GROUP ACTIONS ON RINGS AND THE ČECH COMPLEX

PETER SYMONDS

Abstract. We have previously shown that when a finite group acts on a polynomial ring over a finite field k then only finitely many isomorphism classes of indecomposable kGmodules occur as summands of S. We have also shown that the regularity of the invariant subring S^{G} is at most zero, which has various consequences, for example S^{G} is generated in degrees at most n(|G| − 1) (provided n, |G| ≥ 2). Both of these theorems depend on the Structure Theorem of Karagueuzian and the author, which is proved by means of a long and complicated calculation. The aim of this paper is to prove these results using a more conceptual method.

1. Introduction

We have previously shown with Dikran Karagueuzian that, when a finite group G acts on a polynomial ring S = k[x_{1}, …, x_{n}] over a finite field k by homogeneous linear transformations, only finitely many isomorphism classes of indecomposable kG-modules occur as summands of S [16]. We have also shown that the regularity of the invariant subring S^{G} is at most zero, which has various important consequences: for example, S^{G} is generated in degrees at most n(|G| − 1) (provided n, |G| ≥ 2); more generally, if S is finitely generated over a polynomial subring k[d_{1}, …, d_{n}] < S^{G} then S^{G} is generated in degrees at most ∑_{i}(deg(d_{i})−1) (provided deg(d_{i}) > 1 for at least two i) [20].

Both of these results depend on the Structure Theorem of Karagueuzian and the author [16], which is proved by means of a long and complicated calculation. The aim of this paper is to prove these results using a simpler, more conceptual method, based on considering the Čech complex associated to S and showing that it is split exact over kG in degrees greater than −n. This approach also has the advantage that it applies to a somewhat more general class of rings. Some other results along these lines have been obtained by Bleher and Chinburg [5] by considering Koszul resolutions. We wish to thank Burt Totaro for his helpful comments.

2. Background

Throughout this paper, k is a field. Our results are trivial if k has characteristic zero, so implicitly k has characteristic p > 0. Unless otherwise indicated, all rings will be ℤ-graded noetherian k-algebras, and before any localization it will be assumed that they are 0 in negative degrees and are finite dimensional over k in each degree. For brevity, we will refer to such rings that are also commutative as rings of standard type. A ring such as kG is implicitly in degree 0. Modules will also be graded, and when a group acts on a ring or module, it must preserve the grading. For a module M and an integer N we will write M_{≥N} = ⊕_{i=N}^{∞}M_{i}, and similarly for other inequalities.

1991 Mathematics Subject Classification. Primary: 13A50; Secondary: 13D45, 20C05.

Our basic tool is the Čech complex, strictly speaking the “extended Čech complex” or the “stable Koszul complex”.

Given a ring R, a sequence of homogeneous elements x = x_{1}, …, x_{r} from R and an R-module M, the Čech complex Č(x; M) is a cochain complex

M → ⨁_{i} M_{x_{i}} → ⨁_{i<j} M_{x_{i}x_{j}} → ⋯ → M_{x_{1}···x_{r}},

where M_{x} denotes the localization obtained by inverting x. It can be obtained as follows. Č(x_{i}; R) is the complex

R → R_{x_{i}},

with R in degree 0 and R_{x_{i}} in degree 1 and

Č(x; M) = (⨂_{i=1}^{r}_{R} Č(x_{i}; R)) ⊗_{R} M.

Observe that if G is a finite group and M is an RG-module then Č(x; M) is a complex of graded RG-modules; up to isomorphism it is independent of the ordering of the elements of x.

The jth homology group of this complex is equal to the local cohomology group H_{(x)}^{j}(M), where (x) denotes the ideal in R generated by x. It is known that if y is another sequence such that rad(y) = rad(x) then H_{(y)}^{∗}(M) = H_{(x)}^{∗}(M). We are particularly interested in the case when rad(x) = 𝔪 = 𝔪_{R} := rad(R_{>0}), the radical of the ideal of elements in positive degrees (but this ideal is only maximal if R_{0}/ rad(R_{0}) is a field). For more information, see [7, 9, 15].

One very useful technique is to change categories in such a way that all the kG-modules in question become projective. Before we do this we fix some notation. For any ring Λ, let Λ-Mod denote the category of left Λ-modules and Mod-Λ denote the category of right Λ-modules. The full subcategory of Mod-Λ on the projective modules is denoted by Proj-Λ. Let A be a finite dimensional left kG-module. Then Add(A) is the full subcategory of kG-Mod on the modules that can be expressed as a summand of a direct sum of copies of A. Set E = End_{kG}(A).

There are functors

U = Hom_{kG}(A, −) : Add(A) → Proj-E

and

V = − ⊗_{E} A : Proj-E → Add(A),

which furnish an equivalence of categories. For more details, see [23, Ch. 10] or [19].

These functors take graded modules to graded modules. In addition, they preserve any structure as an R-module for commutative k-algebras R, meaning that if M is an RG-module then U(M) is naturally a right R ⊗_{k} E-module that is projective on restriction to E and similarly for V. The functors also preserve the property of being finitely generated over R.

Lemma 2.1. The functors U and V commute with localization. In other words, if M is an RG-module and x ∈ R then U(M_{x}) ≅ U(M)_{x}, and if N is a right R ⊗_{k} E-module then V (N_{x}) ≅ V (N)_{x}.

Proof. Because A is finite dimensional, Hom_{kG}(A, −) commutes with direct limits, and the localization can be constructed as a direct limit; this proves the lemma for U. Tensor products always commute with direct limits, which proves it for V. □ Definition 2.2. A kG-module is of finite decomposition type if it can be expressed as a sum of finite dimensional indecomposable kG-modules with only finitely many isomorphism classes of indecomposable modules appearing.

A kG-module is of finite decomposition type if and only if it is in Add(A) for some finite dimensional A. Such a module satisfies the Krull-Schmidt-Azumaya property that any direct sum decomposition can be refined into a sum of indecomposables and any two decompositions into indecomposables will involve the same isomorphism classes with the same multiplicities [22]. It follows that any summand of a module of finite decomposition type is also of finite decomposition type. Finite decomposition type is also preserved by induction, restriction and tensor product over k. Lemma 2.3. If M is an RG-module that is of finite decomposition type when considered as a kG-module, then so is M_{x} for any x ∈ R. Proof. As a kG-module, M is in Add(A) for some finite dimensional kG-module A. The right R ⊗_{k} E-module U(M) is projective over E, hence flat over E. Its localization, U(M)_{x}, is a direct limit of flat modules and thus is also flat. But, over a finite dimensional k-algebra, a flat module is projective ([2, Theorem P] or [11, 22.29, 22.31A] or devise a proof by considering the projective cover of the module). Thus U(M)_{x} ∈ Proj-E, and so V (U(M)_{x}) ∈ Add(A). But V (U(M)_{x}) ≅ V (U(M))_{x} ≅ M_{x}, by Lemma 2.1. □ Proposition 2.4. Let R be a ring of standard type, x_{1}, …, x_{r} a sequence of elements such that rad(x) = 𝔪 and let M be a finitely generated RG-module. Then M is of finite decomposition type if and only if

(1) Č(x; M) is split exact over kG in sufficiently high degrees and

(2) each M_{x_{i}} is of finite decomposition type.

Note that when we say that a complex is exact we mean exact at every module, so the homology is 0 everywhere.

Proof. If the two listed conditions are satisfied, then there is a number D such that M_{>D} is a summand of (⊕_{i=1}^{r}M_{x_{i}})_{>D}, thus of finite decomposition type. But M_{≤D} is of finite decomposition type, because it is finite dimensional, and thus M is of finite decomposition type.

Conversely, if M is of finite decomposition type then M is in Add(A) for some finite dimensional kG-module A. The second condition holds, by Lemma 2.3. In order to see that the first condition is also satisfied, apply U to Č(x; M) and note that U(Č(x; M)) ≅ Č(x; U(M)), by Lemma 2.1. Since M is finitely generated, so is U(M); thus we know that H_{𝔪}^{∗}(U(M)) is zero in sufficiently large degrees [7, 15.1.5], [9, 3.6.19]. It follows that Č(x; U(M)) is exact in high degrees. But this is a complex of projective E-modules, so it is split exact as a complex of E-modules in high degrees. Thus V (Č(x; U(M))) ≅ Č(x; M) is split exact as a complex of kG-modules in high degrees. □ Proposition 2.5. Let M be an RG-module of finite decomposition type. If x = x_{1}, …, x_{r} and y = y_{1}, …, y_{s} are two sequences of elements of R such that rad(x) = rad(y), then there is a homotopy equivalence of complexes of graded kG-modules Č(x; M) ≃ Č(y; M).

Proof. Write xy = x_{1}, …, x_{r}, y_{1}, …, y_{s}. We will show that Č(x; M) ≃ Č(xy; M); together with the variation with x and y interchanged, this will prove the result. We can add the y_{i} to x one at a time, so the key case is to show that Č(x; M) ≃ Č(xy; M) for any y ∈ rad(x). Just as in the previous proof, it is easier to apply the functor U and show that Č(x; U(M)) ≃ Č(xy; U(M)) as a complex of right E-modules, then apply V to obtain the original statement.

By construction, Č(xy; U(M)) ≅ Č(y; R) ⊗_{R} Č(x; U(M)). This leads to a short exact sequence of complexes

0 → Č(x; U(M_{y}))(−1) → Č(xy; U(M)) → Č(x; U(M)) → 0,

where the (−1) indicates a shift in complex degree of 1.

The homology of Č(x; U(M_{y})) is H_{x}^{∗}(U(M_{y})), which is (x)-torsion, yet the action of y is invertible and y ∈ rad(x), so this homology is 0. It follows that Č(x; U(M_{y})) is an exact complex of projective right E-modules; hence it is split exact, which means that it is 0 in the homotopy category. The other two complexes in the short exact sequence are thus homotopy equivalent. □

3. Preliminary Results

Given a graded R-module M, let a_{i}(M) denote the highest degree in which H_{𝔪}^{i}(M) is non-zero (possibly ∞ or −∞). The Castelnuovo-Mumford regularity of M is reg(M) := max{a_{i}(M) + i}. In [20] we also considered hreg := max{a_{i}(M)}. Because H_{𝔪}^{i}(M) vanishes for i > dim(M), we have reg(M) ≤ hreg(M) + dim(M). If M is a ring and R is not specified then we take R = M, although any subring of M over which it is still finite will yield the same answer.

Let R be a ring of standard type, x_{1}, …, x_{r} a sequence of homogeneous elements such that rad(x) = 𝔪 and M a finitely generated RG-module. If we know that hreg(M) ≤ N for some integer N then Č(x; M)_{>N} is exact. Notice that (M_{x})^{G} = (M^{G})_{x}, because M^{G} ≅ Hom_{kG}(k, M), which commutes with direct limits; hence Č(x; M)^{G} ≅ Č(x; M^{G}). Thus if we happen to know that Č(x; M)_{>N} is split exact as a complex of kG-modules, then Č(x; M^{G})_{>N} is exact, and so hreg(M^{G}) ≤ N.

We are interested in the case when M is actually a ring of standard type, which we call S, G acts on S by grading-preserving ring automorphisms, and R is S^{G} or some subring of S^{G} over which it is still finite. The aim is to find a bound on hreg(S^{G}). It is this that leads to bounds on the degrees of the generators and relations: see [20] for details.

The fixed point subscheme of Spec(S) under the action of G is the closed subscheme, denoted by Spec(S)^{G}, defined by the ideal I_{G,S} < S generated by all the elements of the form (g − 1)s for g ∈ G and s ∈ S.

We will need the following geometric result due to Fleischmann [12, 5.9] (see also [17]). Recall that for a kG-module M and a subgroup H ≤ G the relative trace is the map tr_{H}^{G} : M^{H} → M^{G} defined by tr_{H}^{G}(m) = ∑_{g∈G/H} gm.

Proposition 3.1. Let S be a commutative ring ( not assumed graded or noetherian here) on which a finite group G acts as ring automorphisms. If (Spec(S))^{G} = ∅, then S^{G} = ∑_{H≨G} tr_{H}^{G}(S^{H}).

The formulation in [12] is more general in that it also describes the case when the fixed point set is not empty, but it is stated only for polynomial rings. For the convenience of the reader we sketch a proof.

Proof. Notice that T := ∑_{H≨G} tr_{H}^{G}(S^{H}) is an ideal in S^{G}; we need to show that it is not contained in any maximal ideal.

Let I be a maximal ideal in S^{G}. Clearly S is integral over S^{G}, so there is a maximal ideal J < S such that J ∩S^{G} = I [1, 5.8, 5.10]. Let H be its stabilizer, and consider the surjection π : S ↠ ∏_{g∈G/H} S/^{g}J. If J is not fixed under the action of G, let e be the element of the product that is 1 at S/J and 0 at the other coordinates, so e^{2} = e and he = e for h ∈ H; then tr_{H}^{G}(e) = π(1). Let e′ ∈ S be such that π(e′) = e and let ẽ = ∏_{h∈H} he′. Then ẽ ∈ S^{H} and π(ẽ) = ∏_{h∈H} π(he′) = ∏_{h∈H} hπ(e′) = ∏_{h∈H} he = e. Thus π(tr_{H}^{G}(ẽ)) = tr_{H}^{G}(π(ẽ)) = 1. It follows that T ≰ I in this case.

If G fixes J, consider the action of G on S/J. The condition on the fixed point subscheme shows that S/J is generated by elements of the form (g−1)s for g ∈ G and s ∈ S, so the action is not trivial; let H be its kernel. By the surjectivity of the trace in Galois theory, there is an element f ∈ S/J such that tr_{H}^{G}(f) = 1. Let p be the characteristic of S/J and let P be the Sylow p-subgroup of H (P = 1 if p = 0). Notice that for any x ∈ S/J, tr_{H}^{G}(x^{p}) = (tr_{H}^{G}(x))^{p} (provided p ≠ 0). Let f′ ∈ S be such that π(f′) = f and let f̃ = tr_{P}^{H}(∏_{g∈P} gf′). Then f′ ∈ S^{H} and π(tr_{H}^{G}(f̃)) = tr_{H}^{G} tr_{P}^{H}(f^{|P |}) = tr_{H}^{G}(|H: P |f^{|P |}) = |H: P |(tr_{H}^{G} f)^{|P |} = |H: P | ≠ 0. Again, it follows that T ≰ I. □

Remark. If S is a k-algebra, G acts by k-algebra automorphisms and k̄ denotes the algebraic closure of k then the fixed point subscheme is empty if and only the action of G on the set of closed points of Spec(k̄ ⊗_{k} S) has no fixed point.

Remark. If G is a p-group and p ∈ rad(S) then the converse statement to that in the proposition is also true. When S is the ring of k-valued functions on a G-set this is a well-known consequence of the properties of the Brauer construction [8].

Definition 3.2. If a group G acts on a commutative k-algebra S, then by an SG-module M we mean an S-module that is also a kG-module in such a way that g(sm) = (gs)(gm) for g ∈ G, s ∈ S and m ∈ M. In other words, M is a module for the twisted group algebra.

When M is a kG-module we can regard End_{k}(M) as a kG-module by setting (gf)(m) = g(f(g^{−1}m)) for g ∈ G, f ∈ End_{k}(M), m ∈ M.

Let C be a set of subgroups of G. Recall that a kG-module M is said to be projective relative to C if the following equivalent conditions hold.

(1) The module M is a summand of a sum of induced modules of the form N ↑_{H}^{G}, where H ∈ C and N is a kH-module.

(2) The module M is a summand of ⊕_{H∈C}M ↓_{H}^{G}↑_{H}^{G}.

(3) Any surjection of kG-modules L ↠ M that splits on restriction to any H ∈ C is also split over kG.

(4) Id_{M} ∈ ∑_{H∈C} tr_{H}^{G}(End_{kH}(M)). The equivalence of these conditions is known as Higman’s criterion and there are proofs in many places for the case when C consists of just one group [14] [3, 3.6.13]. The general case is not much harder and can be found in [18, 3.5.8] or [6, 2.2.3]. In characteristic p, a module is projective relative to its Sylow p-subgroups.

Lemma 3.3. Let S be a commutative k-algebra on which a p-group P acts by algebra automorphisms in such a way that 1 = ∑_{Q≨P} tr_{Q}^{P}(s_{Q}) for some s_{Q} ∈ S^{Q}. Then any SP -module M is projective relative to proper subgroups of P as a kP -module.

Proof. By characterization (4) of relative projectivity, it suffices to show that Id_{M} = ∑_{Q≨P} tr_{Q}^{P}(m_{Q}) for some m_{Q} ∈ End_{kQ}(M). Let m_{Q} ∈ End_{kQ}(M) be multiplication by s_{Q}. □

Lemma 3.4. Let M be an R-module, x a sequence of homogeneous elements of R_{>0} and N an integer. Then (M_{>N})_{x_{i_{1}}···x_{i_{ℓ}}} ≅ M_{x_{i_{1}}···x_{i_{ℓ}}} if x is non-empty, and Č_{x}(M_{>N})_{>N} = Č_{x}(M)_{>N}.

Proof. The first part follows from the short exact sequence M_{>N} → M → M̄_{≤N}, where M̄_{≤N} denotes M/M_{>N}, and the fact that (M̄_{≤N})_{x_{i}} = 0. For the second part, consider the short exact sequence Č_{x}(M_{>N}) → Č_{x}(M) → Č_{x}(M̄_{≤N}). From the first part we see that Č_{x}(M̄_{≤N}) = M̄_{≤N}. □

Lemma 3.5. Let C and D be complexes of graded kG-modules and let M and N be integers such that C_{>M} and D_{>N} are split exact. Then (C ⊗_{k} D)_{>M+N} is split exact.

Proof. We have (C ⊗_{k} D)_{i} = ∑_{u+v=i} C_{u} ⊗_{k} C_{v}. If i > M + N, then either u > M and so C_{u} is split exact, or v > N and so D_{v} is split exact. A split exact complex tensored with any other complex is split exact. □

Lemma 3.6. Let G be a finite group and let M be a kG-module. Let C be a set of subgroups of G.

(1) If M is projective relative to C and M is of finite decomposition type after restriction to any subgroup in C, then M is of finite decomposition type.

(2) If 0 → C_{0} → C_{1} → ⋯ → C_{n} → 0 is a complex of kG-modules that is split exact after restriction to any subgroup in C and C_{1}, …, C_{n} are projective relative to C, then the complex is split exact over kG.

Proof. For (1), use characterization (2) of relative projectivity. For (2), use induction on n and characterization (3) to split the map C_{n−1} ↠ C_{n}. □

Definition 3.7. A kG-module X is semi-invertible if there exist kG-modules A and B such that A ⊗_{k} X ≅ k ⊕ B. Similarly, a complex of kG-modules X is semi-invertible if there exist complexes A and B such that A ⊗_{k} X ≃ k ⊕ B.

The module A in this definition can be assumed to be finite dimensional. For if c is a generator of the copy of k on the right hand side then we have c = ∑_{i=1}^{n} x_{i} ⊗ a_{i} with x_{i} ∈ X, a_{i} ∈ A, and the a_{i} generate a finite dimensional kG-submodule of A that we can use instead of A.

4. Main Theorem

Theorem 4.1. Let S be a graded k-algebra of standard type on which a finite group G acts and let M be an SG-module. Let N be an integer such that hreg(M) ≤ N. Suppose that for each p-subgroup P of G there exist homogeneous elements y_{1}, …, y_{r} and z_{1}, …, z_{s} of S_{>0}^{P} such that:

(a) y_{i} vanishes on (Spec(S))^{P},

(b) rad_{S}(yz) = 𝔪_{S}, (c) there is a k[y]P -module T and a k[z]P -module U such that M_{>N} ≅ (T ⊗_{k} U)_{>N} as

k[yz]P -modules,

(d) (i) either the action of P on U is trivial or

(ii) U is semi-invertible and of finite decomposition type and for some integer d the complex Č_{z}(U)_{>d} is split exact over kP and Č_{z}(U)_{d} is semi-invertible, and

(e) (i) either M is finitely generated over S or

(ii) M_{≤N} is of finite decomposition type and the T_{≤N−d} for each p-subgroup P < G are

also of finite decomposition type. Then:

(1) for any sequence of homogeneous elements x in S_{>0}^{G} such that rad_{S}(x) = 𝔪_{S}, the complex of kG-modules Č_{x}(M)_{>N} is split exact;

(2) hreg(M^{G}) ≤ N;

(3) M is of finite decomposition type.

Proof. Part (2) follows from part (1) by the discussion at the beginning of Section 3. Parts (1) and (3) hold over kG if they hold on restriction to a Sylow p-subgroup, by Lemma 3.6 with C the class of Sylow p-subgroups. We may therefore assume that G is a p-group.

We will prove (1) and (3) together, but with G replaced by one of its p-subgroups, P, using induction on the order of P. The conclusions are clearly valid when |P | = 1, because hreg(M) ≤ N, so assume that P ≠ 1 and the conclusions hold over all proper subgroups of P. Notice that (d.i) implies (d.ii), so we will only consider (d.ii). In fact, (e.i) also implies (e.ii) as follows. If M is finitely generated over S, then M_{>N} is finitely generated over k[yz]; under the isomorphism in (b), there is a finite set of generators m_{i} = ∑_{j} t_{ij} ⊗u_{ij} with t_{ij} ∈ T and u_{ij} ∈ U, all homogeneous. We can replace T by its k[yz]P -submodule generated by the t_{ij}; then T_{≤N−d} is finitely dimensional, so certainly of finite decomposition type.

On restriction to any proper subgroup Q of P, the complex Č_{yz}(M)_{>N} is split exact, by induction; the same must be true for Č_{yz}(T ⊗_{k} U)_{>N} = (Č_{y}(T)⊗_{k} Č_{z}(U))_{>N}, by Lemma 3.4.

The complex Č_{z}(U)_{d} is semi-invertible, so there is a complex A in degree −d such that A ⊗_{k} Č_{z}(U)_{d} ≃ k ⊕ B, with k in degree 0. Hence Č_{y}(T) is homotopy equivalent over kP to a summand of A ⊗_{k} (Č_{y}(T) ⊗_{k} Č_{z}(U)); the latter, on restriction to Q, is split exact in degrees greater than N − d, by the previous paragraph, (d.ii) and Lemma 3.5. Thus Č_{y}(T)_{>N−d} is split exact on restriction to any proper subgroup of P.

By condition (a), y_{i} ∈ I_{P,S}, the ideal defining the fixed point subscheme. But I_{P,S_{y_{i}}} = (I_{P,S})_{y_{i}}, so I_{P,S_{y_{i}}} = S_{y_{i}}; this means that (Spec(S_{y_{i}}))^{P} = ∅. Proposition 3.1 and Lemma 3.3 now imply that M_{y_{i}}, or indeed any M_{y_{i_{1}}···y_{i_{ℓ}}} with ℓ ≥ 1, is projective relative to proper subgroups. But M_{y_{i_{1}}···y_{i_{ℓ}}} ≅ (M_{>N})_{y_{i_{1}}···y_{i_{ℓ}}} ≅ ((T ⊗_{k} U)_{>N})_{y_{i_{1}}···y_{i_{ℓ}}} ≅ (T ⊗_{k} U))_{y_{i_{1}}···y_{i_{ℓ}}} ≅ T_{y_{i_{1}}···y_{i_{ℓ}}} ⊗_{k} U, by Lemma 3.4. Since U is semi-invertible by condition (d.ii), T_{y_{i_{1}}···y_{i_{ℓ}}} must be projective relative to proper subgroups for ℓ ≥ 1.

We can now use Lemma 3.6 with C equal to the class of proper subgroups of P to deduce that Č_{y}(T)_{>N−d} is split exact over kP. From the assumption that Č_{z}(U)_{>d} is split exact over kP and Lemma 3.5, it follows that Č_{yz}(U ⊗_{k} T)_{>N} is split exact; hence so is Č_{yz}(M)_{>N}, by Lemma 3.4. This proves (1) in the case (x) = (yz).

Since U is semi-invertible, there exist kP -modules A and B, with A finite dimensional, such that U ⊗_{k} A ≅ k ⊕ B. Using Lemma 3.4 again, we obtain M_{y_{i}} ⊗_{k} A ≅ T_{y_{i}} ⊗_{k} U ⊗_{k} A ≅ T_{y_{i}} ⊕(T_{y_{i}} ⊗_{k} B). By induction, M is of finite decomposition type on restriction to any proper subgroup; hence so is M_{y_{i}}, by Lemma 2.3, and also M_{y_{i}} ⊗_{k} A, since A is finite dimensional. It follows that T_{y_{i}} is of finite decomposition type after restriction to any proper subgroup.

Because we have seen that T_{y_{i}} is projective relative to proper subgroups, it follows from characterization (2) of relative projectivity that each T_{y_{i}} is of finite decomposition type over kG; hence so is T_{>N−d}, by the splitting of Č_{y}(T)_{>N−d}. Since T_{≤N−d} is of finite decomposition type by (e.ii), it follows that T is of finite decomposition type. Finally, U is assumed to be of finite decomposition type, hence so is T ⊗ U and thus M_{>N}. But M_{≤N} is assumed to be of finite decomposition type; hence so is M, proving (3).

By Proposition 2.5, we can now change yz to x, which proves (1). □

Remark. Condition (d.ii) is rather artificial, but it serves to clarify the proof. Other formulations for non-trivial action on U are possible. The key point arises when Q < P and (writing T_{Q} for the T associated to Q etc.) we need to be able to deduce that Č_{y_{P}} (T_{P}) is split exact over kQ given that Č_{y_{Q}}(T_{Q}) is.

Corollary 4.2. If a finite group G acts by linear substitutions on a polynomial ring S = k[x_{1}, …, x_{n}] with the x_{i} in degree 1, then S is of finite decomposition type and hreg(S^{G}) ≤ −n and reg(S^{G}) ≤ 0.

Proof. The condition on reg is weaker than that on hreg, so it suffices to verify the hypotheses of Theorem 4.1 with N = −n. The method is a variant of that in [16, §6]. Clearly hreg(S) ≤ −n.

Let P be a p-subgroup of G. Let V be the dual space to S_{1}, regarded as a left module in the usual way, and let {x_{i}^{∗}} be the dual basis; we regard S as k[V]. Letting s = dim V^{P} and r = n − s, we may change the basis of V (and of S_{1}) so that the matrices for the action on V are lower triangular and x_{r+1}^{∗}, …, x_{n}^{∗} span V^{P}. For g ∈ P we have

gx_{i}^{∗} = \begin{cases}x_{i}^{∗} + ∑_{i<j} λ_{i,j}(g)x_{j}^{∗}, & if i ≤ r \\ x_{i}^{∗} & if i > r, \end{cases}

and thus

gx_{i} = x_{i} + ∑_{j<i, j≤r} λ_{j,i}(g^{−1})x_{j}.

Let d_{x_{i}} = ∏_{g∈P/ Stab x_{i}} gx_{i} denote the orbit product of x_{i}. When expressed as a sum of

monomials in the x_{i}, d_{x_{i}} involves x_{i}^{deg d_{x_{i}}} and does not contain any x_{j} with j > i. Set y_{i} = d_{x_{i}} for 1 ≤ i ≤ r and z_{i} = d_{x_{i+r}} for 1 ≤ i ≤ s. Let α_{i} = deg z_{i}. Let T be the k-subspace of S spanned by the monomials with x_{i+r}-degree strictly less than

α_{i} for 1 ≤ i ≤ s; it is, in fact a kP -submodule. Let U = k[z_{1}, …, z_{s}], with trivial action of P.

We claim that S ≅ T ⊗_{k} U by the natural map arising from the inclusion of T in S and multiplication by the z_{i} (cf. [16, 6.4]). The two sides have the same graded dimensions, so it suffices to check surjectivity; this can be done by a standard argument used in Gröbner basis theory. We order the basis elements by x_{n} > x_{n−1} > ⋯ > x_{1} and give the monomials the corresponding graded lexicographic order. Thus, if we write z_{i} = x_{i+r}^{α_{i}} + X_{i} for 1 ≤ i ≤ s, each monomial appearing in X_{i} is strictly smaller than x_{i+r}^{α_{i}}.

Suppose that a be the smallest monomial that is not in the image of the map above. If each x_{i+r} for 1 ≤ i ≤ s appears in a to a power strictly less than α_{i}, then a ∈ T so is in the image. Otherwise, let j be the largest value of i with 1 ≤ i ≤ s such that x_{i+r}^{α_{i}} divides a; then a = x_{j+r}^{α_{j}}b = (z_{j} − X_{j})b. Since b < a, b is in the image, so is z_{j}b. Also, each monomial in X_{j}b is strictly smaller than a, so must be in the image. Thus a is in the image, a contradiction.

This proves the claim, and the lettered hypotheses of Theorem 4.1 are satisfied for M = S and any N. □

Remark. The finite decomposition type part of Corollary 4.2 was shown in [16] in the case when k is finite. The regularity part was the main result of [20].

Remark. Given a kG-module M, let T (M) = ∑_{H<G, p||G:H|} tr_{H}^{G}(M^{H}) and set F (M) = M^{G}/T (M), obtaining a functor with values in vector spaces. If M is also an RG-module and x ∈ R then this preserves the R-module structure and it is easy to check that T (M_{x}) ≅ T (M)_{x} and hence F (M_{x}) ≅ F (M)_{x}. It follows that Č_{x}(F (M)) ≅ F (Č_{x}(M)).

In particular, take M to be the polynomial ring S = k[V] of Corollary 4.2, where dim V = n. The complex Č_{x}(S) is split exact over kG in degrees greater than −n, hence so is Č_{x}(F (M)). Thus hreg(F (S)) ≤ −n.

It was shown by Fleischmann that dim F (k[V]) = dim_{k}(V^{P}), where P is a Sylow psubgroup of G. Thus reg(F (S)) ≤ s − n, where s = dim V^{P}. Fleischmann’s proof shows that a set of elements of k[V]^{G} that restricts to a system of parameters on k[V^{P}] also forms a system of parameters on F (k[V]). This allows us to find a bound on the degrees of the generators of F (k[V]) along the lines of [20, 2.1].

Using Dade’s Lemma we can find a system of parameters of degree at most |G|; this leads to the bound max{(s|G| − n, |G|}. If k is the finite field 𝔽_{q} and G is a p-group, then we can use invariants for the full group of lower triangular matrices and obtain the bound max{q^{n−s}(q^{s} −1)/(q −1)−n, q^{n−1}}. If G is not a p-group, we can use the Dickson invariants; it is those of degrees q^{n} − q^{n−1}, …, q^{n} − q^{n−s} that restrict correctly (see e.g. [4, 8.1.2]). This yields the bound max{sq^{n} − q^{n−s}(q^{s} − 1)/(q − 1) − n, q^{n} − q^{n−1}}.

Fleischmann [13, 14.1, 14.2] showed that F (k[V])/ rad(F (k[V])) is Cohen-Macaulay. Totaro [21] has shown that F (k[V]) is Cohen-Macaulay and also suggested this remark.

5. Examples

There are several observations that can help in checking the hypotheses of Theorem 4.1. First of all, we only need to check representatives of the p-subgroups up to conjugacy. Also, if Q < P and (Spec(S))^{Q} = (Spec(S))^{P} then we do not need to check for Q. If we happen to know for some subgroup P and some sequence x_{1}, …, x_{n} in S^{P} that Č_{x}(S)_{>N} is split over kP, then this can replace conditions (a)-(e) for P.

The last of these is particularly useful if we only want to show that S is of finite decomposition type. For then the conditions at each P only have to hold for N large enough, and this is ensured by Proposition 2.4 if S is known to be of finite decomposition type over P, for example if P is cyclic.

Example. There are many interesting examples when S = k[x_{1}, …, x_{n}], with the x_{i} in positive degrees but not all the same, and G acts by k-algebra automorphisms that respect the grading.

For a simple case, let us assume that each element g ∈ G acts with x_{1} fixed and gx_{i} = x_{i} + λ_{i}x_{1}^{r_{i}} for i ≥ 2, λ_{i} ∈ k and r_{i} deg x_{1} = deg x_{i} if λ_{i} ≠ 0. Assuming that the action is faithful, for any non-trivial subgroup the fixed point subscheme of the action on the spectrum is given by x_{1} = 0. By the discussion above, we only have to verify the hypotheses for the whole group.

Let y_{1} = x_{1} and let z_{i} = d_{x_{i+1}} for 1 ≤ i ≤ n − 1 (this is the orbit product again). Let T be the part of S spanned by monomials with x_{i}-degree strictly less that deg d_{x_{i}} for 2 ≤ i ≤ n. Let U = k[z_{1}, …, z_{n−1}]. This satisfies the hypotheses of Theorem 4.1 with N = hreg(S) = − ∑ deg x_{i}. Thus S has finite decomposition type and hreg(S^{G}) ≤ − ∑ deg x_{i}.

In the next two examples it is possible to calculate the invariants and their regularity directly, so we concentrate on finite decomposition type.

Example. Suppose that k has characteristic 2 and let S = k[v, w, x, y, z]/(vx + wy), with all the generators in degree 1. Let G be the Klein four-group with generators a and b and let it act on S fixing v, w, x, y and with a(z) = z + x and b(z) = z + v.

The fixed point subscheme of the action is where v and x vanish. Using the fact that k[w, y] is free over k[w + y, wy] on the basis {1, y}, we obtain k[v, w, x, y, z] ≅ (k[v, x] ⊕ yk[v, x]) ⊗_{k} k[w + y, vx + wy, z]. Set y_{1} = v, y_{2} = w, z_{1} = w + y and z_{2} = d_{z}. Let T be the free k[v, x]-submodule of S spanned by {1, z, z^{2}, z^{3}, y, yz, yz^{2}, yz^{3}} and U = k[z_{1}, z_{2}]. Then S ≅ T ⊗_{k} U, verifying the conditions for G. By the discussion above, this suffices to show that S is of finite decomposition type.

Example. This example is the same as the previous one except that b(z) = z + y. The ring S is graded by total degree and also by v-degree+w-degree. Let A_{n} be the part of S with total degree n+1 and v-degree+w-degree=n. Then A_{n} is a kG-summand of S of dimension 2n+3. We will show that A_{n} is indecomposable, which proves that S is not of finite decomposition type.

The module A_{n} has a monomial basis, after the obvious identifications. The group action can be described by a diagram, as in [10]:

\xymatrix{ & w^{n}z \ar[dr]^{b−1} \ar[dl]_{a−1} &  & vw^{n−1}z \ar[dr]^{b−1} \ar[dl]_{a−1} &  & … &  & v^{n}z \ar[dr]^{b−1} \ar[dl]_{a−1} \\ w^{n}x &  & w^{n}y = vw^{n−1}x &  & vw^{n−1}x = &  & = v^{n}x &  & v^{n}y .}

This is known to be the diagram of an indecomposable module (in fact Ω^{−n−1}k). Another approach is to consider matrices, as in [3, 4.3.3].

References

[1] M.F. Atiyah, I.G. MacDonald, Introduction to Commutative Algebra, Addison-Wesley 1969.

[2] H. Bass, Finitistic dimension and a homological generalization of semi-primary rings, Trans. Amer. Math. Soc. 95 (1960) 466-488.

[3] D.J. Benson, Representations and Cohomology I, Cambridge Studies in Advanced Mathematics 30 (1991), Cambridge Univ. Press, Cambridge.

[4] D.J.Benson, Polynomial Invariants of Finite Groups, London Math. Soc. Lecture Notes 190, Cambridge Univ. Press, Cambridge 1993.

[5] F. Bleher, T. Chinburg, Galois structure of homogeneous coordinate rings, Trans. Amer. Math. Soc. 360 (2008), 6269-6301.

[6] S. Bouc, Tensor induction of relative syzygies, J. reine angewandte Mathematik 523 (2000), 113–171.

[7] M.P. Brodmann, R.Y. Sharp, Local Cohomology, Cambridge Studies in Advanced Mathematics 60, Cambridge Univ. Press, Cambridge 1998.

[8] M. Broué, On Scott modules and p-permutation modules: An approach through the Brauer morphism, Proc. Amer. Math. Soc. 93 (1985), 401–408.

[9] W. Bruns, J. Herzog, Cohen-Macaulay Rings, Cambridge Studies in Advanced Mathematics 39, Cambridge Univ. Press, Cambridge 1993.

[10] S.B. Conlon, Modular representations of C_{2} × C_{2}, J. Austral. Math. Soc. 10 (1969), 363-366.

[11] C. Faith, Algebra II, Ring theory, Grundlehren der Mathematischen Wissenschaften, 191 (1976), Springer-Verlag, Berlin-New York.

[12] P. Fleischmann, Relative trace ideals and Cohen-Macaulay quotients of modular invariant rings, Computational Methods for Representations of Groups and Algebras (Essen, 1997), 211-233, Progr. Math., 173 (1999), Birkhäuser.

[13] P. Fleischmann, On invariant theory of finite groups, CRM Proceedings and Lecture Notes 35 (2004), 43–70.

[14] D.G. Higman, Modules with a group of operators, Duke Math. J. 21 (1954), 369–376.

[15] S.B. Iyengar, G.J. Leuschke, A. Leykin, C. Miller, E. Miller, A.K. Singh, U. Walther, Twenty-Four Hours of Local Cohomology, Graduate Studies in Mathematics 87, Amer. Math. Soc. 2007.

[16] D. Karagueuzian, P. Symonds, The module structure of a group action on a polynomial ring: a finiteness theorem, J. Amer. Math. Soc. 20 (2007), 931–967.

[17] M. Lorenz, J. Pathak, On Cohen-Macaulay rings of invariants, J. Algebra textbf245 (2001), 247-264.

[18] G.O. Michler, Theory of Finite Simple Groups, Cambridge Univ. Press, Cambridge 2006.

[19] P. Symonds, Structure theorems over polynomial rings, Advances in Math. 208 (2007), 408–421.

[20] P. Symonds, On the Castelnuovo-Mumford regularity of rings of polynomial invariants, Ann. Math. 174 (2011), 499–517.

[21] B. Totaro, private communication 2012.

[22] R.B. Warfield Jr., A Krull-Schmidt theorem for infinite sums of modules, Proc. Amer. Math. Soc. 22 (1969), 460–465.

[23] R. Wisbauer, Foundations of Module and Ring Theory, Algebra, Logic and Applications, 3 (1991), Gordon and Breach, Philadelphia.

School of Mathematics, University of Manchester, Manchester M13 9PL, United Kingdom

  E-mail address: Peter.Symonds@manchester.ac.uk