EXTENDING THE COINVARIANT THEOREMS OF CHEVALLEY, SHEPHARD–TODD, MITCHELL, AND

SPRINGER

ABRAHAM BROER, VICTOR REINER, LARRY SMITH, AND PETER WEBB

Abstract. We extend in several directions invariant theory results of Chevalley, Shephard and Todd, Mitchell and Springer. Their results compare the group algebra for a finite reflection group with its coinvariant algebra, and compare a group representation with its module of relative coinvariants. Our extensions apply to arbitrary finite groups in any characteristic.

Contents

1. Statement of Results 4 1.1. Chevalley, Shephard–Todd, Mitchell Type Results 4 1.2. Springer-Type Results 7 1.3. A Further Generalization 8 2. Generalities for Rings, Modules, and Tor 10 2.1. Review of Graded Resolutions 10 2.2. The Group Action on Tor 11 2.3. The Group Action on Resolutions 12 2.4. A Short Review of Brauer theory 15 2.5. The Case of Domains with Trivial Group Action 16 2.6. Proof of Theorem 1.1.1 21 2.7. Remarks on Group Cohomology and other related Constructions 22 3. Proof of Theorems 1.2.1 and 1.3.1 24 3.1. Reduction 1: Removing the Γ-action 25 3.2. Reduction 2: Replacing Special Fiber with General Fiber 26 3.3. Reduction 3: Working Locally via Completions 28 3.4. (Semi-)local Analysis of the Fibers 29 3.5. Finishing the proofs: Incorporating the C-Action 33 3.6. Induced Modules and the Proof of Corollary 1.2.2 36 4. Character Values of the Binary Icosahedral Group 39 4.1. A General Character Value Corollary 39 4.2. The Binary Icosahedral Group 40 References 42

Date: May 27, 2008.

Key words and phrases. reflection group, invariant theory, polynomial invariants, relative invariants, coinvariants, Brauer isomorphism, cyclic sieving phenomenon.

Work of first author supported by NSERC. Work of second author supported by NSF grant DMS-0245379. Work of fourth author supported by MSRI..

Let k be an arbitrary field, V an n-dimensional k-vector space, and G a finite subgroup of GL(V). Then G acts by linear change of coordinates on the algebra k[V] of polynomial functions on V. If x_{1}, …, x_{n} is a basis for the dual vector space V^{∗} we may identify k[V] with the polynomial algebra k[x_{1}, …, x_{n}] in x_{1}, …, x_{n} regarded as formal variables. The coinvariant algebra is the quotient algebra

k[V]/(k[V]_{+}^{G}) ≅ k[V] ⊗_{k[V]^{G}} k

where (k[V]_{+}^{G}) denotes the ideal of k[V] generated by the elements of the invariant subalgebra k[V]^{G} of strictly positive degree, and k = k[V]^{G}/k[V]_{+}^{G} is regarded as a trivial k[V]^{G}-module. The coinvariant algebra is a finite dimensional G-representation. Much of its significance (see, e.g., [4], [13], [25]) derives from the fact that in favorable cases it provides a graded version of the regular representation k(G). This is made precise in the following result, due to Chevalley (and also Shephard and Todd) in characteristic zero, and to Mitchell in positive characteristic.

Theorem(Chevalley [7], Shephard and Todd [24], Mitchell [17]). Let k be an arbitrary field, V an n-dimensional k-vector space, and G a finite subgroup of GL(V). Suppose that k[V]^{G} is a polynomial algebra. Then as k(G)-modules, the coinvariant algebra k[V] ⊗_{k[V]^{G}} k and the regular representation k(G) have the same composition factors counting multiplicities.

Something similar for relative invariants may be deduced under suitable hypotheses as we explain next. Suppose we have a second group Γ and a (k(Γ), k(G))-bimodule U, i.e., a right k(G)-module U which has a commuting left action of k(Γ). We chose to use the terminology of bimodules here as an aid in keeping the different actions distinct. If however no confusion can arise we frequently identify (k(Γ), k(G))-bimodules with left k(Γ × G)-modules; the left action of (γ, g) ∈ Γ × G on an element u ∈ U being given by γug^{−1}. The module M of U-relative invariants is defined by M := (U ⊗_{k} k[V])^{G} where G acts on the tensor product diagonally, viz., g(u ⊗ x) = ug^{−1} ⊗ gx. Note that M has the structure of a graded (k(Γ), k[V]^{G})-bimodule.

Corollary. Let k be a field, V = k^{n} an n-dimensional vector space over k, G a finite subgroup of GL(V), Γ a second finite group, and U a finitedimensional (k(Γ), k(G))-bimodule. Regard the relative invariants

M = (U ⊗_{k} k[V])^{G}

as a (k(Γ), k[V]^{G})-bimodule. If |G| is invertible in k and k[V]^{G} is polynomial algebra, then one has a k(Γ)-module isomorphism

M ⊗_{k[V]^{G}} k ≅ U.

Note that this corollary includes the nonmodular version of the Chevalley, Shephard-Todd, Mitchell Theorem as the special case where Γ = G and U = k(G) as a (k(G), k(G))-bimodule. More generally, if we let H be any subgroup of G and Γ = N_{G}(H), the normalizer of H in G, we obtain a k-linear representation U = k(H\G) from the permutation representation of G on the set H\G of right cosets of H in G. Regarded as a (k(Γ), k(G))- bimodule, the relative invariants M = (U ⊗_{k} k[V])^{G} become the subalgebra of H-invariant polynomials k[V]^{H} regarded as a k(N_{G}(H))-module (see §3.6 below). Certain other cases of relative invariant modules M appear frequently in the literature, such as the i^{th}-exterior power U = ∧^{i}(V^{∗}) of V^{∗} (resp. U = V), and M is the module of G-invariant differential i-forms (resp. G-invariant vector fields) on V, (see e.g., [28], [19, §6.1]), or, for a simple k(G)-module U where the Hilbert series of M ⊗_{k[V]^{G}} k defines the fake degrees for U (see e.g., [11, §1.6]).

Our first main result, Theorem 1.1.1 below, extends the Chevalley, Shephard–Todd, Mitchell result and its corollary by removing the hypothesis that k[V]^{G} be polynomial and |G| lie in k^{×}. Our other main results are inspired by a generalization of the Chevalley-Shephard-Todd Theorem due to Springer [29], which incorporates the action of an extra cyclic group. We recall this next.

Given a finite subgroup G ⊆ GL(V), say that v ∈ V is a regular vector if the orbit Gv is a regular orbit, meaning that the stabilizer in G of v is 1, or equivalently that the orbit achieves the maximum cardinality |Gv| = |G|. An element c ∈ G is a regular element if it has a regular eigenvector v ∈ V, after possibly extending the field k to include the corresponding eigenvalue ω ∈ k^{×}. Letting C = ⟨c⟩ denote the cyclic subgroup generated by c, the group algebra k(G) becomes a (k(G), k(C))-bimodule in which (g, c) acts on the basis element t_{h} of k(G) corresponding to h in G via (g, c) · t_{h} := t_{ghc}. As before we may identify (k(G), k(C))-bimodules with k(G × C)-modules by letting a left action of c ∈ C correspond to a right action of c^{−1}. We also let C act on k[V] by the algebra automorphisms which are a scalar multiplication in each degree and determined by requiring c^{j}(x_{i}) = ω^{j}x_{i} for i = 1, 2, …, n. In this way we obtain the structure of a (k(G), k(C))- bimodule on k[V] as well as k[V]⊗_{k[V]^{G}} k. The following theorem was proven by Springer in characteristic zero, and extended to arbitrary fields in [20].

Theorem(Springer [29], Reiner-Stanton-Webb [20]). Let k be a field, V a finite-dimensional k-vector space, G a finite subgroup G ⊂ GL(V). Suppose that k[V]^{G} is a polynomial algebra, and c is a regular element of G.

Let C = ⟨c⟩ as above. Then one has the equality

[k[V] ⊗_{k[V]^{G}} k] = [k(G)].

in R(k(G × C)), where R(k(G × C)) denotes the Grothendieck ring of finite dimensional (k(G), k(C))-bimodules.

We are concerned with extensions of these results to arbitrary groups in any characteristic. This will require significant reformulation since the naive versions of these results would not be correct. For example, a simple consequence of the Chevalley, Shephard–Todd, Mitchell result is that the coinvariant algebra for G has dimension |G| whenever k[V]^{G} is a polynomial algebra, but this fails when k[V]^{G} is not polynomial (see e.g., [26]).

The authors thank Radha Kessar, Gennady Lyubeznik, Ezra Miller, and Dennis Stanton for helpful conversations and stimulating questions.

1. Statement of Results

In this section we state our main results and illustrate them with a simple example. For background material on invariant theory see [4, 8, 25], on representation theory see [23], particularly Part III, and on reflection groups see [11, 13].

1.1. Chevalley, Shephard–Todd, Mitchell Type Results. We first indicate how to remove the hypothesis that k[V]^{G} be polynomial, and |G| lie in k^{×}, from the Chevalley, Shephard–Todd, Mitchell Theorem and its corollary. Our result, Theorem 1.1.1, compares the ungraded k(Γ)-module U with various graded k(Γ)-modules, showing how, in a sense to be made precise below,

(i) M ⊗_{k[V]^{G}} k is an overestimate for U, (ii) it is the first in a sequence of alternating overestimates and underestimates, and

(iii) these estimates converge to a suitably defined limit. To explain what this means, recall that k(Γ) is not in general semisimple, so one compensates for this by working with composition factors. A convenient way to do this is to introduce the Grothendieck ring R(k(Γ)) of finitedimensional k(Γ)-modules. This is defined to be the ring with one generator [M] for each isomorphism class {M} of finite dimensional k(Γ)-modules, and one relation [M′] − [M] + [M′′] = 0 for each short exact sequence

0 ⟶ M′ ⟶ M ⟶ M′′ ⟶ 0.

Addition in R(k(Γ)) is induced by direct sum and product by tensor product. There is also a partial ordering defined by requiring for two elements x, y ∈ R(k(Γ)) that x ≥ y if x − y = [M] for some (genuine) k(Γ)-module M. For example, given two genuine modules M_{1}, M_{2}, the inequality [M_{1}] ≥ [M_{2}] means that for every simple k(Γ)-module S one has the inequality of composition multiplicities [M_{1} : S] ≥ [M_{2} : S], where [M : S] denotes the number of occurrences of the simple module S as a composition factor in M.

Given a finite group Γ, a (non-negatively) graded k(Γ)-module is one with a direct sum decomposition M = ⊕_{d≥0}M_{d} in which each M_{d} is a finitedimensional k(Γ)-module. Such an M gives rise to an element [M](t) := ∑_{d≥0}[M_{d}]t^{d} in the formal power series ring over the Grothendieck ring R(k(Γ))[[t]] := ℤ[[t]] ⊗_{ℤ} R(k(Γ)). If one forgets the group action we obtain the formal power series ∑_{d≥0} dim_{k}(M_{d})t^{d} ∈ ℤ[[t]] often called the Hilbert series of M.

The motivating example for us of such a graded k(Γ)-module will be

⊕_{i≥0}Tor_{i}^{R}(M, k)

in the situation where R is a finitely generated graded, connected, commutative k-algebra with a grade-preserving action of Γ, and M is a finitely generated graded R-module with a compatible k(Γ)-module structure (see Section 2). These hypotheses imply that Tor_{i}^{R}(M, k) acquires a grading from R and M, and that each graded component Tor_{i}^{R}(M, k)_{j} is finite-dimensional over k. Furthermore, for each fixed i this component is non-zero for only finitely many j and for each fixed j the component is non-zero for only finitely many i. This means that ∑_{j}(−1)^{i}[Tor_{i}^{R}(M, k)_{j}]t^{j} is in fact a polynomial in t for each fixed i. We will also consider the infinite alternating sum ∑_{i≥0}(−1)^{i} [Tor_{i}^{R}(M, k)](t), which as a consequence of these finiteness properties is a well-defined element in R(k(Γ))[[t]]. This sum generalizes the multiplicity symbol of Serre from the case with no group action (see e.g.,

[22] and [27, §3]).

Theorem 1.1.1. Let k be a field, V a finite-dimensional k-vector space, G a finite subgroup of GL(V), and Γ a finite group. Let U be a finitedimensional (k(Γ), k(G))-bimodule, and let M := (U ⊗_{k} k[V])^{G}, regarded as a (k(Γ), k[V]^{G})-bimodule. Set K := k(V)^{G}, the field of G-invariant rational functions on V (the ungraded field of fractions of k[V]^{G}).

(i) In R(k(Γ)), one has the inequality

[M ⊗_{k[V]^{G}} k] ≥ [U].

Furthermore, one has equality if and only if M is k[V]^{G}-free, in which case K ⊗_{k} U has a K(Γ)-module filtration {F_{j}} for which the factor F_{j}/F_{j−1} is isomorphic to the j^{th} homogeneous component (M ⊗_{k[V]^{G}} k)_{j} ⊗_{k} K. (ii) More generally, for any m ≥ 0, in R(k(Γ)), one has the inequality

∑_{i=0}^{m}(−1)^{i} [Tor_{i}^{k[V]^{G}}(M, k)] \{ \begin{matrix}≥ [U] & if m is even, \\ ≤ [U] & if m is odd, \end{matrix}

with equality if and only if Tor_{i}^{k[V]^{G}}(M, k) vanishes for i > m.

(iii) The element ∑_{i=0}^{∞}(−1)^{i} [Tor_{i}^{k[V]^{G}}(M, k)] (t) of R(k(Γ))[[t]] lies in

the subring ℚ(t) ⊗_{ℤ} R(k(Γ)) where each simple module S has composition multiplicity series lying in ℚ(t). Furthermore, t = 1 is a regular value of this rational function and

∑_{i≥0}(−1)^{i} [Tor_{i}^{k[V]^{G}}(M, k)] (t)|_{t=1} = [U] in R(k(Γ)).

Theorem 1.1.1 is proven in Section 2.5, using a homological strengthening of Chevalley’s method from [7], relying ultimately on the Normal Basis Theorem from Galois theory. We illustrate it here with a simple example.

Example 1. Let G be the cyclic group ℤ/2 of order 2 regarded as the subgroup of GL(2, ℂ) generated by the scalar matrix g = −I_{2×2} which is the negative of the identity. Note that the ring of invariants is not a polynomial algebra, rather

R := ℂ[x, y]^{ℤ/2} = ℂ[x^{2}, xy, y^{2}].

There are two simple ℂ(ℤ/2)-modules, U_{+} and U_{−}, both 1-dimensional, with g acting by the scalar +1, −1 on U_{+}, U_{−}, respectively.

Choose Γ to be the trivial subgroup {1} of Aut_{ℂ(ℤ/2)}(U_{±}). In this case, the Grothendieck ring R(ℂ(Γ)) is isomorphic to ℤ, with the isomorphism sending the class [1] of the trivial 1-dimensional ℂ(Γ)-module to the integer 1. Any ℂ-vector space of dimension d then represents the element [ℂ^{d}] = d[1] in R(ℂ(Γ)). In particular, [U_{+}] = [U_{−}] = [1] in R(ℂ(Γ)).

One can easily check that in this case

M_{+} := (U_{+} ⊗_{ℂ} ℂ[x, y])^{ℤ/2} = ℂ[x, y]^{ℤ/2} = R M_{−} := (U_{−} ⊗_{ℂ} ℂ[x, y])^{G} = Rx + Ry.

Here M_{+} is a free R-module of rank 1, and all inequalities asserted in Theorem 1.1.1 become trivial equalities. By contrast, M_{−} has an interesting, infinite, 2-periodic^{1} R-free resolution

⋯ ⟶^{d_{4}} R(−7)^{2} ⟶^{d_{3}} R(−5)^{2} ⟶^{d_{2}} R(−3)^{2} ⟶^{d_{1}} R(−1)^{2} ⟶^{d_{0}} M_{−} → 0,

in which R(−d) denotes a free R-module of rank 1 having a basis element of degree d. Here the differential d_{0} maps the two basis elements of R(−1)^{2} onto x, y in M_{−}, while the differentials d_{i} for i ≥ 1 can be chosen as follows:

d_{i} = [ \begin{matrix}x^{2} & xy \\ xy & y^{2} \end{matrix} ] for even i, d_{i} = [ \begin{matrix}y^{2} & −xy \\ −xy & x^{2} \end{matrix} ] for odd i.

R = k[V]^{G} ≅ k[f_{1}, …, f_{n}, f_{n+1}]/(h)

1In fact, whenever R is a hypersurface, i.e., for a single homogeneous relation h among the f_{i}’s, there will always be such an R-free resolution of M which is eventually 2-periodic (see e.g., [31, §6] or [10]).

For any m ≥ 0, this gives the following strict inequalities in R(ℂ(Γ)) ≅ ℤ

[∑_{i=0}^{m}(−1)^{i}Tor_{i}^{R}(M_{−}, k)] = [ℂ^{2}]= 2[1]> [1]= [U_{−}] if m is even, [0] = 0[1]< [1]= [U_{−}] if m is odd,

as predicted by Theorem 1.1.1 (i, ii). In the limit as m → ∞, one makes sense of this by noting that Tor_{i}^{R}(M_{−}, ℂ)_{j} vanishes unless the internal degree j and homological degree i satisfy j = 2i + 1. Hence one can calculate in R(ℂ(Γ))[[t]] ≅ ℤ[[t]] that

\begin{aligned}∑_{i≥0}(−1)^{i} [Tor_{i}^{R}(M_{−}, ℂ)] (t) & = 2[1]t^{1} − 2[1]t^{3} + 2[1]t^{5} − 2[1]t^{7} + ⋯ \\ & = \frac{2t}{1 + t^{2}} [1]. \end{aligned}

This is a rational function of t with t = 1 as a regular value, and upon substituting t = 1, one obtains in R(ℂ(Γ))

∑_{i≥0}(−1)^{i} [Tor_{i}^{R}(M_{−}, ℂ)] (t)|_{t=1} = \frac{2t}{1 + t^{2}} [1]|_{t=1} = \frac{2}{2}[1] = [1] = [U_{−}]

as predicted by Theorem 1.1.1(iii).

1.2. Springer-Type Results. Let k be a field, V a finite-dimensional k-vector space, G a finite subgroup of GL(V). Suppose that k[V]^{G} is a polynomial algebra, and c is a regular element of G. As with the Chevalley, Shephard-Todd, Mitchell Theorem, we wish to deduce a more general version of Springer’s Theorem that applies to an arbitrary (k(Γ), k(G))- bimodule U for any finite group Γ. The k(Γ)-module of relative invariants M := (U ⊗_{k} k[V])^{G} carries a commuting C-action in which C acts on k[V] by scalars as before, and does nothing to the factor of U in U ⊗_{k} k[V]. In this way M becomes a graded (k(Γ), k(C))-bimodule. It is also possible to view U itself as a (k(Γ), k(C))-bimodule in a different way, namely with the action of C coming as the restriction of the action of G, of which C is a subgroup. From the theorem of Springer, it is not hard to deduce that if k[V]^{G} is a polynomial algebra and both |G|, |Γ| lie in k^{×}, then as (k(Γ), k(C))-bimodules,

M ⊗_{k[V]^{G}} k ≅ U.

Our next main result shows how to remove the hypothesis that |G|, |Γ| lie in k^{×}.

Theorem 1.2.1. Let k be any field, V a finite-dimensional k-vector space of k and G a finite subgroup of GL(V) with k[V]^{G} a polynomial algebra. Let C = ⟨c⟩ be the cyclic subgroup generated by a regular element c in G with regular eigenvalue ω and U a finite-dimensional (k(Γ), k(G))-bimodule for some finite group Γ. Regard M := (U ⊗_{k} k[V])^{G} as a k[V]^{G}-module and as a (k(Γ), k(C))-bimodule, where C acts on k[V] via scalar multiplication determined by c^{j}(x_{i}) = ω^{j}x_{i} for i = 1, 2, …, n and x_{1}, …, x_{n} is a basis for V^{∗}.

Then

∑_{i≥0}(−1)^{i} [Tor_{i}^{k[V]^{G}}(M, k)] = [U].

in R(k(Γ × C)); the sum being finite since k[V]^{G} is a polynomial algebra.

Proving Theorem 1.2.1 (in particular, with no hypothesis on the field k) was one of our original motivations. It easily implies our main application, Corollary 1.2.2 below, which resolves in the affirmative both Conjecture 3 and Question 4 in [20].

Corollary 1.2.2. Let k be a field, V a finite-dimensional k-vector space over k and H ⊂ G two nested finite subgroups of GL(V). Assume k[V]^{G} is a polynomial algebra, and let C = ⟨c⟩ be the cyclic subgroup generated by a regular element c in G, with eigenvalue ω on some regular vector in V and ω̂ ∈ ℂ a complex lift of ω. Let Γ := N_{G}(H)/H, where N_{G}(H) denotes the normalizer of H in G. Give k[V]^{H} the structure of a (k(Γ), k(C))-bimodule in which c scales the variables x_{1}, …, x_{n} in V^{∗} by ω, and Γ acts by linear substitutions. Let k(H\G) be the k-vector space with basis the right costs of H in G regarded as a (k(Γ), k(C))-bimodule where

γ · Hg · c := γHgc = Hγgc.

Then in R(k(Γ × C)) one has the equality

∑_{i≥0}(−1)^{i} [Tor_{i}^{k[V]^{G}}(k[V]^{H}, k)] = [k(H\G)] .

Ignoring the Γ-action, this implies that the quotient of Hilbert series

X(t) := \frac{[k[V]^{H}](t)}{[k[V]^{G}](t)},

is a polynomial in t, and together with the C-action on the set X = H\G, gives a triple (X, X(t), C) that exhibits the cyclic sieving phenomenon of [21]: Namely, for each element c^{j} in C the cardinality of the fixed point set X^{c^{j}} ⊂ X is given by evaluating X(t) at the complex root-of-unity ω̂^{j} of the same multiplicative order as c^{j}. In other words,

|X^{c^{j}}| = [X(t)]_{t=ω̂^{j}} .

1.3. A Further Generalization. Theorem 1.2.1 follows from the more general Theorem 1.3.1, which does not make the assumption that k[V]^{G} is polynomial and has other applications. The guiding principle in this generalization is to replace the condition that k[V]^{G} be a polynomial algebra with the assumption it contains a Noether normalization R ⊆ k[V]^{G} fulfilling certain key technical conditions. Recall that a Noether normalization for k[V]^{G} is a polynomial subalgebra R ⊂ k[V]^{G} over which k[V]^{G} is a finitely generated module. To state our result requires some preliminaries.

Let V be a (k(G), k(C))-bimodule with G and C finite groups, so that C acts on k[V] and k[V]^{G}. Suppose there exists a homogeneous and Cstable Noether normalization R ⊂ k[V]^{G} ⊂ k[V], with the added property that the fiber Φ_{v} := ϕ^{−1}(ϕ(v)) over the point ϕ(v) in the ramified covering V → Spec(R) has free (but not necessarily transitive) G-action, and is stable under C. This means that C preserves the tower of inclusions

𝔪_{ϕ(v)} ⊂ R ⊂ k[V]^{G} ⊂ k[V],

where 𝔪_{ϕ(v)} is the (generally inhomogeneous) maximal ideal in R corresponding to ϕ(v) in Spec(R). The quotient ring k[V]/𝔪_{ϕ(v)}k[V] =: A(Φ_{v}) can be thought of as the coordinate ring for the fiber Φ_{v} regarded as a (possibly non-reduced) subscheme of the affine space V. This ring A(Φ_{v}) carries an interesting (k(G), k(C))-bimodule structure, whose precise description we defer until §3.5 where the extra generality is exploited.

However, it is worth mentioning here what this bimodule structure looks like under the hypotheses of Theorem 1.2.1, that is, if k[V]^{G} is polynomial, so that we can choose R = k[V]^{G}, and where c is a regular element of G with eigenvalue ω on a regular eigenvector v. In this special case, if we take for C the group of scalar matrices in GL(V) generated by ω times the identity matrix, then A(Φ_{v}) ≅ k(G) carries the same (k(G), k(C))-bimodule structure as was described on k(G) in Springer’s theorem (Theorem 1.2.1).

Back in the general setting, given a (k(Γ), k(G))-bimodule U, one lets C act trivially on U and Γ act trivially on k[V]. In this way, the relative invariants M := (U ⊗_{k} k[V])^{G} carry the structure of a graded (k(Γ), k(C))- bimodule, compatible with its R-module structure. Similarly (U ⊗_{k} A(Φ_{v}))^{G} carries the structure of a (k(Γ), k(C))-bimodule in which Γ acts only on the U factor, and C acts only on the A(Φ_{v}) factor.

Theorem 1.3.1. Let k be a field, G, Γ, and C finite groups, and V a finite dimensional (k(G), k(C))-bimodule on which G acts faithfully (so G ⊂ GL(V)). Regard V, k[V] and k[V]^{G} as trivial k(Γ)-modules. Suppose there is a Noether normalization R ⊂ k[V]^{G} that is stable under the action of C on k[V]. Suppose further that there is a vector v in V such that the fiber Φ_{v} := ϕ^{−1}(ϕ(v)) containing v for the map ϕ : V → Spec(R) both carries a free (but not necessarily transitive) G-action and is stable under C. Denote by 𝔪_{ϕ(v)} the maximal ideal in R corresponding to ϕ(v) in Spec(R) and set A(Φ_{v}) = k[V]/𝔪_{ϕ(v)}k[V] which is a k(C)-module. Let U be a finite-dimen-sional (k(Γ), k(G))-bimodule regarded as a trivial k(C)-module.

Then the relative invariants M := (U ⊗_{k} k[V])^{G} satisfy

∑_{i≥0}(−1)^{i} [Tor_{i}^{R}(M, k)] = [(U ⊗_{k} A(Φ_{v}))^{G}]

in R(k(Γ × C)); the sum being finite since R is a polynomial algebra.

From this we easily deduce Theorem 1.2.1 in Section 3.5. Section 4 illustrates a different application of Theorem 1.3.1, to character values of the binary icosahedral group.

2. Generalities for Rings, Modules, and Tor

We record here some issues surrounding Tor^{R}(M, k) and the action of a group on R-resolutions of M. Without the group action most of these results can be found in [27]. Because the group varies in the applications we denote it by the new symbol G. It will always be assumed throughout Section 2 that

(i) R = ⊕_{d≥0}R_{d} is a commutative, ℕ-graded, connected (R_{0} = k), Noetherian k-algebra (i.e., finitely generated as an algebra over k) and (ii) G is a group which acts on R by graded k-algebra automorphisms.

In addition we will consider finitely generated R-modules with a compatible homogeneous action of G in a sense conveniently described in terms of the skew group algebra R ⋊ G (see e.g., [2]). This is the free R-module with elements {t_{g}}_{g∈G} indexed by G as a basis, and whose multiplication is determined by the rule rt_{g} · st_{h} = r · g(s)t_{gh} and bilinearity, for all r, s ∈ R and g, h ∈ G. We put a grading on R ⋊ G by requiring that an element rt_{g} has the same degree as r. An R ⋊ G-module is the same thing as an R-module M with an action of G on M regarded as a graded abelian group by grading preserving group endomorphisms satisfying g(rm) = g(r)g(m) and g_{1}(g_{2}m) = (g_{1}g_{2})m. This is what we mean by a compatible action of G. So the third standing assumption in this section is that

(iii) M = ⊕_{d≥0}M_{d} is an ℕ-graded R ⋊ G-module which is Noetherian

(i.e., finitely generated) as an R-module. The kind of graded R ⋊ G-modules we will consider arises, for example

in the situation discussed in the introduction: We have a (k(G), k(C))- bimodule V, so that C acts on k[V], k[V]^{G} and possibly also on a Noether normalization R ⊂ k[V]^{G}, as well as compatibly on the R-module M := (U ⊗_{k} k[V])^{G} for any (k(Γ), k(G))-bimodule U. In fact putting G = Γ × C, M becomes an R ⋊ G-module.

2.1. Review of Graded Resolutions. Recall that, ignoring group actions, there always exist graded R-free resolutions F of M in which all terms are finitely generated, that is, an exact sequence

⋯ →^{d_{i+1}} F_{i} →^{d_{i}} F_{i−1} →^{d_{i−1}} ⋯ F_{1} →^{d_{1}} F_{0} →^{d_{0}} M → 0

with each F_{i} a graded free R-module of finite rank, and grade-preserving differentials d_{i}. From any such resolution one can compute the bigraded kvector space Tor^{R}(M, N) = {Tor_{i}^{R}(M, N)_{j}} for any graded R-module N, by taking the homology of the tensored complex F⊗_{R}N. For i ∈ ℕ Tor_{i}^{R}(M, N) is a graded k-vector space, the index i is called the homological grading, and the grading on Tor_{i}^{R}(M, N), namely the index j, is called the internal grading. Depending on the context we will use the notation Tor^{R}(M, N) for the bigraded Tor-functor, or its ungraded analog obtained by taking the direct sum of the homogeneous components {Tor_{i}^{R}(M, N)_{j}} .

It is possible to choose the resolution F to be minimal in the sense that the ranks β_{i} of the resolvents F_{i} ≅ R^{β_{i}} are simultaneously all minimized; this turns out to be equivalent to each differential d_{i} having entries in R_{+} = ⊕_{i>0}R_{i}. In particular, when N = k = R/R_{+} is the trivial R-module, if the complex is minimal then F ⊗_{R} N becomes a complex of k-vector spaces with all zero differentials, showing that

β_{i} = dim_{k} Tor_{i}^{R}(M, k).

The length of a minimal resolution is called the homological dimension hd_{R}(M), that is, hd_{R}(M) := min{i : Tor_{i}^{R}(M, k) ≠ 0}. Note that hd_{R}(M) need not be finite. However, Hilbert’s syzygy theorem asserts that when R is a polynomial algebra on n generators, one always has hd_{R}(M) ≤ n.

Given an ℕ-graded k-vector space U = ∑_{d≥0} U_{d}, let

start(U) := min{d : U_{d} ≠ 0}.

The usual construction of a minimal free R-resolution F of M shows that it enjoys the property start(F_{i+1}) > start(F_{i}). We will show in Section 2.3 that a similar property holds after incorporating a finite group action.

2.2. The Group Action on Tor. We start by pointing out that for R ⋊Gmodules M and N (where G is a group which acts on R by graded kalgebra automorphisms) there are diagonal actions of R and G on M ⊗_{R} N and also on Tor_{i}^{R}(M, N) making them into R ⋊ G-modules. Since R is commutative, we allow ourselves to take the tensor product of two left Rmodules. We claim that for each g ∈ G the map M ×N → M ⊗_{R} N given by (m, n) ↦ g(m) ⊗ g(n) is R-balanced. To establish this we must show that for each r ∈ R we have g(rm) ⊗ g(n) = g(m) ⊗ g(rn). This is so because

g(rm) ⊗ g(n) = g(r)g(m) ⊗ g(n) = g(m) ⊗ g(r)g(n) = g(m) ⊗ g(rn).

From this we obtain the diagonal action of R ⋊ G on M ⊗_{R} N.

In fact for each g ∈ G we have a natural transformation from the functor

− ⊗_{R} N : R ⋊ G-mod → R-mod

to itself, giving a functor

− ⊗_{R} N : R ⋊ G-mod → R ⋊ G-mod.

We next show that the diagonal action of G extends to an action on Tor_{i}^{R}(M, N) for i > 0. Regard Tor_{i}^{R}(−, N) as a functor R ⋊ G-mod → R-mod. For each g ∈ G and i ∈ ℕ we construct a natural transformation η_{g,i} from Tor_{i}^{R}(−, N) to itself so that these maps also commute with the boundary maps in the long exact sequences which arise from any short exact sequence of R ⋊ G-modules 0 → M_{1} → M_{2} → M_{3} → 0. For i = 0 the natural transformation will be the one already constructed. To extend this for arbitrary i, we may take any complex of projective R ⋊ G-modules P = (⋯ → P_{2} → P_{1} → P_{0} → 0) which is acyclic except in degree zero where its homology is M. Since R ⋊ G is free as an R-module this is also a projective resolution of M as an R-module, so Tor_{i}^{R}(M, N) = H_{i}(P ⊗_{R} N). As the action of G is by natural transformations of the functor − ⊗_{R} N it passes to an action on the complex P ⊗_{R} N and hence to an action on its homology. The verification that this action commutes with the boundary homomorphisms is a standard argument in homological algebra.

We finally observe that if higher natural transformations exist with these properties, they must be unique. This follows from a homological degree shifting argument (cf [6, Chapter III]), since given any R ⋊ G-module M one may take a short exact sequence 0 → K → P → M → 0 where P is a projective R ⋊ G-module. This gives a long exact sequence

0 = Tor_{i}^{R}(P, N) → Tor_{i}^{R}(M, N) → Tor_{i−1}^{R}(K, N) → Tor_{i−1}^{R}(P, N) → ⋯

and the specification of η_{g,i−1} on Tor_{i−1}^{R}(K, N) and Tor_{i−1}^{R}(P, N) (which is only non-zero for i = 1) determine the specification of η_{g,i} on Tor_{i}^{R}(M, N) since these maps commute with the connecting homomorphisms. Then the uniqueness implies that Tor_{i}^{R}(M, N) becomes an R ⋊ G-module, since the relations on the action of G which are needed for this hold if i = 0, hence also for higher values of i.

2.3. The Group Action on Resolutions. Given these preliminaries we can state and prove our first rationality result. This proposition can be regarded as an equivariant generalization of the theorem of Hilbert–Serre (see e.g., [27, Theorem 4.2]) on the rationality of Poincaré series of graded Noetherian modules over Noetherian k-algebras.

Proposition 2.3.1. Let R be a commutative ℕ-graded Noetherian k algebra, G is a group which acts on R by graded k-algebra automorphisms, and M a Noetherian R ⋊ G-module.

(i) There exists an R-resolution F = {F_{i}}_{i≥0} of M in which each resolvent F_{i} is not only a free R-module but also a k(G)-module, the maps are k(G)-module morphisms, and start(F_{i+1}) > start(F_{i}). Consequently, the infinite sum

∑_{i≥0}(−1)^{i}[Tor_{i}^{R}(M, k)](t)

gives rise to a well-defined element in R(k(G))[[t]].

(ii) When k(G) is semisimple (i.e., |G| ∈ k^{×}), the resolution in (i) can in addition be chosen minimal as an R-resolution.

(iii) If hd_{R}(M) is finite, the resolution in (i) or in (ii) can in addition be chosen with length hd_{R}(M).

(iv) In R(k(G))[[t]], the series [R](t) is invertible, and one has the following relation:

∑_{i≥0}(−1)^{i}[Tor_{i}^{R}(M, k)](t) = \frac{[M](t)}{[R](t)}.

(v) All three series

[R](t), [M](t), ∑_{i≥0}(−1)^{i}[Tor_{i}^{R}(M, k)](t)

in R(k(G))[[t]] actually lie in the subring ℚ(t) ⊗_{ℤ} R(k(G)). That is, for each simple k(G)-module S, the power series which is the coefficient of [S] actually lies in ℚ(t).

(vi) For each simple k(G)-module S, the coefficient series of [S] in [M](t) and [R](t) have poles at t = 1 of order at most the Krull dimension of R.

Proof. (i): Because M is finitely-generated as an R-module, the quotient M/R_{+}M is a finite-dimensional, graded k-vector space. Any homogeneous k-basis {m̄_{j}} for M/R_{+}M lifts to a minimal homogeneous generating set {m_{j}} for M as an R-module, by the graded version of Nakayama’s Lemma. Let U be the k(G)-submodule of M generated by any choice of such lifts {m_{j}}. Then U is a graded finite-dimensional subspace of M because G is finite and acts in a degree-preserving fashion on M. Start a resolution F with the surjection

F_{0} := R ⊗_{k} U ⟶^{d_{0}} M , where r ⊗ u ⟼ ru.

Observe that the diagonal action of k(G) on F_{0} := R ⊗_{k} U is required here, both to make d_{0} a k(G)-module morphism, and to make the R-module structure on F_{0} compatible with the k(G)-module structure of R. Observe also that U ≅ k ⊗_{R} F_{0}, a relationship which will be used in proving (iv).

Replacing M by ker(d_{0}), we can iterate this process, and produce the desired resolution F, provided we can show that the inequality

start(ker(d_{0})) > μ := start(M) = start(F_{0})

holds. However, this follows easily from the observation that the restriction of the above map d_{0} to the μ^{th} homogeneous components is the k-vector space isomorphism

R_{0} ⊗_{k} U_{μ} = k ⊗_{k} U_{μ} ⟶^{d_{0}} U_{μ} = M_{μ}, where 1 ⊗ u ⟼ u,

and hence ker(d_{0}) is nonzero only in degrees strictly larger than μ.

(ii): If k(G) is semisimple, then in the construction of F in (i), the k(G)- submodule U ⊂ M is a direct summand so there is a k(G)-module direct sum decomposition M = U ⊕ R_{+}M. Since U ≅ M/R_{+}M, iterating this construction will produce a minimal resolution.

(iii): This we prove by induction on hd_{R}(M). In the base case, i.e., where hd_{R}(M) = 0, M is a free R-module and the assertion is trivial. In the inductive step, note that after one step of the construction in (i) or (ii), there is a short exact sequence

0 → ker(d_{0}) → F_{0} → M → 0

whose long exact sequence in Tor^{R}(−, k) shows that

hd_{R}(ker(d_{0})) = hd_{R}(M) − 1.

Applying the inductive hypothesis to ker(d_{0}) gives the result.

(iv): The fact that [R](t) lies in R(k(G))[[t]]^{×} follows from the assumption that R is a connected k-algebra with G acting trivially on R_{0} = k. This means that the series expansion of [R](t) begins [1] + [R_{1}]t^{1} + [R_{2}]t^{2} + ⋯, and [1] is a unit of R(k(G)).

For the remaining assertion, start with a resolution F of M produced as in (i). There is the following string of equalities in R(k(G))[[t]], which are justified below.

[M](t) = ∑_{i≥0}(−1)^{i}[F_{i}](t)

= ∑_{i≥0}(−1)^{i}[R ⊗_{k} (k ⊗_{R} F_{i})](t)

= ∑_{i≥0}(−1)^{i}[R](t) · [k ⊗_{R} F_{i}](t)

= [R](t) ∑_{i≥0}(−1)^{i} · [k ⊗_{R} F_{i}](t)

= [R](t) ∑_{i≥0}(−1)^{i} · [Tor_{i}^{R}(M, k)](t)

The first equality comes from looking at the Euler characteristic for the (finite) exact sequence in each homogeneous component.

The second equality comes from the fact that F_{i} is constructed as R⊗_{k} U_{i} and U_{i} ≅ k ⊗_{R} F_{i}, so F_{i} ≅ R ⊗_{k} (k ⊗_{R} F_{i}).

The third equality comes from the fact that in the isomorphism F_{i} ≅ R⊗_{k} (k ⊗_{R} F_{i}) the action of G on the tensor product on the right is diagonal, and this tensor product defines the product in R(k(G))[[t]].

The fourth equality is trivial.

The fifth equality holds because Tor^{R}(M, k) is the homology of the complex F ⊗_{R} k. In each homogeneous component (F ⊗_{R} k)_{d} one has a finite complex of finite-dimensional k-vector spaces, and the alternating sum F_{i} ⊗_{R} k over i represents the same element in R(k(G)) as the alternating sum of the Tor_{i}^{R}(M, k)_{d} in that component (i.e., taking homology preserves Euler characteristics).

(v): It suffices to prove the assertion for [M](t); one then takes M = R to deduce it for [R](t), and uses (iv) to deduce it for ∑_{i≥0}(−1)^{i}[Tor_{i}^{R}(M, k)](t).

To prove [M](t) is invertible, one can reduce to the case where R is a polynomial algebra A = k[f_{1}, …, f_{n}] with trivial G-action as follows. Note that since R is Noetherian, by a result of Emmy Noether R^{G} is also Noetherian. Hence by the Noether Normalization Lemma, R^{G} contains a homogeneous system of parameters f_{1}, …, f_{n}, and we put A = k[f_{1}, …, f_{n}]. The ring extensions A ↪ R^{G} ↪ R are both integral (i.e. module-finite), and hence M is also a finitely generated A-module.

In this case, one can apply (iv) to give

(2.3.1)  [M](t) = [∑_{i≥0}(−1)^{i}[Tor_{i}^{R}(M, k)](t)] · [A](t).

Note that the sum on the right is finite, i.,e., lies in R(k(G))[t], because Hilbert’s syzygy theorem says Tor^{A}(M, k) is finite dimensional. Note also that because G acts trivially on A, one has

[A](t) = (∏_{i=1}^{n} \frac{1}{1 − t^{deg(f_{i})}} ) [1],

which is an element of ℚ(t) times the class [1] of the trivial module. Hence [M](t) has ℚ(t) coefficients.

(vi): Again it suffices to prove it for [M](t), and then take M = R to deduce it for [R](t). For [M](t) it is implied by equation (2.3.1) and the comments after it, as the sum ∑_{i≥0}(−1)^{i}[Tor_{i}^{R}(M, k)](t) has ℤ[t] coefficients and the pole at t = 1 in [A](t) is the Krull dimension of A. This is the same as the Krull dimension of R since A ↪ R is an integral extension. □

2.4. A Short Review of Brauer theory. At several points we will need facts about the Grothendieck ring R(k(Γ)) for a finite group Γ which can be conveniently deduced from the theory of Brauer characters. We review this theory here and refer to [23, Part III] for details.

Let p denote the characteristic of the ground field k. We say that an element γ in Γ is p-regular if its order lies in k^{×}. Let m be the least common multiple of the orders of all p-regular elements of Γ, and let ζ be a primitive m^{th} root of unity in some extension field of k. Then every pregular element γ in Γ acting on a finite-dimensional k(G)-module U has all its eigenvalues among the group of m^{th} roots of unity μ_{m}(k(ζ)^{×}) = ⟨ζ⟩. Pick a primitive complex m^{th} root of unity ζ̂ ∈ ℂ which will lift ζ, and consider the resulting homomorphism which lifts m^{th} roots of unity from k(ζ) to ℂ:

\begin{matrix}μ_{m}(k(ζ)^{×}) & ⟶^{lift} μ_{m}(ℂ^{×}) \\ ζ^{j} & ⟼ ζ̂^{j}. \end{matrix}

The Brauer character value χ_{U}(γ) for γ is then defined to be the sum of the lifts of the eigenvalues of g on U. Furthermore the field k(ζ) is a splitting field for Γ by a theorem of Brauer (see e.g., [23, §12.3, Theorem 24]).

The Brauer character χ_{U} of a k(Γ)-module U determines the composition factors of U, and this has several important consequences. To begin with, the collection of restriction homomorphisms

R(k(Γ)) ⟶ R(k⟨γ⟩),

where γ ranges over all p-regular elements γ, determines elements of R(k(Γ)) uniquely; that is, the map R(k(Γ)) → ⨁_{γ} R(k⟨γ⟩) is injective. It implies

also that whenever one has a field extension k ↪ K, the map R(k(Γ)) →^{ψ_{k,K}} R(K(Γ)) that is induced by extension of scalars U ↦ K ⊗_{k} U is injective, since the Brauer character of a module remains the same after extending scalars. So to prove an equality in R(k(Γ)) it suffices to prove the equality in R(k⟨γ⟩) for the p-regular elements γ in Γ.

If γ ∈ Γ is a p-regular element then k⟨γ⟩ is semisimple. Over a splitting field k(ζ) the simple k⟨γ⟩-modules U_{j} are all 1-dimensional and are indexed by j ∈ ℤ/dℤ, where d is the order of γ, with γ acting as the scalar ω^{j} for some primitive d^{th} root of unity ω in k^{×}. An element in R(k⟨γ⟩) is determined by the (virtual) composition multiplicities of each U_{j}. If this element is of the form [U] for some genuine k(Γ)-module U, then [U] will be determined by the dimensions dim_{k}(U ⊗_{k} U_{j})^{⟨γ⟩} of its U_{j}-isotypic components.

Observe also that if one has a commuting action of another finite group Γ′, and one wants to prove an equality in R(k(Γ × Γ′)), it suffices to prove it in R((k⟨γ⟩×Γ′)) for each p-regular γ ∈ Γ. Furthermore, this can be done for genuine k(Γ × Γ′)-modules by proving equality in R(k(Γ′)) for each isotypic component.

2.5. The Case of Domains with Trivial Group Action. We assume the notations introduced in Proposition 2.3.1, and in addition require that the graded k-algebra R be an integral domain on which Γ-acts trivially. We let K be the fraction field of R and recall from the discussion of Section 2.4 that extension of scalars gives an inclusion of Grothendieck rings ψ_{k,K} : R(k(Γ)) → R(K(Γ)). We will make various assertions about isomorphism of K(Γ)-modules, but where these modules are in fact defined over k we may also deduce a corresponding result for k(Γ) modules which we leave to the reader to formulate.

The main result in this section is an abstract version of Theorem 1.1.1, from which Theorem 1.1.1 will immediately follow. Before we state it, we present a lemma which will be needed in the proof.

Lemma 2.5.1. Let M be a finitely generated graded R(Γ)-module, where R is a commutative graded Noetherian k-algebra on which Γ acts trivially. Assume that R is an integral domain and let R′ ⊆ R be a graded subring over which R is integral. Let the fields of fractions of R and R′ be K and K′, respectively. Then the map

φ : K′ × M ⟶ K ⊗_{R} M

sending (a, m) ∈ K′ × M to a ⊗_{R} m ∈ K ⊗_{R} M induces an isomorphism of K′(Γ)-modules

K′ ⊗_{R′} M → K ⊗_{R} M .

Note that here K is regarded as a (K′, R)-bimodule, and may even be regarded as a (K′(Γ), R(Γ))-bimodule, on which Γ acts trivially.

Proof. Note that the map φ is R′-balanced simply because R′ ⊂ R. Hence it induces a well-defined map φ : K′ ⊗_{R′} M → K ⊗_{R} M, which one can see is K′-linear and even a K′(Γ)-morphism.

It remains to show that it is a K′-vector space isomorphism, which is facilitated by first observing that the K′-dimensions of the domain and range are the same, viz.,

dim_{K′} (K′ ⊗_{R′} M) = rank_{R′}(M)
= rank_{R′}(R) · rank_{R}(M)
= [K : K′] · dim_{K} (K ⊗_{R} M)
= dim_{K′} (K ⊗_{R} M) .

Thus it suffices to show φ is surjective. For this one need only check for any s, r ∈ R with r ≠ 0, and any m ∈ M, that the decomposable tensor \frac{s}{r} ⊗_{R} m is in the image of φ. For this we use that R is integral over R′, so there is a dependence

r^{n} + a_{n−1}r^{n−1} + ⋯ + a_{2}r^{2} + a_{1}r + a_{0} = 0

with a_{i} ∈ R′. We may assume a_{0} ≠ 0, since one can divide by r in the domain R. Let

m′ := −(r^{n−1} + a_{n−1}r^{n−2} + ⋯ a_{2}r + a_{1})m,

which is an element of M, satisfying

r · m′ = a_{0} · m.

Hence

\frac{s}{r} ⊗_{R} m = \frac{s}{ra_{0}} ⊗_{R} a_{0}m = \frac{s}{ra_{0}} ⊗_{R} rm′ = \frac{1}{a_{0}} ⊗_{R} sm′ = φ (\frac{1}{a_{0}} ⊗_{R′} sm′)

lies in the image of φ. □

Theorem 2.5.2. Let M be a finitely generated graded R(Γ)-module, where R is a commutative graded connected Noetherian k-algebra on which Γ acts trivially. Assume in addition that R is an integral domain.

(i) In R(K(Γ)) we have

[K ⊗_{k} (M ⊗_{R} k)] ≥ [K ⊗_{R} M].

Furthermore, one has equality if and only if M is R-free, in which case the K(Γ)-module K ⊗_{R} M has a filtration

0 = F_{−1} ⊆ F_{0} ⊆ F_{1} ⊆ ⋯ ⊆ F_{d} = K ⊗_{R} M

so that for each j there is an isomorphism of K(Γ)-modules

K ⊗_{k} (M ⊗_{R} k)_{j} → F_{j}/F_{j−1}

where (M ⊗_{R} k)_{j} denote the j^{th} homogeneous component of M ⊗_{R} k. (ii) More generally, for any m ≥ 0, in R(k(Γ)),

∑_{i=0}^{m}(−1)^{i} [K ⊗_{k} Tor_{i}^{R}(M, k)] \{ \begin{matrix}≥ [K ⊗_{R} M] & if m is even, \\ ≤ [K ⊗_{R} M] & if m is odd, \end{matrix}

and equality holds if and only if hd_{R}(M) ≤ m, that is, if and only if Tor_{i}^{R}(M, k) vanishes for i > m.

(iii) Even if hd_{R}(M) is not finite, the formal power series

∑_{i≥0}(−1)^{i} [K ⊗_{k} Tor_{i}^{R}(M, k)] (t)

of R(K(Γ))[[t]] is a rational function of t so lies in ℚ(t)⊗_{ℤ}R(K(Γ)). Moreover, t = 1 is a regular value of this rational function and

∑_{i≥0}(−1)^{i} [K ⊗_{k} Tor_{i}^{R}(M, k)] (t)|_{t=1} = [K ⊗_{R} M] in R(K(Γ)).

Observe in the statement of this result that the action of R on K coming from the inclusion R ⊆ K is not the same as the action coming from the composite homomorphism R → k ↪ K, thus distinguishing K ⊗_{R} M from K ⊗_{k} (M ⊗_{R} k).

Proof. We begin by proving (i). Let {e_{α}} be a minimal homogeneous Rspanning subset for M. Then {e_{α} ⊗ 1} forms a k-basis for M ⊗_{R} k, by the graded version of Nakayama’s Lemma. Hence {1⊗(e_{α} ⊗1)} forms a K-basis for K ⊗_{k} (M ⊗_{R} k).

Also {1 ⊗ e_{α}} is a K-spanning set for K ⊗_{R} M. Filter K ⊗_{R} M by letting F_{j} be the K-span of 1 ⊗ e_{α} for which e_{α} has degree at most j. Then K ⊗_{k} (M ⊗_{R} k) has a direct sum decomposition coming from its inherited grading, and there is a composite mapping

defined by  \begin{matrix}(K ⊗_{k} (M ⊗_{R} k))_{j} & → & F_{j} & → & F_{j}/F_{j−1} \\ 1 ⊗ (e_{α} ⊗ 1) & ↦ & 1 ⊗ e_{α} & ↦ & \overline{1 ⊗ e_{α}} \end{matrix}

where e_{α} is assumed to have degree exactly j. This composite mapping is a surjection of K(Γ)-modules.

These surjections show the inequality asserted in (i). One has equality if and only if all these surjections are isomorphisms, that is, if and only if the {1 ⊗ e_{α}} are a K-basis for K ⊗_{R} M, which happens if and only if they are K-linearly independent. This in turn happens if and only if the {e_{α}} are R-linearly independent, and hence an R-basis for M, so that M is free over R.

We turn next to the proof of (ii) and (iii). In both of these proofs, it is convenient to reduce to the case where |Γ| lies in k^{×} and hence k(Γ) is semisimple: Recall from Section 2.4 that virtual modules in R(k(Γ)) are determined by their restrictions to the cyclic subgroups generated by pregular elements γ ∈ Γ, so one may replace Γ with ⟨γ⟩ without loss of generality.

For the proof of (ii), semisimplicity of k(Γ) allows us to write down the first m steps in a minimal R-free resolution of M as in Proposition 2.3.1 (ii), so that all differentials are k(Γ)-module maps. Let L denote the kernel after the m^{th} stage, so that one has the exact sequence

0 → L → F_{m} → F_{m−1} → ⋯ → F_{1} → F_{0} → M → 0.

Applying the functor K ⊗_{R} (−) is the same as a localization, and hence gives rise to an exact sequence, whose i^{th} term for i = 0, 1, …, m looks like

K ⊗_{R} F_{i} ≅ K ⊗_{R} R ⊗_{k} F_{i}/R_{+}F_{i} ≅ K ⊗_{k} Tor_{i}^{R}(M, k)

due to minimality of the resolution. This shows that

(−1)^{m} (∑_{i=0}^{m}(−1)^{i} [K ⊗_{k} Tor_{i}^{R}(M, k)] − [K ⊗_{R} M]) = [K ⊗_{R} L] ≥ 0

in K(Γ), which gives the inequality in R(K(Γ)) asserted in (ii), with equality if and only if K ⊗_{R} L = 0. Since L is a submodule of the free R-module F_{m}, it is torsion-free as an R-module and hence K ⊗_{R} L = 0 if and only if L = 0. Minimality of the resolution then shows that vanishing of L (which is sometimes called the (m + 1)^{st} syzygy module for M) is equivalent to hd_{R}(M) ≤ m.

For the proof of (iii), we note by Hilbert’s syzygy theorem that it holds when R is a polynomial algebra as a special case of (ii).

To deduce the general case of (iii), extend the field k if necessary in order to pick a graded Noether normalization R′ ⊆ R, that is a graded polynomial subalgebra R′ over which R is module-finite, and let K′ ⊆ K be the associated extension of fraction fields with degree [K : K′]. We will take advantage of the injective ring homomorphisms from Section 2.4

R(k(Γ)) ↪^{ψ_{k,K′}}  R(K′(Γ)) ↪^{ψ_{K′,K}}  R(K′(Γ))
R(k(Γ))[[t]] ↪^{ψ_{k,K′}} R(K′(Γ))[[t]] ↪^{ψ_{K′,K}} R(K′(Γ))[[t]].

that arise by extension of scalars in each case. Applying Proposition 2.3.1 (iv) twice, one has in R(k(Γ))[[t]] that

\begin{aligned}∑_{i≥0}(−1)^{i}[Tor_{i}^{R}(M, k)](t) & = \frac{[M](t)}{[R](t)} = \frac{[M](t)}{[R′](t)} · \frac{[R′](t)}{[R](t)} \\ & = (∑_{i≥0}(−1)^{i}[Tor_{i}^{R′}(M, k)](t)) · \frac{[R′](t)}{[R](t)}. \end{aligned}

Applying the map ψ_{k,K′}, one concludes that in R(K′(Γ))[[t]] one has

(2.5.1) \begin{matrix}∑_{i≥0} & (−1)^{i}[K′ ⊗_{k} Tor_{i}^{R}(M, k)](t) \\ = (∑_{i≥0}(−1)^{i}[K′ ⊗_{k} Tor_{i}^{R′}(M, k)](t)) · ψ_{k,K′} (\frac{[R′](t)}{[R](t)}) . \end{matrix}

The first factor on the right in (2.5.1) has t = 1 as a regular value in R(K′(Γ)), because R′ is a polynomial algebra, and the case already proven shows that the value taken there is [K′ ⊗_{R′} M]. For the second factor on the right in (2.5.1), note that both R′, R carry trivial Γ-actions, and hence either [4, Lemma 2.4.1 (iii) on page 20] or [25, Prop. 5.5.2 on page 123] shows that this factor also has t = 1 as a regular value in R(K′(Γ)), taking the value \frac{1}{[K:K′]}[1]. Consequently, the left side of (2.5.1) has t = 1 as a regular value in R(K′(Γ)), taking the the value

\begin{aligned}∑_{i≥0}(−1)^{i}[K′ ⊗_{k} Tor_{i}^{R}(M, k)](t)|_{t=1} & = \frac{1}{[K : K′]}[K′ ⊗_{R′} M] \\ & = \frac{1}{[K : K′]}[K ⊗_{R} M] \end{aligned}

where the second equality uses Lemma 2.5.1. Applying the map ψ_{K′,K}, one concludes that the element

∑_{i≥0}(−1)^{i}[K ⊗_{k} Tor_{i}^{R}(M, k)](t)

in R(K(Γ))[[t]] also has t = 1 as a regular value in R(K(Γ)), taking the value

∑_{i≥0}(−1)^{i}[K ⊗_{k} Tor_{i}^{R}(M, k)](t)|_{t=1} = \frac{1}{[K : K′]}ψ_{K′,K}[K ⊗_{R} M]

= \frac{1}{[K : K′]}[K ⊗_{K′} K ⊗_{R} M]

= [K ⊗_{R} M].

Here the last equality uses the fact that both K′, K carry trivial Γ-actions, and K ⊗_{K′} K ≅ K^{[K:K′]}. □

Remark 2.5.3. Simple examples with Tor^{R}(M, k) infinite show that the assumption that Γ act trivially on R in Theorem 2.5.2 cannot in general be removed. For example, consider the following inclusions of algebras.

\underbrace{ℂ[x^{2}, y^{2}]}_{R′} ⊂ \underbrace{ℂ[x^{2}, xy, y^{2}]}_{R} ⊂ \underbrace{ℂ[x, y]}_{M}

Let Γ = ℤ/4ℤ act compatibly on R′, R, M via the scalar substitution given by x, y ↦ ix, iy (where i^{2} = −1 in ℂ), so that their d^{th} homogeneous components are scaled by i^{d}. Then R(k(Γ)) ≅ ℤ[α]/(α^{4} − 1), where α represents the class of the 1-dimensional k(Γ)-module that is scaled by i. In R(k(Γ))[[t]] there are equalities

[Tor^{R}(M, k)](t) = \frac{[M](t)}{[R](t)} ( by Proposition 2.3.1(iv) above)

= \frac{[R′](t)[Tor^{R′}(M, k)](t)}{[R′](t)[Tor^{R′}(R, k)](t)} ( again by Proposition 2.3.1(iv) above)

= \frac{(1 + αt)^{2}}{1 + α^{2}t^{2}}

= 1 + \frac{2αt}{1 + α^{2}t^{2}}

= 1 + α · \frac{2t}{1 − t^{4}} + α^{2} · 0 + α^{3} · \frac{−2t^{3}}{1 − t^{4}}

and the last of these does not have t = 1 as a regular value. Note that here R is not fixed by Γ.

2.6. Proof of Theorem 1.1.1. To prove Theorem 1.1.1, one applies Theorem 2.5.2 in the special case where R = k[V]^{G}. Then K is its fraction field k(V)^{G}, and one has the R-module M = (U ⊗_{k} k[V])^{G}, with action by Γ coming from the fact that U is a (k(Γ), k(G))-bimodule. What remains to show in this situation is that

[K ⊗_{R} M] = [K ⊗_{k} U] in R(K(Γ)).

In fact, these two K(Γ)-modules are isomorphic as the following string of equalities and isomorphisms proves.

K ⊗_{R} M =^{(1)} k(V)^{G} ⊗_{k[V]^{G}} (U ⊗_{k} k[V])^{G}

≅^{(2)} (U ⊗_{k} k(V))^{G}

≅^{(3)} (U ⊗_{k} KG)^{G}

≅^{(4)} K ⊗_{k} (U ⊗_{k} k(G))^{G}

≅^{(5)} K ⊗_{k} U.

These may be justified as follows. Equality =^{(1)} is just a definition. The

isomorphism ≅^{(2)} is given by the map

\begin{matrix}k(V)^{G} ⊗_{k[V]^{G}} (U ⊗_{k} k[V])^{G} & → & (U ⊗_{k} k(V))^{G} \\ \frac{g}{h} ⊗ ∑_{i}(u_{i} ⊗ f_{i}) & ⟼ & ∑_{i} u_{i} ⊗ f_{i} \frac{g}{h}. \end{matrix}

whose inverse sends

∑_{i} u_{i} ⊗ \frac{g_{i}}{h_{i}} = ∑_{i} u_{i} ⊗ \frac{ĝ_{i}}{h} ⟼ \frac{1}{h} ⊗ (∑_{i} u_{i} ⊗ ĝ_{i})

where ĝ_{i}, h are chosen so that \frac{ĝ_{i}}{h} = \frac{g_{i}}{h_{i}} and h is G-invariant (e.g. choose h to be the product of the finitely many images g(h_{i}) as i varies and as g varies

through the finite group G). The isomorphism ≅^{(3)} comes from the Normal Basis Theorem [15, Theorem VIII.13.1] applied to the Galois extension

K = k(V)^{G} ⊂ k(V),

which asserts k(V) ≅ K(G) as k(G)-modules. The isomorphism ≅^{(4)} comes

from the fact that G acts trivially on K = k(V)^{G}. The isomorphism ≅^{(5)} comes from the k-linear map defined by

\begin{matrix}U & → & (U ⊗_{k} k(G))^{G} \\ u & ⟼ & ∑_{g∈G} ug^{−1} ⊗ t_{g} \end{matrix}

where t_{g} is the k-basis element in k(G) indexed by g.

This completes the proof of Theorem 1.1.1.

2.7. Remarks on Group Cohomology and other related Constructions. We reinterpret some of the foregoing results and comment on their implications for the higher group cohomology H^{j}(G, U ⊗_{k} k[V]).

Given a finitely generated graded integral domain R over the field k and a finite group Γ, we let Γ act trivially on R and denote by R(Γ)-mod the abelian category of finitely generated non-negatively graded R(Γ)-modules M. The assignment M ↦ Tor^{R}(M, k) is a functor from the category R(Γ)- mod to the category bigraded-k(Γ)-mod of bigraded-k(Γ)-modules that are finite-dimensional in each bidegree. One can compose this with the forgetful functor bigraded-k(Γ)-mod → graded-k(Γ)-mod that forgets the internal grading by taking the direct sum over it, and leaves the homological grading. Theorem 2.5.2(iii) asserts that the composite of these two functors followed by taking the alternating sum over the homological grading yields a well-defined homomorphism of Grothendieck rings R(R(Γ)) → R(k(Γ)), which sends [M] ↦ ψ^{−1}[K ⊗_{R} M], where K is the field of fractions of R and ψ is induced from the inclusion k ↪ K.

Now we place ourselves in the context of Theorem 1.1.1 (iii) where R = k[V]^{G} and we are considering the relative invariants functor in the form k(Γ × G)-mod → R(Γ)-mod that sends U ↦ (U ⊗_{k} k[V])^{G}. Then Theorem 1.1.1 (iii) says that if we follow this by the composite functor described above, we get a well-defined homomorphism R(k(Γ × G)) → R(k(Γ)) which sends [U] ↦ [U], that is, it coincides with the restriction homomorphism that simply forgets the k(G)-structure.

This is a bit surprising, as the relative invariants functor is not exact; it is the case j = 0 of a family of (generally nontrivial) group cohomology functors

U ↦ M_{j}(U) := H^{j}(G, U ⊗_{k} k[V])

for j ≥ 0, which measure the inexactness of taking relative invariants. In fact, it is not hard to show using the same ideas as above, that for any strictly positive j, when one follows M_{j}(−) by the composite functor and takes an alternating sum (i.e., Euler characteristic) as discussed above, it induces the zero homomorphism R(k(Γ × G)) → R(k(Γ)). That is, one has the following conclusion.

Theorem 2.7.1. Let G be a finite subgroup of GL(V), let R = k[V]^{G} and let U be any finite-dimensional k(G × Γ)-module. For all j ≥ 0, let

M_{j}(U) := H^{j}(G, U ⊗_{k} k[V])

be the j^{th} cohomology group of G with coefficients in U ⊗_{k} k[V], considered as a R(Γ)-module. Then for strictly positive j one has

∑_{i≥0}(−1)^{i} Tor_{i}^{R}(M_{j}(U), k)|_{t=1} = 0

in R(k(Γ))

Before proving this, we quote a standard fact about the behavior of group cohomology under change of ground rings, which will also be of use further on in the proof of Proposition 3.3.1.

Proposition 2.7.2. Let G be a finite group and let R → S be a homomorphism of commutative rings which is flat, so that M → S ⊗_{R} M is an exact functor from R-modules to S-modules. Then for any R(G)-module M, one has

S ⊗_{R} H^{j}(G, M) ≅ H^{j}(G, S ⊗_{R} M)

as S-modules.

Proof. See [12, Sec. 10.3, Prop. 7]. □

Proof of Theorem 2.7.1. Applying Theorem 2.5.2(iii), and the remarks in Section 2.4 concerning extension of scalars, it suffices for us to show that

K ⊗_{R} H^{j}(G, U ⊗_{k} k[V]) = 0 for j > 0,

where K is the field of fractions of R. Note that K ⊗_{R} (−) is exact, because it is a localization, so Proposition 2.7.2 implies the first isomorphism in the following string of isomorphisms and equalities.

K ⊗_{R} H^{j}(G, U ⊗_{k} k[V]) ≅ H^{j}(G, K ⊗_{R} (U ⊗_{k} k[V]))

≅ H^{j}(G, U ⊗_{k} k(V))

≅ H^{j}(G, U ⊗_{k} KG)

≅ H^{j}(G, K ⊗_{k} (U ⊗_{k} k(G))) = 0

The equality on the second line is by definition, while the next three isomor-

phisms appeared as ≅^{(2)}, ≅^{(3)}, ≅^{(4)} in Section 2.6. The last vanishing assertion comes from the fact that U ⊗_{k} k(G) is always free as a k(G)-module, and hence K ⊗_{k} (U ⊗_{k} k(G))) is free as a K(G)-module, so its higher cohomology will vanish. □

We close this section by noting that whereas we have been dealing with the fixed points of G and its derived functors, we could instead have worked with the fixed quotients by the action of G. Specifically, the same arguments that we have used in the proof of Theorem 1.1.1 can also be used if we replace M by M′ := U ⊗_{k(G)} k[V], namely the fixed quotient of G acting on U ⊗_{k} k[V] rather the fixed points which we have used before. Of course, when |G| is invertible in k these constructions are isomorphic, and so they are both valid interpretations of the notion of relative invariants in the modular case. With this definition of M′ the crucial chain of equations in the proof of Theorem 1.1.1 becomes

K ⊗_{R} M′ = k(V)^{G} ⊗_{k[V]^{G}} (U ⊗_{k(G)} k[V])
= U ⊗_{k(G)} (k[V] ⊗_{k[V]^{G}} k(V)^{G})
≅ (U ⊗_{k(G)} k(V))
≅ (U ⊗_{k(G)} KG)
≅ K ⊗_{k} (U ⊗_{k(G)} k(G))
≅ K ⊗_{k} U

and this shows that it is valid to replace M by M′ in the statement of Theorem 1.1.1.

3. Proof of Theorems 1.2.1 and 1.3.1

Let us recall the setting and statement of Theorem 1.3.1. We suppose k is an arbitrary field, G, Γ, and C finite groups, and V a finite dimensional (k(G), k(C))-bimodule on which G acts faithfully, so G ⊂ GL(V). We regard V, k[V] and k[V]^{G} as trivial k(Γ)-modules.

Suppose there is a Noether normalization R ⊂ k[V]^{G} that is stable under the action of C on k[V]. Further suppose that one has a vector v in V such that the fiber Φ_{v} := ϕ^{−1}(ϕ(v)) containing v for the map ϕ : V → Spec(R) both carries a free (but not necessarily transitive) G-action, and that this fiber Φ_{v} is stable under C. Denote by 𝔪_{ϕ(v)} the maximal ideal in R corresponding to ϕ(v) in Spec(R) and introduce the coordinate ring of Φ_{v} namely, A(Φ_{v}) = k[V]/𝔪_{ϕ(v)}k[V], which is a k(C)-module.

Let U be a finite-dimensional (k(Γ), k(G))-bimodule which we regard as a trivial k(C)-module.

In this situation Theorem 1.3.1 asserts that the relative invariants M := (U ⊗_{k} k[V])^{G} satisfy the equation

∑_{i≥0}(−1)^{i} [Tor_{i}^{R}(M, k)] = [(U ⊗_{k} A(Φ_{v}))^{G}] .

in R(k(Γ × C)). Note that Hilbert’s syzygy theorem tells us that the sum is finite since R is a polynomial algebra.

In the subsections that follow we make various reductions leading up to the proof of Theorem 1.3.1, with the goal of separating out the different ideas involved.

3.1. Reduction 1: Removing the Γ-action. The goal here is to prove the following lemma.

Lemma 3.1.1. Theorem 1.3.1 follows from its special case where Γ acts trivially on U; that is, the case with only a C-action and no Γ-action.

This will essentially be a consequence of the Brauer theory reviewed in Section 2.4, applied to the k(Γ)-modules (U ⊗_{k} k[V])^{G} and (U ⊗_{k} A(Φ_{v}))^{G}.

Given a p-regular element γ in Γ, let G̃ := ⟨γ⟩ × G. For any (k(Γ), k(G))- bimodule W, which we regard as a left k(Γ×G)-module, its G-fixed subspace W^{G} is a semisimple k⟨γ⟩-module, and one can express its k⟨γ⟩-isotypic direct sum decomposition in terms of G̃-fixed subspaces:

(3.1.1)  W^{G} ≅ ⨁_{j∈ℤ/dℤ}(W ⊗_{k} U_{j})G̃

If W happens also to have an action of C commuting with the actions of Γ and G this becomes a direct sum of k(C)-modules.

Proof of Lemma 3.1.1. Apply the preceding discussion to the (k(G), k(Γ))- bimodules W = U ⊗_{k} A(Φ_{v}) and W = U ⊗_{k} k[V]. Using the fact that functors like tensor product and Tor commute with direct sums, along with the Brauer theory from Section 2.4, one sees that Theorem 1.3.1 is equivalent to showing for each p-regular element γ ∈ Γ

(3.1.2)  ∑_{i≥0}(−1)^{i} [Tor^{R}((Ũ_{j} ⊗_{k} k[V])G̃, k)] = [(Ũ_{j} ⊗_{k} A(Φ_{v}))G̃]

in R(k(C)), where U_{j} ranges over the simple k(⟨γ⟩)-modules and Ũ_{j} := U ⊗_{k} U_{j}. However, the equality (3.1.2) is an instance of Theorem 1.3.1 with trivial Γ-action: Simply replace G by G̃, and U by Ũ_{j}. □

3.2. Reduction 2: Replacing Special Fiber with General Fiber. A key idea in the proof of Theorem 1.3.1 goes back to Borho and Kraft [5], before that to Kostant [14], and perhaps earlier: Given the finite ramified cover ϕ : V → Spec(R), one should compare the group actions on the special fiber Φ_{0} and its coordinate ring k[V]/R_{+}k[V] with the (often better understood) actions on a more general fiber Φ_{v} and its coordinate ring k[V]/𝔪_{ϕ(v)}k[V].

To set this up, we return to the situation of Section 2 with an R-module M and compatible finite group G acting as above. Then k = R/R_{+} carries the trivial R-module and k(G)-action. Let k′ be a different, possibly nongraded, R-module structure on the field k. In other words, k′ = R/𝔪′ where 𝔪′ is some (generally inhomogeneous) maximal ideal of R which happens to be G-stable. Note that the action of G on k′ remains trivial, since k′ is spanned by the image of 1, on which G acts trivially.

Proposition 3.2.1. Let G be a finite group, R an ℕ-graded Noetherian algebra over the field k and M a finitely generated R-module. Assume that both R and M have compatible G-actions and that hd_{R}(M) is finite. Denote by 𝔪 the (tautological) maximal ideal R_{+} of R (so R/𝔪 ≅ k is the tautological R-module structure on k), and by 𝔪′ is some (possibly inhomogenous) maximal ideal of R which is G-stable. Set k′ = R/𝔪′.

Then one has the following (ungraded) equality in R(k(G)) of two (finite) sums:

∑_{i,j≥0}(−1)^{i}[Tor_{i}^{R}(M, k)_{j}] = ∑_{i,j≥0}(−1)^{i}[Tor_{i}^{R}(M, k′)_{j}].

Proof. Compute either of the two Tor’s by starting with a finite (not necessarily minimal) free R-resolution F produced as in Proposition 2.3.1(iii), tensoring over R with k or k′, and then taking the homology of either F ⊗_{R}k or F ⊗_{R} k′. Each term F_{i} ⊗_{R} k or F_{i} ⊗_{R} k′ is a finite-dimensional k-vector space, and taking alternating sums in R(k(G)) gives the following equalities

\begin{aligned}∑_{i,j≥0}(−1)^{i} [Tor_{i}^{R}(M, k)_{j}] & = ∑_{i≥0}(−1)^{i}[F_{i} ⊗_{R} k] \\ ∑_{i,j≥0}(−1)^{i} [Tor_{i}^{R}(M, k′)_{j}] & = ∑_{i≥0}(−1)^{i}[F_{i} ⊗_{R} k′]. \end{aligned}

Note the sums are finite because M is finitely generated and hd_{R}(M) is finite.

It therefore suffices to show that as k(G)-modules only (disregarding their R-module structure), one has [F_{i} ⊗_{R} k] = [F_{i} ⊗_{R} k′] in R(k(G)), which we prove by a filtration argument. Given a homogeneous R-basis {e_{α}} for F_{i}, one has filtrations A, A′ on the two k-vector spaces F_{i} ⊗_{R} k, F_{i} ⊗_{R} k′ defined as follows: let A_{j}, A′_{j} be the k-span of those k-basis elements e_{α}⊗_{R}1 in which deg(e_{α}) ≤ j. Since G acts in a grade-preserving fashion, it will preserves these filtrations. We claim that there is also a k(G)-module isomorphism A_{j}/A_{j−1} → A′_{j}/A′_{j−1} sending the k-basis element e_{α} ⊗ 1 to the k-basis element e_{α} ⊗ 1. To check that this isomorphism is G-equivariant, given g ∈ G, let g(e_{α}) = ∑_{β} r_{β,α}(g)e_{β} for some homogeneous elements r_{β,α}(g) in R. One then has the same computation in either of A_{j}/A_{j−1} or A′_{j}/A′_{j−1}:

g(e_{α} ⊗ 1) = ∑_{β:deg(e_{β})=deg(e_{α})} r_{β,α}(g) (e_{β} ⊗ 1)

Note that in this last sum, the coefficient r_{β,α}(g) in R represents the same element in the quotient fields k or k′, since it must be of degree zero by homogeneity considerations. □

We comment that in the last part of the proof of Proposition 3.2.1 we do not necessarily get an isomorphism of k(G)-modules, as can be seen by considering an example where R = 𝔽_{2}[x] acted on trivially by a cyclic group G of order 2. We may take F_{i} = Re_{0} ⊕ Re_{1} to be free of rank 2, where e_{0} lies in degree 0 and e_{1} lies in degree 1. Let G act on F_{i} via the matrix

[ \begin{matrix}1 & x \\ 0 & 1 \end{matrix} ]

and let k′ = R/(x − 1). Then G acts on F ⊗_{R} k via

[ \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ]

and on F ⊗_{R} k′ via

[ \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ] .

These two actions are non-isomorphic.

The particular case of Proposition 3.2.1 that will interest us most is where one has a finite group G ⊆ GL(V), with R ⊆ k[V]^{G} an integral extension of graded algebras. Then given any finite-dimensional k(G)-module U, as in the Introduction, one can form the R-module U ⊗_{k} k[V], with diagonal k(G)-action, having G-fixed subspace M := (U ⊗_{k} k[V])^{G} which retains the structure of an R-module (because G acts trivially on R).

Proposition 3.2.2. Let G be a finite group that acts on the finite dimensional k-vector spaces V and R ⊂ k[V]^{G} a Noether normalization. Then for any finite dimensional k(G)-module U the module of relative invariants M := (U ⊗_{k} k[V])^{G} is finitely-generated as an R-module.

Proof. Recall the tower of integral extensions

R ⊆ k[V]^{G} ⊆ k[V].

Note that U ⊗_{k} k[V] is finitely-generated as a k[V]-module, hence also finitely-generated as a k[V]^{G}-module. Hence it is a Noetherian k[V]^{G}- module, and its k[V]^{G}-submodule M = (U ⊗_{k} k[V])^{G} will be Noetherian, that is, finitely-generated over k[V]^{G}. But then M is also finitely-generated over R. □

In the situation of Theorem 1.3.1, with R, M, Γ, C as defined there, choose

k′ = k_{v} := R/𝔪_{ϕ(v)}.

Note hd_{R}(M) is finite and bounded above by dim_{k}(V) via Hilbert’s syzygy theorem since R is a polynomial algebra. Thus Proposition 3.2.1 together with Lemma 3.1.1 show that to prove Theorem 1.3.1 reduces to showing

(3.2.1)  ∑_{m≥0}(−1)^{i} [Tor_{i}^{R}(M, k_{v})] = [(U ⊗_{k} A(Φ_{v}))^{G}]

in R(k(C)).

The plan for proving (3.2.1) is pretty clear: Try to prove the vanishing result

Tor_{i}^{R}(M, k_{v}) = 0 for i > 0,

and then try to identify

Tor_{0}^{R}(M, k_{v}) ≅ (U ⊗_{k} A(Φ_{v}))^{G}

as k(C)-modules.

3.3. Reduction 3: Working Locally via Completions. Both parts in the aforementioned plan for proving (3.2.1) in Section 3.2 have the advantage that one can work with local and semi-local rings, by taking completions with respect to 𝔪_{ϕ(v)}-adic topologies. Completion will be shown to commute with Tor, reducing the vanishing statement Tor_{i}^{R}(M, k_{v}) = 0 for i > 0 to showing that the completion M̂ of M is free as a module over the completion R̂ of R. Working locally also allows us to take advantage of the Chinese Remainder Theorem for semi-local rings.

We begin with a review of some properties of completion in a general setting–much of this material can be found in [1, Chap. 10], [9, Chap. 7], [16, §8]. Let R be a commutative Noetherian ring, M an R-module, and 𝔩 any ideal of R. Denote by R̂ the completion of R with respect to the 𝔩-adic topology, and M̂ ≅ R̂ ⊗_{R} M the corresponding completion of M.

Proposition 3.3.1. Let R be a commutative Noetherian ring, M an Rmodule, 𝔩 an ideal of R, and G a finite group acting on M by R-module homomorphisms. Then taking fixed points commutes with 𝔩-adic completion:

\binom{}{ˆM}^{G} ≅ M̂^{G} that is,

(R̂ ⊗_{R} M)^{G} ≅ R̂ ⊗_{R} M^{G}

as R̂-modules.

Proof. Completion is an exact functor from R-modules to R̂-modules, so commutes with the cohomology functors H^{j}(G, −) by Proposition 2.7.2. In particular, this holds for j = 0, the fixed-point functor (−)^{G}. □

If 𝔪 is a maximal ideal of R, then the ideal 𝔪̂ := 𝔪R̂ is also maximal in R̂, and the two residue fields are the same, viz.,

k_{𝔪} = R/𝔪 = R̂/𝔪̂.

Thus k_{𝔪}-vector spaces can be regarded as both R-modules and R̂-modules. Furthermore, R̂ is a local ring. The next proposition exploits this to translate the vanishing of Tor_{i} for all i > 0 into freeness of the completed module.

Proposition 3.3.2. Let 𝔪 be a maximal ideal of a commutative Noetherian ring R, and M an R-module. Then as k_{𝔪}-vector spaces one has

TorR̂ ; i(M̂, k_{𝔪}) ≅ Tor_{i}^{R}(M, k_{𝔪}).

Consequently Tor_{i}^{R}(M, k_{𝔪}) vanishes for all i > 0 if and only if M̂ is a free R̂-module.

Proof. The asserted isomorphism is a consequence of the string of isomorphisms that follows.

TorR̂ ; i(M̂, k_{𝔪}) ≅ TorR̂ ; i(R̂ ⊗_{R} M̂, R̂ ⊗_{R} k_{𝔪})
≅ R̂ ⊗_{R} Tor_{i}^{R}(M, k_{𝔪})
≅ Tor_{i}^{R}(M, k_{𝔪})

The second of these uses the fact that R̂ ⊗_{R} (−) is exact, and the last that Tor^{R}(M, N) is an R/Ann_{R}(N)-module, so a vector space over k_{𝔪} = R/𝔪 = R̂/𝔪̂, and R̂ ⊗_{R} k_{𝔪} = k_{𝔪}.

Since R̂ is a local ring with residue field k_{𝔪}, the R̂-module M̂ is free if and only if TorR̂ ; i(M̂, k_{𝔪}) = 0 for i > 0. So the previous isomorphism implies Tor_{i}^{R}(M, k_{𝔪}) = 0 for all i > 0. □

3.4. (Semi-)local Analysis of the Fibers. Let k be a field and V a finitedimensional k-vector space. Suppose that G is a finite subgroup of GL(V) and R ⊂ k[V]^{G} a Noether normalization, so we have integral extensions of graded algebras

R ⊆ k[V]^{G} ⊆ k[V].

Let U be a finite-dimensional (k(Γ), k(G))-bimodule for some finite group Γ with module of relative invariants M := (U ⊗_{k} k[V])^{G}. Suppose that there exists a vector v ∈ V whose fiber Φ_{v} for the composite ϕ in the tower of ramified coverings

V → V/G → Spec(R)

is permuted freely (but not necessarily transitively) by G. In this section we let Γ act trivially on V and ignore all C-actions. The C-actions will be put back in Section 3.5.

For any w ∈ V, denote by k_{w} the k[V]-module structure on k defined by f · α := f(w)α for all f ∈ k[V] and α ∈ k. The corresponding maximal ideal of polynomials vanishing at w is denoted by 𝔪_{w} ⊂ k[V]. By restriction we also consider k_{w} as module over k[V]^{G}, R, etc. The notations for the corresponding maximal ideals of k[V]^{G} and R are

\begin{aligned}𝔪_{Gw} & = 𝔪_{w} ∩ k[V]^{G} \\ 𝔪_{ϕ(w)} & = 𝔪_{w} ∩ R. \end{aligned}

The ideal 𝔪_{Gw} can also be characterized as the G-invariant polynomials that vanish at the G-orbit Gw regarded as a point of V/G. Likewise, 𝔪_{ϕ(w)} can be characterized as the polynomials in R that vanish on the whole fiber

Φ_{w} := ϕ^{−1}(ϕ(w)) = {u ∈ V : ϕ(u) = ϕ(w)},

and the generators are easy to describe: If R = k[f_{1}, …, f_{m}] then

𝔪_{ϕ(w)} = (f_{1} − f_{1}(w), f_{2} − f_{2}(w), …, f_{m} − f_{m}(w))R.

Let R̂ denote the complete local ring obtained by completing R at the maximal ideal 𝔪_{ϕ(w)}.

An important property of completion, the Chinese remainder theorem (see e.g., [9, Corollary 7.6], [16, Theorem 8.15]), gives cartesian product decompositions of complete semi-local rings. Thus in the context just described one has the following table in which the first column lists various R-modules M, the second column lists their associated semi-local completed R̂-modules M̂ := R̂ ⊗_{R} M along with Chinese remainder theorem isomorphisms, and the third column lists their quotient k_{w}-modules , viz., M/𝔪_{ϕ_{(}w)}M ≅ M̂/𝔪̂_{ϕ(w)}M̂.

R-module M  R̂-module M̂  k_{w}-module M/𝔪_{ϕ(w)}M
A := k[V]  Â ≅ ∏_{w∈Φ_{w}} Â_{w}  A(Φ_{w})
B := k[V]^{G} B̂(= Â^{G}) ≅ ∏_{Gw∈Φ_{w}/G} B̂_{Gw}  B(Φ_{w}/G)
R  R̂  k_{w}

Table 3.4.1: Chinese Remainder Table

The heart of the matter now lies in using this to prove the following lemma.

Lemma 3.4.1. Let k be a field and V a finite-dimensional k-vector space. Suppose that G is a finite subgroup of GL(V) and R ⊂ k[V]^{G} a Noether normalization. Assume that there is a v ∈ V whose fibre Φ_{v} carries a free Gaction. Let U be a finite-dimensional (k(Γ), k(G))-bimodule for some finite group Γ with module of relative invariants M := (U ⊗_{k} k[V])^{G} considered as a B(Γ)-module in the notations of Table 3.4.1. Then with the notations of that table the following hold.

(i) As B̂(Γ)-modules, one has isomorphisms

M̂ ≅ (U ⊗_{k} B̂(G))^{G} ≅ B̂ ⊗_{k} (U ⊗_{k} k(G))^{G}.

In particular, Â ≅ B̂(G) as B̂(G)-modules; this is the special case U = k(G).

(ii) M̂ is a free R̂-module, and hence

(Tor_{i}^{R}(M, k) =) TorR̂ ; i(M̂, k) = 0 for i > 0.

(iii) As k_{v}(Γ)-modules there are isomorphisms

Tor_{0}^{R}(M, k) ≅ (U ⊗_{k} A(Φ_{v}))^{G} ≅ (U ⊗_{k} B(Φ_{v}/G)(G))^{G}.

In particular, A(Φ_{v}) ≅ B(Φ_{v}/G)(G) as B(Φ_{v}/G)(G)-modules; this is the special case U = k(G).

Proof. We begin with some preparations making use of the isomorphism Â ≅ ∏_{w∈Φ_{v}} Â_{w} from Table 3.4.1. Here the right side has componentwise multiplication, and has a k(G)-module structure given via the isomorphisms Â_{w} →^{g} Â_{gw} which permute the factors. As a consequence of the assumption that G acts freely on Φ_{v}, there is a decomposition of the fiber

Φ_{v} = Gw_{1} ⊔ ⋯ ⊔ Gw_{r}

into free G-orbits. This gives two ways to regroup the factors, viz.,

Â ≅ ∏_{g∈G} \underbrace{(∏_{i=1}^{r} Â_{gw_{i}})}_{:=Â_{g}}  and  Â ≅ ∏_{i=1}^{r} \underbrace{(∏_{g∈G} Â_{gw_{i}} )}_{:=Â_{Gw_{i}}} .

Note that here Â_{g} = g(Â_{e}) where Â_{e} = ∏_{i=1}^{r} Â_{w_{i}}.

View Â as an Â_{e}-algebra and Â_{Gw_{i}} as a Â_{w_{i}}-algebra via the diagonal embeddings

\begin{aligned}Â_{e} & ⟶^{Δ} & ∏_{g∈G} Â_{g} & ≅ Â \\ Â_{w_{i}} & ⟶^{Δ} & ∏_{g∈G} Â_{gw_{i}} & ≅ Â_{Gw_{i}} \end{aligned}

defined by mapping a to the product (g(a))_{g∈G}. These embeddings give ring isomorphisms of Â_{e} and Â_{Gw_{i}} onto their diagonal images, which are exactly the G-invariant subrings of the relevant completions, or the completions of the relevant G-invariant subrings by Proposition 3.3.1. To wit,

(3.4.1)  \begin{aligned}Â_{e} & ≅ Δ(Â_{e}) & ≅ Â^{G} & ≅ B̂ \\ Â_{w_{i}} & ≅ Δ(Â_{w_{i}}) & ≅ (Â_{Gw_{i}})^{G} & ≅ B̂_{Gw_{i}} \end{aligned}

Next consider the group algebras Â_{e}(G) and Â_{w_{i}}(G) for the group G, with either Â_{e} or Â_{w_{i}} as coefficients; in either case, denote by t_{g} the basis element corresponding to the element g in G. We claim that there are isomorphisms of Â_{e}(G)-modules and Â_{w_{i}}(G)-modules

(3.4.2)  \begin{aligned}Â_{e}(G) & ⟶^{α} & ∏_{g∈G} Â_{g} & ≅ Â \\ Â_{w_{i}}(G) & ⟶^{α} & ∏_{g∈G} Â_{gw_{i}} & ≅ Â_{Gw_{i}} \end{aligned}

defined in both cases by

at_{g} ⟼ g(a)e_{g}

where e_{g} is the standard basis vector/idempotent corresponding to the factor in the product indexed by g, and where a lies either in Â_{e} or Â_{w_{i}} (so that g(a) lies either in g(Â_{e}) = Â_{g} or g(Â_{w_{i}}) = Â_{gw_{i}}). The inverse isomorphism α^{−1} in either case is defined by

(a^{(g)})_{g∈G} = ∑_{g∈G} a^{(g)}e_{g} ⟼ ∑_{g∈G} g^{−1}a^{(g)}t_{g}.

With this preparation, we can start to prove the assertions of the Lemma, beginning with assertion (i). In light of the first isomorphism Â_{e} ≅ B̂ in (3.4.1) the first isomorphism in (3.4.2) shows Â ≅ B̂(G) as a B̂(G)-modules. Since G acts trivially on R, using Proposition 3.3.1 again, one has

M̂ := R̂ ⊗_{R} ((U ⊗_{k} A)^{G}) ≅ (R̂ ⊗_{R} (U ⊗_{k} A))^{G}
≅ (U ⊗_{k} Â)^{G}
≅ (U ⊗_{k} B̂(G))^{G}
≅ B̂ ⊗_{k} (U ⊗_{k} k(G))^{G}

as B̂-modules. However, these are also B̂(Γ)-module isomorphisms since the Γ-action occurs entirely in the U factor and acts trivially on R, B, A and their completions.

For (ii), it suffices to show that M̂ is a free R̂-module, and then to apply Proposition 3.3.2(ii). Since (i) implies M̂ is a free B̂-module it suffices to show that B̂ is a free R̂-module. From Table 3.4.1, one has B̂ ≅ ∏_{i=1}^{r} B̂_{Gw_{i}}, and hence it is enough to verify that each B̂_{Gw_{i}} is a free R̂-module. Note that the ring B̂_{Gw_{i}} is a finite extension of the ring R̂. It turns out that both of these are regular local rings, because they are isomorphic to completions of polynomial algebras at maximal ideals ([9, Corollary 19.14]): in the case of R̂ this is due to the assumption that R is polynomial, and in the case of B̂_{Gw_{i}} this is due to the second isomorphism B̂_{Gw_{i}} ≅ Â_{w_{i}} of (3.4.1). Hence according to the Auslander-Buchsbaum Theorem [9, Theorem 19.9], using dim(−) to indicate Krull dimension of − , we obtain

hdR̂(B̂_{Gw_{i}}) = dim(R̂) − depthR̂(B̂_{Gw_{i}})
= dim(R̂) − dim(B̂_{Gw_{i}})
= dim(R̂) − dim(R̂)
= 0.

The second equality here is due to the fact that regular local rings are Cohen-Macaulay, so their depth and Krull dimension are the same, while the third follows from the fact that B̂_{Gw_{i}} is a finite extension of R̂. Thus B̂_{Gw_{i}} is R̂-free, and hence so is M̂.

For (iii), note that

Tor_{0}^{R}(M, k) ≅ M/𝔪_{ϕ(v)}M
≅ M̂/𝔪̂_{ϕ(v)}M̂
≅ (U ⊗_{k} Â)^{G}) /𝔪̂_{ϕ(v)} (U ⊗_{k} Â)^{G})
≅ (U ⊗_{k} (Â/𝔪̂_{ϕ(v)}Â))^{G}
≅ (U ⊗_{k} (A/𝔪̂_{ϕ(v)}A))^{G} =: (U ⊗_{k} A(Φ_{v}))^{G}.

This gives the first isomorphism in (iii). For the second, note that

Â/𝔪̂_{ϕ(v)}Â ≅ B̂(G)/𝔪̂_{ϕ(v)}B̂(G)
≅ (B̂/𝔪̂_{ϕ(v)}B̂)(G)
≅ (B/𝔪̂_{ϕ(v)}B)(G)
= B(Φ_{v}/G)(G)
□

3.5. Finishing the proofs: Incorporating the C-Action. Assertions (ii) and (iii) of Lemma 3.4.1 complete the proof of Theorem 1.3.1, except that we have not yet accounted for the C-action. We rectify this here, and explain how Theorem 1.3.1 implies Theorem 1.2.1.

So we assume there is also a finite subgroup C ⊂ GL(V), commuting with G, that preserves R and the maximal ideal 𝔪_{ϕ(v)}, making C act on the fiber Φ_{v}. One then has compatible C-actions on

\begin{aligned}R & ⊆ & B = k[V]^{G} & ⊆ & A = k[V] \\ R̂ & ⊆ & B̂ & ⊆ & Â \\ k_{v} & ⊆ & B(Φ_{v}/G) & ⊆ & A(Φ_{v}). \end{aligned}

We wish to describe these actions more explicitly under the assumption that the G-orbits {Gw_{i}}_{i=1}^{r} in Φ_{v}/G are all regular. Note that C permutes these G-orbits, since it commutes with G, and if c ∈ C stabilizes some G-orbit Gw_{i}, then there will be a unique element g_{c,w_{i}} ∈ G for which

(3.5.1)  cw_{i} = g_{c,w_{i}}w_{i}.

One checks that this element g_{c,w_{i}} depends only on the choice of the representative w_{i} for the orbit Gw_{i} up to conjugacy as follows. First

g_{c,hw_{i}} = hg_{c,w_{i}}h^{−1}.

However, once a choice of representative w_{i} is made, one has

cgw_{i} = gcw_{i} = gg_{c,w_{i}}w_{i}

for all g ∈ G.

Also recall that for each i = 1, 2, …, r, there is an isomorphism (see (3.4.1) of §3.4) of the completed local rings Â_{w_{i}} ≅ B̂_{Gw_{i}}, which are finite extensions of the local ring R̂_{𝔪_{ϕ(v)}}. Let

B_{(Gw_{i})} := B_{Gw_{i}}/𝔪_{ϕ(v)}B_{Gw_{i}}
≅ B̂_{Gw_{i}}/𝔪̂_{ϕ(v)}B̂_{Gw_{i}}
≅ Â_{w_{i}}/𝔪̂_{ϕ(v)}Â_{w_{i}}
≅ A_{w_{i}}/𝔪_{ϕ(v)}A(w_{i}).

This ring is a finite-dimensional k_{v}-vector space; it may be viewed either as the coordinate ring for the (possibly non-reduced) structure on the fiber Φ_{v} local to the point w_{i}, as a subscheme of V, or for the structure on the orbit space Φ_{v}/G local to the orbit Gw_{i}, as a subscheme of V/G.

Lemma 3.5.1. Assume the notation and hypotheses of Lemma 3.4.1. Further assume, as in this section, that there is a finite subgroup C ⊂ GL(V) which preserves the Noether normalization R and the fiber Φ_{v}, Choose for each G-orbit Gw_{i} within Φ_{v} a representative w_{i}, and define g_{c,w_{i}} in G via (3.5.1) whenever cGw_{i} = Gw_{i}.

Then the isomorphism of B(Φ_{v}/G)(G)-module asserted in part (iii) of that lemma, viz.,

Tor_{0}^{R}(M, k) ≅ (U ⊗_{k} A(Φ_{v}))^{G} ≅ (U ⊗_{k} B(Φ_{v}/G)(G))^{G}.

is also a k(C)-module isomorphism, obtained by using the k(C)-module structure induced from the following isomorphisms:

A(Φ_{v}) ≅ B(Φ_{v}/G)(G) ≅ ∏_{i=1}^{r} B_{Gw_{i}}(G).

If c ∈ C has cGw_{i} = Gw_{j} for j ≠ i, then there is a ring isomorphism B_{Gw_{i}} →^{c} B_{Gw_{j}} that extends to a ring isomorphism B_{Gw_{i}}(G) →^{c} B_{Gw_{j}}(G).

For c ∈ C with cGw_{i} = Gw_{i}, the ring automorphism B_{Gw_{i}} →^{c} B_{Gw_{i}} extends to a ring automorphism

\begin{matrix}B_{Gw_{i}}(G) & ⟶^{c} & B_{Gw_{i}}(G) \\ at_{g} & ⟼ & c(a)t_{gg_{c,w_{i}}} . \end{matrix}

Consequently, there is the following identity relating Brauer character values:

(3.5.2)  χ_{Tor_{0}^{R}(M,k)}(c) = ∑_{i:cGw_{i}=Gw_{i}} χ_{B_{Gw_{i}}} (c) · χ_{U}(g_{c,w_{i}}^{−1}).

Proof. The assertions will be derived by passing to the quotient by 𝔪_{ϕ(v)} from the analogous statement for the k(C)-module structures on the completed rings Â, B̂, etc.

Note that the Chinese Remainder Theorem isomorphism

Â ≅ ∏_{w∈Φ_{v}} Â_{w}

translates the C-action on Â to a C-action by isomorphisms Â_{w} →^{c} Â_{c(w)} permuting the factors on the right. From this, and the isomorphisms

B̂ ≅ ∏_{i=1}^{r} B̂_{Gw_{i}}

Â ≅ B̂(G) ≅ ∏_{i=1}^{r} B̂_{Gw_{i}}(G),

it is straightforward to check the assertions for the case when c ∈ C has cGw_{i} = Gw_{j} for j ≠ i.

If c ∈ C has cGw_{i} = Gw_{i}, then there is an automorphism c of Â_{Gw_{i}} = ∏_{g∈G} Â_{gw_{i}}, which acts by isomorphisms Â_{gw_{i}} →^{c} Â_{gg_{c,w_{i}}w_{i}} between the components. This action of c translates over to Â_{w_{i}}(G) using the isomorphism α from the proof Lemma 3.4.1 which sends at_{g} ↦^{α} g(a)e_{g}. So c sends g(a)e_{g} to cg(a)e_{gg_{c,w_{i}}}, and this maps under α^{−1} to

(gg_{c,w_{i}})^{−1}cg(a)t_{gg_{c,w_{i}}} = g_{c,w_{i}}^{−1}g^{−1}cg(a)t_{gg_{c,w_{i}}} = g_{c,w_{i}}^{−1}c(a)t_{gg_{c,w_{i}}} ,

where the last equality uses the fact that C and G commute within GL(V). Thus c(at_{g}) = g_{c,w_{i}}^{−1}c(a)t_{gg_{c,w_{i}}}.

From here one can translate the C-action to B̂_{Gw_{i}} ≅ ∏_{g∈G} Â_{gw_{i}} using the

diagonal embedding Â_{w_{i}} →^{Δ} B̂_{Gw_{i}} that maps a ↦ Δ(a) = (g(a))_{g∈G}. One then finds that g_{c,w_{i}}^{−1}c(a) ↦^{Δ} c(Δ(a)). In other words, a typical element bt_{g} in B̂_{Gw_{i}}(G) is sent by c to c(b)t_{gg_{c,w_{i}}}, as claimed.

The assertion about Brauer characters is then a consequence of Lemma 3.5.2 which follows. □

Lemma 3.5.2. Given an element g_{0} in the finite group G and a finitedimensional k(G)-module U, let a cyclic group C = ⟨c⟩ act on U via c(u) := g_{0}^{−1}(u), and let C act on (k(G) ⊗_{k} U)^{G} via

c (∑_{g∈G} t_{g} ⊗ u_{g}) := ∑_{g∈G} t_{gg_{0}} ⊗ u.

Then (k(G) ⊗_{k} U)^{G} ≅ U as k(C)-modules, and consequently, if g_{0} is p-regular, the Brauer character value for c acting on (k(G) ⊗_{k} U)^{G} is

χ_{(k(G)⊗U)^{G}}(c) = χ_{U}(g_{0}^{−1}).

Proof. The map

\begin{matrix}U & ⟶ & (k(G) ⊗_{k} U)^{G} \\ u & ↦ & ∑_{g∈G} t_{g} ⊗ g(u) \end{matrix}

is easily checked to give the desired k(C)-module isomorphism. □

Proof of Theorem 1.2.1. Assume that k[V]^{G} is polynomial, and c in G is a regular element, so that c(v) = ωv for some vector v whose G-orbit Gv is free.

We wish to apply Theorem 1.3.1 with R = k[V]^{G}, so that Φ_{v} consists of only the regular G-orbit Gv (that is, r = 1 and w_{1} = v is the representative of the unique G-orbit on Φ_{v}). In this case, the local rings R̂_{ϕ(v)} = B̂_{Gv} are the same, and their quotient B(Gv) by the maximal ideal 𝔪_{ϕ(v)} = 𝔪_{Gv} is the field k_{v} ≅ k. Thus as k(G)-modules, one has A(Φ_{v}) ≅ k(G), and Theorem 1.3.1 implies that

∑_{i≥0}(−1)^{i} [Tor^{R}(M, k)] = [(k(G) ⊗_{k} U)^{G}] = [U]

as k(Γ)-modules, where M = (U ⊗_{k} k[V])^{G}, for any finite-dimensional k(G)- module U and finite subgroup Γ of Aut_{k(G)}U.

We wish to also take into account the action of a cyclic group C = ⟨τ⟩ ⊂ Aut_{k(G)}V whose generator τ acts as the scalar ω^{−1} on V. Then τ scales V^{∗} by ω, and acts on the graded rings and modules R = k[V]^{G}, k[V], M := (U ⊗k[V])^{G} by the scalar ω^{j} in their j^{th} homogeneous component, exactly as in the C-action considered in Theorem 1.2.1. Note that τ acts on Φ_{v} = Gv by τ(v) = c^{−1}(v) and more generally τ(g(v)) = ω · g(v) = gc^{−1}(v). Thus Lemma 3.5.1 shows that the k(C)-structure on A(Φ_{v}) ≅ k(G) has τ(t_{g}) = t_{gc^{−1}}, in agreement with the k(G × C)-structure on k(G) that appeared in Springer’s Theorem, and Lemma 3.5.2 then shows that the k(Γ × C) structure on U agrees with the one that appears in Theorem 1.2.1. □

3.6. Induced Modules and the Proof of Corollary 1.2.2. We would like to apply Theorems 1.2.1 and 1.3.1 to modules M which are (relative) invariants for a subgroup H of G, rather than for G itself; in particular, we would like to draw conclusions about the H-invariant subring k[V]^{H}, as in Corollary 1.2.2.

This is made possible by a suitable notion of induction of k(H)-modules to k(G)-modules. Regard k(G) as a k(G × H)-module via the action

(g, h) · t_{g′} := gg′h^{−1}.

Then given any finite-dimensional k(H)-module W, define its induced k(G)- module to be

Ind_{H}^{G}(W) := Hom_{k(H)}(k(G), W)
= {f ∈ Hom_{k}(k(G), W) : f(t_{gh^{−1}}) = h(f(t_{g}))}.

That this is a k(G)-module follows from the equality (g · f)(t_{g′}) = f(t_{g^{−1}g′}). We next explain how this construction converts relative invariants for G into relative invariants for H.

Proposition 3.6.1 (Frobenius Reciprocity). For finite-dimensional k(G)- modules V and k(H)-modules W,

(V ⊗_{k} Ind_{H}^{G}(W))^{G} ≅ (Res_{H}^{G}(V) ⊗_{k} W)^{H}.

Proof. This is a standard isomorphism which follows from [23, §3.3, Lemma 1] or [3, 3.3.1, 3.3.2], giving

(V ⊗_{k} Ind_{H}^{G}(W))^{G} ≅ (Ind_{H}^{G}(Res_{H}^{G}(V) ⊗_{k} W))^{G} ≅ (Res_{H}^{G}(V) ⊗_{k} W)^{H}.

as required. □

We will be particularly interested in the situation where the k(H)-module W is the trivial module, so that Ind_{H}^{G}(W) = Ind_{H}^{G}(k) is the permutation module for G on the cosets H\G, and the H-relative invariants are simply the H-invariants, namely, (V ⊗_{k} W)^{H} = (V ⊗_{k} k)^{H} = V^{H}. In this situation, the isomorphism in Proposition 3.6.1 becomes a k(Γ)-module isomorphism for the action of the group Γ := N_{G}(H)/H. This is a special case of the following result.

Proposition 3.6.2. Consider nested subgroups H ⊂ L ⊂ G of a finite group G, and assume that W = Res_{H}^{L}(W̃) for some finite-dimensional k(L)-module W̃. Then

(i) The group Γ := N_{L}(H)/H acts on U := Ind_{H}^{G}(W) via

(γ · f)(t_{g}) := γ(f(t_{gγ}))

for γ ∈ N_{L}(H). (ii) For W̃ ≠ 0 this action is faithful, that is, it injects Γ ↪ Aut_{k(G)}(U). (iii) The isomorphism

(V ⊗_{k} Ind_{H}^{G}(W))^{G} ≅ (Res_{H}^{G}(V) ⊗_{k} W)^{H}.

of Proposition 3.6.1 is also a k(Γ)-isomorphism, assuming one lets k(Γ) act as follows:

• on the left, solely in the factor Ind_{H}^{G}(W), while

• on the right, diagonally in both factors of Res_{H}^{G}(V) ⊗_{k} W.

Proof. It is straightforward to check that the definition in (i) describes an action of N_{L}(H) on Ind_{H}^{G}(W) with H acting trivially, and that this action commutes with the k(G)-module structure.

To see that W̃ ≠ 0 implies the action is faithful, pick any w̃ ≠ 0 in W̃ and consider the element f of Ind_{H}^{G}(W) defined by

f(t_{g}) = \begin{cases}g^{−1}(w) & if g ∈ H, \\ 0 & otherwise. \end{cases}

An element γ in N_{L}(H) sends f to the function

(γ · f)(t_{g}) = \begin{cases}γ((gγ)^{−1}(w)) = g^{−1}(w) & if gγ ∈ H, \\ 0 & otherwise. \end{cases}

Hence γ · f = f if and only if γ lies in H.

It is also not hard to check that the isomorphism given in the proof of Proposition 3.6.1 is k(Γ)-equivariant for the actions described. □

We next provide the proof of our main application, Corollary 1.2.2, resolving in the affirmative both Conjecture 3 and Question 4 in [20].

Proof of Corollary 1.2.2. Recall that the ground field k is arbitrary and H ⊂ G are two nested finite subgroups of GL(V) where V is a finite dimensional k-vector space. We have assumed that the ring of invariants k[V]^{G} is a polynomial algebra, so it is its own Noether normalization and may be taken as R in the results established in this section. The group C = ⟨c⟩ is a cyclic subgroup generated by a regular element c in G, with eigenvalue ω on some regular eigenvector in V. We set Γ := N_{G}(H)/H and have given k[V]^{H} the k(Γ × C)-structure in which c scales the variables in V^{∗} by ω, and Γ acts by linear substitutions.

What we must prove is that in R(k(Γ × C))

(3.6.1)  ∑_{i≥0}(−1)^{i} [Tor^{k[V]^{G}}(k[V]^{H}, k)] = [k(H\G)] .

So ignoring the action of Γ this will imply that

X(t) = \frac{[k[V]^{H}] (t)}{[k[V]^{G}] (t)} ∈ ℤ[[t]],

is a polynomial in t, and the action of C on the set X = H\G then gives a triple (X, X(t), C) that exhibits the cyclic sieving phenomenon of [21]. Specifically, for any element c^{j} in C, the cardinality of its fixed point subset X^{c^{j}} ⊂ X is given by evaluating X(t) at a complex root-of-unity ω̂^{j} of the same multiplicative order as c^{j}, viz., |X^{c^{j}}| = [X(t)]_{t=ω̂^{j}} .

In this context, we apply Theorem 1.2.1, together with the results of this section, taking W̃ = k to be the trivial k(G)-module. Then W := Res_{H}^{G}(W̃) = k is the trivial k(H)-module, and

U := Ind_{H}^{G}(W) = Ind_{H}^{G}k = k(H\G)

where k(H\G) carries the k(Γ × C)-module structure as described in the statement of Corollary 1.2.2. So we have

M = (U ⊗_{k} k[V])^{G} ≅ (k ⊗_{k} k[V])^{H} = k[V]^{H} ,

and unraveling the notations, Theorem 1.2.1 tells us that equality (3.6.1) holds in R(k(Γ × C)) as required. If we ignore the Γ-action and compare Brauer character values for each element c^{−j} in C on the two sides of equality (3.6.1) we find, on the left side

∑_{i≥0}(−1)^{i} [Tor^{k[V]^{G}}(k[V]^{H}, k)] (t)|_{t=ω̂^{j}} = \frac{[k[V]^{H}] (t)}{[k[V]^{G}] (t)}|_{t=ω̂^{j}} = X(ω̂^{j}),

where ω̂ is the Brauer lift to ℂ^{×} of ω ∈ k^{×}, while on the right it is the number of fixed points for c^{j} permuting the elements of the set X = H\G. □

4. Character Values of the Binary Icosahedral Group

We note here an interesting corollary of Theorem 1.3.1, relating the Brauer character value χ_{U}(g) of a regular element g to evaluations of the Hilbert series for the module of relative invariants (U ⊗_{k} k[V])^{G} at the lift of a regular eigenvalue for g. We then apply this to the binary octahedral group.

4.1. A General Character Value Corollary. Given a graded algebra R and graded R-module M, as in Section 2, define

X_{M,R}(t) = \frac{[M] (t)}{[R] (t)} = ∑_{i≥0}(−1)^{i} [Tor_{i}^{R}(M, k)] (t).

where the second equality comes from Proposition 2.3.1(iv).

Proposition 4.1.1. Let G be a finite subgroup of GL(V), and g a regular element of G with regular eigenvector v and associated eigenvalue ω. Let U be a k(G)-module and denote its Brauer character by χ_{U} : G → ℂ. Let ω̂ ∈ ℂ^{×} be the Brauer lift of ω.

Assume there exists a graded Noether normalization R ⊂ k[V]^{G} having the following properties.^{2}

(i) G acts freely on the fiber Φ_{v} = ϕ^{−1}(ϕ(v)) for the map ϕ : V →

Spec(R).

(ii) For every G-orbit Gw_{i} in Φ_{v} that has ωGw_{i} = Gw_{i}, the unique element g_{ω,w_{i}} ∈ G for which ωw = g_{ω,w_{i}}w has the same Brauer

character value χ_{U}(g_{ω,w_{i}}) = χ_{U}(g).

(iii) X_{k[V]^{G},R}(ω̂) ≠ 0. Then

χ_{U}(g) = X_{M,k[V]^{G}}(ω̂)

where M := (U ⊗_{k} k[V])^{G}.

Proof. First note that Proposition 2.3.1(iv) implies

X_{M,R}(t) = X_{M,k[V]^{G}}(t) · X_{k[V]^{G},R}(t)

2These hypotheses would hold if k[V]^{G} were polynomial and one took R = k[V]^{G}, but we do not assume this.

and hence

(4.1.1)  X_{M,R}(ω̂) = X_{M,k[V]^{G}}(ω̂) · X_{k[V]^{G},R}(ω̂).

Consider the cyclic group C = ⟨τ⟩ ⊂ Aut_{k(G)}V whose generator τ acts as the scalar ω^{−1} on V, as in the proof of Theorem 1.2.1. Then τ scales V^{∗} by ω, and acts on the graded rings and modules

R = k[V]^{G}, k[V], M := (U ⊗ k[V])^{G}

by the scalar ω^{j} on their j^{th} homogeneous component. Hypothesis (i) allows us to apply Theorem 1.3.1. If we consider only the k(C)-module structure then in R(k(C))

[Tor^{R}(M, k)] = [(U ⊗_{k} A(Φ_{v}))^{G}] .

Therefore

(4.1.2)  \begin{aligned}X_{M,R}(ω̂) & = χ_{Tor^{R}(M,k)}(c) \\ & = χ_{(U⊗_{k}A(Φ_{v}))^{G}}(c) \\ & = ∑_{i:ωGw_{i}=Gw_{i}} χ_{B_{Gw_{i}}} (c) · χ_{U}(g_{c,w_{i}}^{−1}) \\ & = χ_{U}(g^{−1}) · ∑_{i:ωGw_{i}=Gw_{i}} χ_{B_{Gw_{i}}} (c) \end{aligned}

where the first equality uses the definition of the scalar action on R, M, Tor^{R}(M, k), the second equality is equation (3.5.2) of Lemma 3.5.1 and the last equality used hypothesis (ii) above.

In particular, taking U = k the trivial k(G)-module in (4.1.2) gives

(4.1.3)  X_{k[V]^{G},R}(ω̂) = ∑_{i:ωGw_{i}=Gw_{i}} χ_{B_{Gw_{i}}} (c).

Putting together equations (4.1.1), (4.1.2), and (4.1.3), one obtains

(4.1.4) X_{M,k[V]^{G}}(ω̂) · X_{k[V]^{G},R}(ω̂) = X_{M,R}(ω̂) = χ_{U}(g) · X_{k[V]^{G},R}(ω̂)

Since by hypothesis (iii) X_{k[V]^{G},R}(ω̂) is a nonzero factor of the extreme left and right terms of this string of equalities, we may divide by it yielding the desired equality. □

4.2. The Binary Icosahedral Group. We give here an application of Proposition 4.1.1 that highlights the flexibility of using more than one Noether normalization R ⊂ k[V]^{G}.

The finite subgroups G inside SL(2, ℂ) are called the binary polyhedral groups; see Springer [29, §4.4] for a discussion, including their classification. These groups consist of:

• two infinite families, the binary cyclic and dihedral groups, and • three exceptional groups, the binary tetrahedral, octahedral, and

icosahedral groups.

The binary icosahedral group is the largest of the exceptions, having 120 elements.

Proposition 4.2.1. Let G be the binary icosahedral group, and g any element of G with eigenvalues ω, ω^{−1}, Then for any ℂ(G)-module U one has

χ_{U}(g) = X_{M,ℂ[V]^{G}}(ω)

where V = ℂ^{2} and M := (U ⊗_{ℂ} ℂ[V])^{G}.

Proof. We wish to apply Proposition 4.1.1 with v an ω-eigenvector for g. To do so, note that for any finite subgroup G of SL(2, ℂ), any nonzero vector w ∈ ℂ^{2} always has a regular G-orbit Gw because, if g ∈ G fixes w, then both its eigenvalues will be 1, and g being semisimple must be the identity. Thus any choice of a graded Noether normalization R ⊂ ℂ[V]^{G} has G acting freely on the fiber Φ_{v}, and hypothesis (i) of Proposition 4.1.1 is satisfied.

Springer [29, §4.5] computes a presentation for the G-invariants, namely

ℂ[V]^{G} = ℂ[f_{12}, f_{20}, f_{30}]/(f_{12}^{5} + f_{20}^{3} + f_{30}^{2})

where f_{d} is a homogeneous generator of degree d, and hence

[ℂ[V]^{G}] (t) = \frac{1 − t^{60}}{(1 − t^{12})(1 − t^{20})(1 − t^{30})}.

The elements g ∈ G have orders 1, 2, 3, 4, 5, 6, 10, and there are unique conjugacy classes of elements with each of the orders 1, 2, 3, 4, 6. To verify the remaining hypotheses (ii) and (iii) in Proposition 4.1.1 for each element of G, we use two different Noether normalizations R ⊂ ℂ[V]^{G}, the choice depending on the order of g.

For elements g of order dividing 20 (that is, orders 1, 2, 4, 5, 10), we use

\begin{aligned}R & = ℂ[f_{12}, f_{30}] \\ X_{ℂ[V]^{G},R}(t) & = \frac{1 − t^{60}}{1 − t^{20}} = 1 + t^{20} + t^{40} \end{aligned}

Note that X_{ℂ[V]^{G},R}(ω̂) = 3 ≠ 0 for such elements g, so hypothesis (iii) of Proposition 4.1.1 will be satisfied. Springer’s discussion (loc. cit.) of the above presentation ℂ[V]^{G} shows that the ring R is in fact the invariant ring ℂ[V]^{Γ} for the subgroup Γ = ⟨G, ζ⟩ of GL(2, ℂ), where ζ is the scalar matrix for a primitive cube root-of-unity. Since ζ commutes with ω and G, this means that the fiber Φ_{v} either consists of

• the single G-orbit Gv, having g_{ω,v} = g, or

• three orbits Gv, Gζv, Gζ^{2}v, in which case all three correspond to the same element g_{ω,ζ^{i}v} whose existence is asserted in Proposition 4.1.1 (ii), viz.,

g = g_{ω,v} = g_{ω,ζv} = g_{ω,ζ^{2}v} ,

if one uses the three orbit representatives v, ζv, ζ^{2}v.

So in these cases hypothesis (ii) of Proposition 4.1.1 is satisfied.

For elements of orders 3, 6, we use

\begin{matrix}R = & ℂ[f_{12}, f_{20}] \\ X_{ℂ[V]^{G},R}(t) & = \frac{1 − t^{60}}{1 − t^{30}} = 1 + t^{30} \end{matrix}

Note that X_{ℂ[V]^{G},R}(ω̂) = 2 ≠ 0 for such elements g, so hypothesis (iii) of Proposition 4.1.1 will be satisfied. On the other hand, hypothesis (ii) will always be satisfied in this case, because all the elements g_{ω,w_{i}} for the various orbits Gw_{i} in Φ_{v} have the same order as ω, namely 3 or 6. So they will all be conjugate since there are unique conjugacy classes of elements of these orders. □

Remark 4.2.2. One might be tempted to apply the same method used to prove Proposition 4.2.1 for finite subgroups G of SL(2, k) with other fields k, even in positive characteristic. However, in order to check the hypothesis (ii) of Proposition 4.1.1, one needs to be careful that the ω-eigenvector v for g and the Noether normalization R avoid having any vector w in the fiber Φ_{v} which is fixed by a transvection [25, §8.2] of G.

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Département de mathémathiques et de statistique, Université de Montréal, C.P. 6128, succursale Centre-ville, Montréal (Québec), Canada H3C 3J7

   E-mail address: broera@DMS.UMontreal.CA

School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA

   E-mail address: reiner@math.umn.edu

AG-Invariantentheorie, Mittelweg 3, D 37133 Friedland, Federal Republic of Germany

   E-mail address: larry smith@GMX.net

School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA

   E-mail address: webb@math.umn.edu