.5 ([10, Corollary 9]). Let G be a linear algebraic group over an algebraically closed field K. Then G is linearly reductive if and only if the following two conditions are satisfied:

(a) If G acts on an affine K-variety X by a morphism G × X → X, then the invariant ring K[X]^{G} (with K[X] the coordinate ring) is finitely generated, and

(b) for every G-module V the invariant ring S(V)^{G} is Cohen-Macaulay.

Proof. Suppose that G is linearly reductive. Then (a) holds by the theorem of Hilbert and Nagata (see [16]), and (b) follows from Theorem 2.9. Conversely, if (a) and (b) are satisfied, then G is reductive by Popov [18]. Therefore Theorem 4.2 applies and yields that G is linearly reductive.

4.2 Lie algebras and abelian group schemes

Let us call a Lie algebra 𝔤 linearly or geometrically reductive if the universal enveloping algebra Λ = U(𝔤) is C-linearly or C-geometrically reductive, respectively, where C is the category of all finite dimensional Λ-modules. An analysis of these properties yields:

Proposition 4.6. Let 𝔤 be a finite dimensional Lie algebra over a field K.

(a) If the characteristic of K is zero, we have the equivalence

𝔤 is linearly reductive ⇔ 𝔤 is geometrically reductive ⇔ 𝔤 is semisimple.

(b) If K has positive characteristic, the 𝔤 is geometrically reductive. Moreover, it is linearly reductive if and only if 𝔤 = 0.

Proof. We start with characteristic 0. If 𝔤 is semisimple, it is linearly reductive by a theorem of Hermann Weyl (see Bourbaki [2, Chap. I, § 6, Theorem 2]). Linear reductivity implies geometric reductivity by Proposition 3.1. Hence we only have to show that geometric reductivity implies semisimplicity. We will not give the proof here but instead refer the reader to Kemper [11, Proposition 2.8].

Now we consider the case of positive characteristic. If 𝔤 is not the zero-algebra, then by Jacobson [9] there exists a representation which is not completely reducible, hence 𝔤 is not linearly reductive. Now let 𝔤 be an arbitrary finite dimensional Lie algebra in positive characteristic p, and V a finite dimensional 𝔤-module. By the definition of the comultiplication (see Example 1.2(b)), it follows that 𝔤 acts by derivations on S(V^{∗}). Therefore f^{p} ∈ S(V^{∗})^{𝔤} for any f ∈ S(V^{∗}). This implies the geometric reductivity of 𝔤.

It is quite surprising how different the behavior is whether we are in characteristic 0 or in positive characteristic. In one case, both notions of reductivity coincide, and in the other, they fall as far apart as they possibly can. From Proposition 4.6 we see that Theorem 2.11 does not yield any result for Lie algebras in characteristic 0. For positive characteristic, we obtain the following result.

Theorem 4.7. Let 𝔤 ≠ 0 be a finite dimensional Lie algebra over a field of positive characteristic. Then there exists a finite dimensional 𝔤-module V such that S(V)^{𝔤} is not Cohen-Macaulay.

We only report on the reductivity of commutative, cocommutative Hopf algebras with antipodes, i.e., coordinate rings of abelian affine group schemes (see Example 1.2).

Proposition 4.8 ([11, Proposition 2.20]). Let Λ = K[G] be the coordinate ring of an affine, smooth, abelian group scheme G of finite type (see Mumford et al. [14, Definition 0.2]), and let C be the category of all finite dimensional Λ-modules.

(a) Λ is C-linearly reductive if and only if dim_{K}(Λ) < ∞, i.e., G is zero-dimensional.

(b) If char(K) = 0, then Λ is C-geometrically reductive if and only if it is C-linearly reductive.

(c) If K has positive characteristic, then Λ is C-geometrically reductive.

The proof makes essential use of the theorem of Lie-Kolchin. The proof of part (c) is quite similar to the proof of (b) in Proposition 4.6. So again both concepts of reductivity coincide in characteristic 0, and fall apart drastically in positive characteristic. Having made this observation for all classes of cocommutative Hopf algebras with antipode that we considered prompts the following conjecture.

Conjecture 4.9. Let Λ be a cocommutative Hopf algebra with antipode over a field of characteristic 0 and C an admissible subcategory of MOD(Λ). Then Λ is C-linearly reductive if and only if it is C-geometrically reductive.

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