ON THE CASTELNUOVO-MUMFORD REGULARITY OF RINGS OF POLYNOMIAL INVARIANTS

PETER SYMONDS

Abstract. We show that when a group acts on a polynomial ring over a field the ring of invariants has Castelnuovo-Mumford regularity at most zero. As a consequence, we prove a well-known conjecture that the invariants are always generated in degrees at most n(|G|−1), where n > 1 is the number of polynomial generators and |G| > 1 is the order of the group. We also prove some other related conjectures in invariant theory.

The main result of this paper is the following theorem.

Theorem 0.1. Let k be any field and S = k[x_{1}, …, x_{n}] a graded polynomial ring. Let the group G act on S by homogeneous linear substitutions. Then the invariant subring S^{G} has Castelnuovo-Mumford regularity at most zero.

As a consequence, we can prove a conjecture of Gregor Kemper [19] on the degrees of the generators of S^{G}.

Corollary 0.2. The invariant subring S^{G} is generated in degrees at most n(|G|−1) (provided that n > 1, |G| > 1). The relations between the the generators are generated in degrees at most 2n(|G| − 1).

That just |G| is a bound on the degrees of the generators when k has characteristic 0 is a result of Noether (for this reason such a bound is called a Noether bound in the literature). This was generalized to the case of coprime characteristic by Fleischmann [13] and Fogarty [14], with a much simplified proof by Benson. However, in general, no bound depending only on |G| is possible, as was shown by Richman [21, 22]. The bound above was shown to hold when S_{1} is a trivial source module by Göbel [15]. Some very much weaker bounds have been shown to hold by Derksen and Kemper [8] and by Karagueuzian and the author [18].

This corollary follows from the next corollary, which in turn follows easily from the Main Theorem 0.1 and elementary properties of regularity.

Corollary 0.3. Let k[d_{1}, …, d_{n}] < S^{G} be a subring of primary invariants. Then: (1) the secondary invariants are bounded in degree by ∑_{i}(deg(d_{i}) − 1),

(2) the relations as a ring are generated in degrees at most 2 ∑_{i}(deg(d_{i}) − 1).

The first part was conjectured by Kemper [19]. It is well known when k has characteristic zero [23] and was proved in the case when the ring of invariants is Gorenstein by Campbell, Geramita, Hughes, Shank and Wehlau [7], then in the Cohen-Macauley case by Broer [5].

The Hilbert series of S^{G} is the formal power series H(S^{G}, t) = ∑_{i} dim_{k}(S_{i}^{G})t^{i}. It is known to be a rational function, by the Hilbert-Serre Theorem. The next result (strictly speaking only a corollary of the proof of 0.1) was also conjectured by Kemper [19].

1991 Mathematics Subject Classification. Primary: 13A50; Secondary: 13D45, 18G25.

Partially supported by a grant from the Leverhulme Trust.

Corollary 0.4. The degree of H(S^{G}, t) as a rational function is at most −n.

This corollary is easily seen to be equivalent to the previous one in the Cohen-Macaulay case. It was proved for a reductive algebraic group over an algebraically closed field of characteristic zero by Knop [20].

There is an excellent survey of results and conjectures on degree bounds by Wehlau [26]. The proof of the Main Theorem 0.1 is quite short, but it depends heavily on the details

of the Structure Theorem that we proved with Karagueuzian in [18] and on the relatively projective resolutions of [24]. We also need to develop some of the properties of Castelnuovo-Mumford regularity.

I wish to thank Dikran Karagueuzian, without whom this project would have been impossible, and Gregor Kemper for making these conjectures and for his hospitality. Burt Totaro showed me how to extend these results to infinite fields and David Wehlau provided some calculations which revealed an error in a preliminary version of this paper. I also thank Ergün Yalçin for introducing me to the concept of regularity and Luchezar Avramov for patiently explaining it to me.

1. Castelnuovo-Mumford Regularity

We will work in categories of ℤ-graded rings and modules throughout, so M = ⊕_{i∈ℤ}M_{i}. We will write M_{≥d} = ⊕_{i≥d}M_{i} and similarly for other inequalities. We also use M(d) to denote a shift down in grading by d, so that M(d)_{i} = M_{i+d}.

Let k be a field and let R be a finitely generated commutative graded k-algebra in nonnegative degrees with dim_{k} R_{0} < ∞. Let I be a (homogeneous) ideal in R and let M be an R-module (graded, by assumption). The I-torsion in M is Γ_{I}(M) = {m ∈ M | ∃n ∈ ℕ I^{n}m = 0}. The local cohomology, H_{I}^{i}(R, M), is then defined to be the ith right derived functor of Γ_{I}(M) (in the category of graded R-modules); frequently the ring R is suppressed from the notation and just H_{I}^{i}(M) is written. It follows easily from the definitions that H_{I}^{i}(R, M) = H_{√\overline{I}}^{i}(R, M).

For more information on local cohomology see [6], [4], [11] or [17].

Let 𝔪 = R_{>0} be the ideal of positively graded elements of R; usually we will have R_{0} = k, so 𝔪 is the unique maximal homogeneous ideal. We will be interested in H_{𝔪}^{i}(M).

Let a_{i}(R, M) denote the maximum degree of a non-zero element of H_{𝔪}^{i}(R, M) (possibly ∞ if unbounded or −∞ if H_{𝔪}^{i}(R, M) = 0). The Castelnuovo-Mumford regularity (or just regularity) of M over R is, by definition,

reg(R, M) = sup_{i}{a_{i}(R, M) + i}.

The number reg(R, R) is important and we denote it by just reg(R).

The Independence Theorem for local cohomology ([16] 5.7, [4] 13.1.6) states that if R′ is another ring satisfying the same conditions as R, I′ < R′ is an ideal and f : R′ → R is a ring homomorphism (all graded) then f induces an isomorphism H_{I′}^{i}(R′, M) → H_{I′R}^{i}(R, M), (where we regard M as an R′-module via f and I′R is the ideal in R generated by f(I′)). Let 𝔪′ = R′_{>0}; if R is finite over R′ then it is easy to see that \sqrt{𝔪′R} = \sqrt{𝔪}. Combining these facts we obtain H_{𝔪′}^{i}(R′, M) ≅ H_{𝔪}^{i}(R, M), so reg(R′, M) = reg(R, M).

We will use this theory when R′ is a Noether normalization of R, that is a polynomial subring over which R is finitely generated. These always exist and their generators are often referred to as a homogeneous system of parameters, or as primary invariants in the case of invariant theory (for more information on this see the references mentioned above or [1]). Restricting to any normalization will yield the same value for the regularity and we will usually write just reg(M).

Now suppose that R = k[x_{1}, …, x_{n}] is a polynomial ring in which the generators have arbitrary positive degree |x_{i}| = deg(x_{i}). We set σ(R) = ∑_{i=1}^{n}(|x_{i}|−1) although traditionally one considers the a-invariant, a(R) = −σ(R) − n = − ∑_{i=1}^{n} |x_{i}|.

Let M be an R-module and consider the minimal (graded) projective resolution of M (projective is equivalent to free in this case)

⋯ → P_{1} → P_{0} → M → 0.

Let ρ_{i}(R, M) be the maximum degree of a non-zero element of R/𝔪 ⊗_{R} P_{i} (possibly ∞ or −∞), which is equal to the maximum degree of a generator of P_{i} (Benson [2, 3] uses β_{i} instead of ρ_{i}, but this can be confused with the Betti numbers). Define

Preg(R, M) = sup_{i}{ρ_{i}(R, M) − i} − σ(R).

This form of the definition first appeared in a paper of Benson [2]. The usual definition does not contain a σ-term, because all the d_{i} are supposed to be in degree 1 and so σ(R) = 0; but the necessity of using this form will become apparent.

We consider Hom-groups between graded modules to be graded modules as well. The homogeneous part in degree i consists of the homomorphisms that increase the grading by i. In this way the Ext groups are also graded modules.

Now define ϵ_{i}(R, M) to be the minimum degree of a non-zero element of Ext_{R}^{i}(M, R) (possibly ∞ or −∞). Define

Extreg(R, M) = sup_{i}{−ϵ_{i}(R, M) − i} − σ(R).

Lemma 1.1. Assume that M is finitely generated over R (which is still a polynomial ring). Then Preg(R, M) = Extreg(R, M).

Proof. A proof is given in [10] 20.16 (it is assumed there that σ(R) = 0, but the argument still holds). For the convenience of the reader we sketch the proof.

That Preg(R, M) ≥ Extreg(R, M) follows easily from the definitions. For the reverse inequality, we will show that ρ_{i}(R, M) ≤ Extreg(R, M) + σ(R) + i by downward induction on i. Since R is a polynomial ring, the minimal resolution has finite length, so the induction certainly starts.

Let ⋯ → P_{r} →^{d_{r}} P_{r−1} → ⋯ → M → 0 denote the minimal projective resolution of M. We claim that there is no map f : P_{r} → R(−u), for any r or u, such that fd_{r+1} is onto. For then fd_{r+1} would split, and P_{r+1} would contain a summand R(−u) that mapped isomorphically to its image in P_{r}. We could then factor out the two copies of R(−u) in P_{•} and obtain a smaller resolution of M, a contradiction.

Suppose that the inequality is established for i > r, but that P_{r} contains a summand R(−u) with u > Extreg(R, M) + σ(R) + r. In particular, u > −ϵ_{r}(R, M). Let f be the corresponding projection of P_{r} onto R(−u) and consider the map (fd_{r+1})_{u} : (P_{r})_{u} → R(−u)_{u} ≅ k. If it is onto then fd_{r+1} is onto, which is impossible by the discussion above. Thus (fd_{r+1})_{u} = 0 and since, by the induction hypothesis, P_{r+1} contains no summands R(−v) with v > u, the map fd_{r+1} = 0. Thus f determines an element of Ext_{R}^{r}(M, R(−u))_{0} ≅ Ext_{R}^{r}(M, R)_{−u}.

But this Ext-group is zero by the condition on u, so f factors through d_{r}, again contradicting the minimality of the resolution. □

Note that we might well have that ρ_{i}(R, M) ≠ ϵ_{i}(R, M).

The Local Duality Theorem ([16] 6.3, [6] 3.6.19/3.6.11) states that for R a polynomial ring in n variables as above and M a finitely generated R-module we have

Hom_{k}(H_{𝔪}^{i}(M), k) ≅ Ext_{R}^{n−i}(M, R(a(R))).

Recall that k is in degree 0 and that R(a(R)) denotes a copy of R that has been shifted down in degree by a(R) or, equivalently, up in degree by σ(R) + n.

Proposition 1.2. If R is a polynomial ring over a field and M is a finitely generated Rmodule, then Preg(R, M) = reg(R, M).

Proof. Combine the Local Duality Theorem with Lemma 1.1. □

This result is well known when σ(R) = 0 (see e.g. [10] A4.2). It is stated in this generality in [3] 2.3.

We are really only concerned with Preg in this paper, but we need the connection with local cohomology in order to see that if R is a noetherian ring and R′ and R′′ are two different Noether normalisations of R, then for any finitely generated R-module M we have Preg(R′, M) = Preg(R′′, M).

For example, if R is a ring of polynomial invariants, then Preg(R′, R) does not depend on the choice of primary invariants R′ and it is equal to reg(R).

Remark. If R is a polynomial ring then reg(R) = Preg(R; R) = −σ(R), so the regularity of a ring of invariants can certainly be negative. However, in characteristic zero the ring of invariants has regularity zero if and only if the representation in degree one has trivial determinant (see [23] 3.9).

This is in contrast to the case of the cohomology of a finite group, where Benson shows in [3] that reg(H^{∗}(G, 𝔽_{p})) ≥ 0 and conjectures that equality holds. This has now been proved [25].

2. Generators and Relations

Given a finitely generated graded k-algebra S in non-negative degrees and an integer N, let τ_{N}^{k}S be the k-algebra determined by the generators and relations of S that occur in degrees at most N. We will normally write just τ_{N}S. There is a canonical map τ_{N}S → S, which is an isomorphism in degrees up to and including N.

For a more abstract setting, consider the functor S ↦ S/S_{>N} on graded k-algebras in non-negative degrees; τ_{N} is its left adjoint.

It is not hard to see that if ℓ is an extension field of k then τ_{N}^{ℓ}(ℓ ⊗_{k} S) ≅ ℓ ⊗_{k} τ_{N}^{k}S.

Proposition 2.1. Let R = k[d_{1}, …, d_{m}] and suppose that there is a map f : R → S such that S is finitely generated over R (e.g. if R is a Noether normalisation of S). Then:

(1) if N ≥ max{reg(S) + σ(R), deg(d_{i})}, then τ_{N}S → S is a surjection;

(2) if N ≥ max{2(reg(S) + σ(R)), reg(S) + σ(R) + 1, deg(d_{i})}, then τ_{N}S → S is an isomorphism; (3) if N ≥ max{reg(S) + σ(R) + 1, deg(d_{i})} and if τ_{N}S, considered an R-module, is

generated in degrees at most N, then τ_{N}S → S is an isomorphism.

Proof. Let ⋯ → P_{1} → P_{0} → S → 0 be the minimal projective resolution of S as an R-module.

It is clear from the definitions that S, considered as an R-module, is generated in degrees at most ρ_{0}(R, S) and is presented in degrees at most max{ρ_{0}(R, S), ρ_{1}(R, S)}.

If N ≥ max{deg(d_{i})}, then f can be lifted uniquely to τ_{N}S, making τ_{N}S into a finitely generated R-module.

Let {v_{i}} be a set of homogeneous generators of S as an R-module with minimum degrees. These have degrees not exceeding ρ_{0}(R, M), which is bounded by reg(S) + σ(R) according to the definition of Preg. This proves part (1).

The R-module relations between the v_{i} are generated in degrees at most reg(S)+σ(R)+1. The only information still needed in order to determine the structure of S as a ring is an expression for each of the products v_{j}v_{k} as an R-linear combination of the v_{i}. Such a formula will lie in degree at most 2(reg(S) + σ(R)). This proves part (2).

For part (3), consider the following commutative diagram of R-modules with exact rows, where the vertical arrows can be filled in since the P_{i} are projective.

⟶ P_{1} ⟶ P_{0} ⟶ S ⟶ 0
↓  ↓  ∥
0 ⟶ K ⟶ τ_{N}S ⟶ S ⟶ 0

We know that K_{≤N} = 0 and that P_{1} is generated in degrees at most ρ_{1}(R, S) ≤ N. Thus the composite map P_{1} → τ_{N}S is zero. It follows that the bottom row is split as a sequence of R-modules. Since τ_{N}S is generated as an R-module in degrees at most N, by hypothesis, so is K, which implies that K = 0. □

Remark. There is a similar result when both S and τ_{N}S are taken to be graded commutative rings (although R remains strictly commutative).

Proofs of 0.2 and 0.3. Corollary 0.3 now follows directly from the Main Theorem 0.1 and Proposition 2.1. It is easy to check that the maximum in the formulas of Proposition 2.1 is achieved by the first term, except in the trivial case when all the d_{i} have degree 1.

The remark about τ_{N} commuting with field extensions shows that, in order to prove Corollary 0.2, we may extend the field. But then a result of Dade in [23] shows that for some finite field extension we can find a set of primary invariants of degree at most the order of the group, so from Corollary 0.3 we obtain the bound max{n(|G| − 1), |G|} for the degrees of the generators. But the |G| term is only larger than the other in the trivial cases that are excluded in the statement of the Corollary. Similarly, the bound obtained on the degrees of the relations is always 2n(|G| − 1), except in the case when G = 1 and there are no relations anyway.

Remark. In the case when G is a p-group, one of the primary invariants can be taken to be in degree 1 and we obtain the bound (n − 1)(|G| − 1).

Remark. Benson [2] §10 defines τ_{N}S and proves a version of 2.1 in the case when S is the cohomology of a group.

3. Relatively Projective Resolutions

Let M be an RG-module for some finite group G. A relatively projective resolution P_{•} of M relative to kG is a complex of RG-modules

⋯ → P_{2} → P_{1} → P_{0} → M → 0

that is split exact over kG and in which each P_{i} is a sum of terms R ⊗_{k} V (−d), where V (−d) is a kG-module considered to be in degree d. For brevity we will call this just an RG/kG-resolution. Here R acts on the first term of R ⊗_{k} V (−d) and G acts on the second. We could also write R ⊗_{k} V (−d) as RG ⊗_{kG} V (−d), with RG acting on the left in the usual way.

If N is another RG-module then we obtain R-modules Ext_{RG/kG}^{i}(M, N) by applying Hom_{RG}(−, N) to the resolution of M and taking homology.

For more information on general relative homological algebra see [9, 12, 27]. We will closely follow our treatment in [24]. In particular, the next result is taken from [24] 4.2.

Theorem 3.1. If M is a finitely generated RG-module, then M has a unique minimal RG/kG-resolution P_{•}. For any indecomposable kG-module V and any integer d, the number of summands of the form R ⊗_{k} V (−d) in any given term P_{i} is finite and is zero if V is not a summand of M as a kG-module. If R is polynomial then this resolution has finite length.

The question of minimality and uniqueness is not explicitly addressed in [24], but the construction used there proceeds by changing the problem to one about projective resolutions in another category, in which modules bounded below have projective covers, so the existence of a unique minimal projective resolution is guaranteed. For a theoretical framework in the context of relative homological algebra see [12] chapter 8.

From now on, assume that R is polynomial and that M is finitely generated over R. For any indecomposable kG-module V, define ρ_{i}(R, G; M, V) to be the largest d for which R ⊗_{k} V (−d) appears in the P_{i} term of the minimal RG/kG-resolution of M. Define

ρ_{i}(R, G; M) = sup_{V}{ρ_{i}(R, G; M, V)},
Preg(R, G; M) = sup_{i}{ρ_{i}(R, G; M) − i} − σ(R).

It is clear that if H is a subgroup of G, then an RG/kG-resolution of M restricts to an RH/kH-resolution of M ↓_{H}. This fact together with the next lemma is key to our strategy.

Lemma 3.2. If ⋯ → P_{1} → P_{0} → M → 0 is an RG/kG-resolution of M, then ⋯ → P_{1}^{G} → P_{0}^{G} → M^{G} → 0 is a projective resolution of the invariants M^{G} over R. As a consequence,

ρ_{i}(R, M^{G}) ≤ ρ_{i}(R, G; M),
Preg(R, M^{G}) ≤ Preg(R, G; M).

Proof. Since the resolution is split over kG, taking fixed points preserves exactness. Each P_{i} is a sum of terms of the form R⊗_{k}V (−d); thus P_{i}^{G} is a sum of terms of the form R⊗_{k}V^{G}(−d), so is free over R. □

The next lemma is a sort of generalization of 1.1, but notice that the bound is horizontal instead of diagonal.

Lemma 3.3. For any integer N, the following are equivalent:

(1) ρ_{i}(R, G; M) ≤ N for all i;

(2) Ext_{RG/kG}^{i}(M, R ⊗_{k} V (−d))_{0} = 0 for all kG-modules V, all i and all d > N;

(3) Ext_{RG/kG}^{i}(M, R ⊗_{k} V (−d))_{0} = 0 for all indecomposable kG modules V that occur as a summand of M, all i and all d > N;

(4) for each indecomposable kG-module V, either ρ_{i}(R, G; M, V) ≤ N for all i or Ext_{RG/kG}^{i}(M, R⊗_{k} V (−d))_{0} = 0 for all i and all d > N.

Proof. This is essentially what is proved in [24] §5, where it is shown that condition (3’) there is equivalent to the other conditions, although the given proof does not explicitly keep track of N, so we do so here.

Condition (4) is implied by each of the other conditions and, using the definition of Ext_{RG/kG}^{∗}, we see that condition (1) implies all the others, so we concentrate on (4)⇒(1). We assume (4) and prove (1) by downward induction on i. Since the minimal projective resolution is of finite length, the induction starts.

We suppose that ρ_{i}(R, G; M) ≤ N for all i > r. Consider an indecomposable kG-module V; if ρ_{r}(R, G; M, V) ≤ N then there is nothing to prove, so we assume that there is a summand R ⊗_{k} V (−u) of P_{r} for some u > N. Let f be a projection of P_{r} onto R ⊗_{k} V (−u). Since P_{r+1} is generated as an R-module in degrees at most N, the map fd_{r+1} is zero. Thus f determines an element of Ext_{RG/kG}^{r}(M, R ⊗_{k} V (−u))_{0}.

But this Ext-group is zero, by hypothesis, so f factors through d_{r}. It follows that R ⊗_{k} V (−u) →^{d_{r}} d_{r}(R ⊗_{k} V (−u)) is a summand of the minimal resolution as a complex of RGmodules, a contradiction. We must have ρ_{r}(R, G; M, V) ≤ N, as required. □

Lemma 3.4. If H < G and W is a kH-module, then

Ext_{RG/kG}^{i}(M, R ⊗_{k} (W (−d)↑_{H}^{G})) ≅ Ext_{RH/kH}^{i}(M, R ⊗_{k} W (−d)).

As usual, ↑ denotes induction and ↓ denotes restriction.

Proof. This is an easy adaptation of the usual Eckmann-Shapiro Lemma. □

Lemma 3.5. If R′ < R and both are polynomial rings, R is finitely generated over R′ and V is an indecomposable kG-module then

ρ_{i}(R′, G; M, V) − σ(R′) ≤ ρ_{i}(R, G; M, V) − σ(R).

As a consequence,

Preg(R′, G; M) ≤ Preg(R, G; M).

Proof. Take a minimal RG/kG-resolution of M and restrict it to R′. By the basic theory of Cohen-Macauley rings (see e.g. [1] or [6]), R is free of finite rank over R′ with a basis of homogeneous elements {z_{j}} bounded in degree by σ(R′) − σ(R). For each summand R ⊗_{k} V (−d) in P_{i} in the original resolution, we now have ⊕_{j}R′z_{j} ⊗_{k} V (−d) and the result follows. □

Lemma 3.6. If R′ < R, M is a finitely generated R′G-module and V is an indecomposable kG-module, then

ρ_{i}(R, G; R ⊗_{R′} M, V) ≤ ρ_{i}(R′, G; M, V).

Proof. Tensoring an R′G/kG-resolution of M with R yields an RG/kG-resolution of R ⊗_{R′} M. □

Lemma 3.7. If H < G then ρ_{i}(R, H; M ↓_{H}^{G}) ≤ ρ_{i}(R, G; M) and so Preg(R, H; M ↓_{H}^{G}) ≤ Preg(R, G; M).

Proof. Take an RG/kG-resolution of M and restrict it to H. □

Lemma 3.8. If P is a Sylow p-subgroup of G (where p is the characteristic of k), then ρ_{i}(R, P; M ↓_{P}^{G}) = ρ_{i}(R, G; M) and so Preg(R, P; M ↓_{P}^{G}) = Preg(R, G; M).

Proof. The inequality ≤ follows from 3.7.

For ≥, note that M is a summand of M ↓_{P} ↑^{G}as an RG-module; thus ρ_{i}(R, G; M ↓_{P} ↑^{G}) ≥ ρ_{i}(R, G; M). But an RP/kP -resolution of M ↓_{P} induces to an RG/kG-resolution of M ↓_{P} ↑^{G} and so ρ_{i}(R, P; M ↓_{P}) ≥ ρ_{i}(R, G; M ↓_{P} ↑^{G}). □

Lemma 3.9. Let 0 → A → B → C → 0 be a short exact sequence of RG-modules that is split over kG and let V be an indecomposable kG-module. Then

ρ_{i}(R, G; B, V) ≤ max{ρ_{i}(R, G; A, V), ρ_{i}(R, G; C, V)}.

As a consequence,

Preg(R, G; B) ≤ max{Preg(R, G; A), Preg(R, G; C)}.

Proof. A relatively projective resolution for B can be constructed from ones for A and C (see [12] 8.2.1) just as in the Horseshoe Lemma for ordinary projective resolutions (cf. [27] 2.2.8). □

Lemma 3.10. If M is a finitely generated RG-module and d is an integer such that M_{>d} = 0, then Preg(R, G; M) ≤ d.

Proof. Filter M by its submodules M_{≥r}; since M is finitely generated and M_{>d} = 0, this is a finite filtration and the composition factors are just the homogeneous pieces M_{r}. Clearly M is the sum of its composition factors over kG. By repeated use of 3.9, we see that Preg(R, G; M) ≤ max_{r}{Preg(R, G; M_{r})}; hence it will be sufficient to show that Preg(R, G; M_{r}) ≤ r.

Let R = k[d_{1}, …, d_{n}], where |d_{1}| ≥ |d_{2}| ≥ ⋯ ≥ |d_{n}| ≥ 1. Since the d_{i} annihilate M_{r}, we can resolve M_{r} by tensoring it with the Koszul resolution on the d_{i}; this is an RG/kGresolution.

It is now easy to calculate that ρ_{i}(R, G; M_{r}) ≤ r + ∑_{j=1}^{i} |d_{j}|. But ∑_{j=1}^{i} |d_{j}| ≤ ∑_{j=1}^{i} |d_{j}| + ∑_{j=i+1}^{n}(|d_{j}|−1) = σ(R)+i. Hence ρ_{i}(R, G; M_{r})−i−σ(R) ≤ r and so Preg(R, G; M_{r}) ≤ r. □

4. The Structure Theorem

Here we summarize the material that we will require from our paper with Karagueuzian [18].

From now until Section 6, k will always be a finite field. Let S = k[x_{1}, …, x_{n}] be a polynomial ring with all the generators in degree 1. For any subset I ⊆ {1, 2, …, n − 1}, let U_{I} denote the group of upper-triangular matrices over k with 1’s on the diagonal and non-zero off-diagonal entries only in rows corresponding to the elements of I. The group U_{I} acts on S in the natural way, i.e. so that S_{1} is the canonical module. The invariants form a polynomial ring generated by the orbit powers of the x_{i}; we denote the latter by d_{i}(I) (here our notation differs slightly from that of [18]).

The Main Theorem of [18] is as follows.

Theorem 4.1. As a graded kU_{I}-module,

S ≅ ⨁_{J⊆I} k[d_{i}(I); i ∉ J] ⊗_{k} X̄_{J}(I),

where X̄_{J}(I) is a finite dimensional graded kU_{I}-submodule of S and the map from right to left is induced by multiplication in S.

We also have some information about the modules X̄_{J}(I). Let us write k[d(I)] for k[d_{i}(I); i = 1, …, n] and σ(I) for σ(k[d(I)]).

Proposition 4.2. (1) X̄_{J}(I) is induced from U_{J}.

(2) X̄_{∅}(I) is homogeneous of degree σ(I).

(3) X̄_{I}(I) lies in degrees at most σ(I) − ∑_{i∈I} |d_{i}(I)|.

Proof. We refer by numbers in parenthesis to statements in [18].

Part (1) is by construction (10.1(2)).

For part (2), notice that X̄_{∅}(I)^{U_{I}} is 1-dimensional in a degree that is denoted by deg_{I}(p⃗) (10.1(3) and 5.14). The fact that deg_{I}(p⃗) = σ(I) can be verified by direct calculation from the definitions or, more conceptually, by observing that deg X̄_{∅}(I) = deg(G(I, ∅)) by construction (10.1(2)), where G(I, ∅) is a polynomial that clearly has degree σ(I) from its definition (9.1).

Part (3) follows from part (2) and (5.22). □

We also record one other fact.

Proposition 4.3. The ring S contains a k[d_{i}(I); i ∈ I]U_{I}-submodule T (I) such that

S ≅ k[d_{i}(I); i ∉ I] ⊗_{k} T (I) ≅ k[d(I)] ⊗_{k[d_{i}(I);i∈I]} T (I)

as k[d(I)]U_{I}-modules.

This is (6.4) in [18]; although there the map is only stated to be a k[d_{i}(I); i ∈ I]U_{I}-module isomorphism, it is clearly a k[d(I)]U_{I}-isomorphism, by construction.

5. Proof of the Main Theorem

What we would like to do is to construct an explicit k[d(I)]U_{I}/kU_{I}-resolution of S. There is an obvious candidate for a description of what the modules in this resolution ought to be. For each X_{J}(I) = k[d_{i}(I); i ∉ J] ⊗_{k} X̄_{J}(I) in the Structure Theorem, there should be a contribution that looks like X_{J}(I) tensored with the Koszul complex on the d_{i}(I), i ∈ J. If we knew that there existed a filtration of S by k[d(I)]U_{I}-modules with the X_{J}(I) as composition factors, then this could be verified. However, the existence of such a filtration is not clear.

We will content ourselves with proving some of the bounds that would be implied by the existence of such a resolution.

Our key result is the following proposition. Notice that the bound is not the diagonal one that might be expected.

Proposition 5.1. In the context of the Structure Theorem,

ρ_{i}(k[d(I)], U_{I}; S) ≤ σ(I)

for all i ≥ 0. As a consequence,

Preg(k[d(I)], U_{I}; S) ≤ 0.

Proof. Use induction on |I|; the case I = ∅ is clear since k[d(∅)] = S and σ(∅) = 0.

Let V be an indecomposable kU_{I}-module; by Lemma 3.3, it is sufficient to verify condition 3.3(4) with N = σ(I). First consider the case when V is projective relative to some proper subgroup U_{J}, J ⫋ I; say V is a summand of W ↑_{U_{J}}^{U_{I}} for some indecomposable kU_{J}-module W. Then

Ext_{k[d(I)]U_{I}/kU_{I}}^{i}(S, k[d(I)] ⊗_{k} V) is a summand of Ext_{k[d(I)]U_{J} /kU_{J}}^{i} (S, k[d(I)] ⊗_{k} W),

by 3.4.

By the induction hypothesis, ρ_{i}(k[d(J)], U_{J}; S, W) ≤ σ(J), so, by 3.5, ρ_{i}(k[d(I)], U_{J}; S, W) ≤ σ(I).

This implies that the Ext group on the right above vanishes in degrees less than −σ(I) for all i, so the same is true for the one on the left. Since

Ext_{k[d(I)]U_{I}/kU_{I}}^{i}(S, k[d(I)] ⊗_{k} V (−d))_{0} ≅ Ext_{k[d(I)]U_{I}/kU_{I}}^{i}(S, k[d(I)] ⊗_{k} V)_{−d},

this Ext-group vanishes for d > σ(I), as required.

In the case that V is not projective relative to any proper subgroup U_{J}, we use 4.3 to write

S ≅ k[d(I)] ⊗_{k[d(I)′]} T (I),

where k[d(I)′] = k[d_{i}(I); i ∈ I]. Thus

ρ_{i}(k[d(I)], U_{I}; S, V) ≤ ρ_{i}(k[d(I)′], U_{I}; T (I), V),

by 3.6. We need to show that the right-hand side is bounded by σ(I).

Since V is not projective relative to any proper subgroup U_{J}, 4.2(1) shows that the only X̄_{J}(I) in which it can appear is X̄_{I}(I). As a consequence, V does not appear in T (I)_{>b}, where b is the maximum degree of an element of X̄_{I}(I) (finite, by 4.2(3)).

From the short exact sequence, split over kU_{I},

0 → T (I)_{>b} → T (I) → T (I)/T (I)_{>b} → 0,

it follows, by 3.9, that

ρ_{i}(k[d(I)′], U_{I}; T (I), V) ≤ ρ_{i}(k[d(I)′], U_{I}; T (I)/T (I)_{>b}, V).

But, since T (I)/T (I)_{>b} is bounded in degree by b, we know, from 3.10, that

Preg(k[d(I)′], U_{I}; T (I)/T (I)_{>b}) ≤ b.

Thus we have

ρ_{i}(k[d(I)′], U_{I}; T (I)/T (I)_{>b}, V) − i ≤ b + ∑_{i∈I}(|d_{i}(I)| − 1).

But, by 4.2(3), b ≤ σ(I) − ∑_{i∈I} |d_{i}(I)|. Thus ρ_{i}(k[d(I)′], U_{I}; T (I), V) ≤ σ(I), as required.

□

We can now finish the proof of the Main Theorem 0.1 in the case of a finite field.

Since the field k is finite, the largest possible group that can act on S is Gℓ_{n} = Gℓ_{n}(k), which is finite. Its Sylow p-subgroup is U_{n} = U_{{1,2,...,n−1}}, in the notation of Section 4. Let k[c] denote the ring of Dickson invariants in S for the action of Gℓ_{n} and let k[d] be the invariants for U_{n}.

For our given group G < Gℓ_{n} we may compute reg(S^{G}) by treating S^{G} as a k[c]-module.

reg(S^{G}) = Preg(k[c], S^{G})  by 1.2
≤ Preg(k[c], G; S)  by 3.2
≤ Preg(k[c], Gℓ_{n}; S) by 3.7
= Preg(k[c], U_{n}; S)  by 3.8
≤ Preg(k[d], U_{n}; S) by 3.5
≤ 0  by 5.1.

This completes the proof.

The last result that remains to be proved is Corollary 0.4. Let ⋯ → P_{1} → P_{0} → S^{G} → 0 be the minimal k[c]-resolution of S^{G}. Then

H(S^{G}, t) = ∑_{i}(−1)^{i}H(P_{i}, t).

Now

H(P_{i}, t) = \frac{f_{i}(t)}{∏_{j}(1 − t^{|c_{j}|})},

where f_{i}(t) is a polynomial in which the coefficient of t^{u} is the multiplicity of k[c](−u) in P_{i}. Thus the degree of f_{i}(t) is equal to ρ_{i}(k[c], S^{G}).

Because

H(S^{G}, t) = \frac{∑_{i}(−1)^{i}f_{i}(t)}{∏_{j}(1 − t^{|c_{j}|})},

we see that it suffices to show that deg(f_{i}) ≤ σ(k[c]). But we have just seen that deg(f_{i}) = ρ_{i}(k[c], S^{G}), and ρ_{i}(k[c], S^{G}) ≤ ρ_{i}(k[c], G; S), by 3.2. The proof concludes with the next lemma.

Lemma 5.2. ρ_{i}(k[c], G; S) ≤ σ(k[c]).

Proof.

ρ_{i}(k[c], G; S) ≤ ρ_{i}(k[c], Gℓ_{n}; S)  by 3.7
= ρ_{i}(k[c], U_{n}; S)  by 3.8
≤ ρ_{i}(k[d], U_{n}; S) − σ(k[d]) + σ(k[c]) by 3.5
≤ σ(k[c])  by 5.1.
□

6. Infinite Fields

We now explain how the case of Theorem 0.1 for finite fields implies the result for all fields. This argument was shown to us by Burt Totaro and we are grateful to him for permission to include it here.

Of course, if the representation of G on S_{1} can be written in the algebraic closure of the prime field, then it can be written in a finite field, so our results for finite fields still hold. It is not so clear what might happen if the field contains transcendental elements.

We have a finite group G that acts on S_{K} = K[x_{1}, …, x_{n}] for some infinite field K. The representation of G on (S_{K})_{1}, the part in degree 1, can be written over a finitely generated subring A of K (the ℤ-subalgebra generated by the coefficients of the matrices with respect to some basis); hence the same is true in all degrees and G acts on S_{A} = A[x_{1}, …, x_{n}].

Let S_{A}^{(G)} denote ∏_{g∈G} S_{A} and let Δ_{A}^{(G)} : S_{A} → S_{A}^{(G)} be the map that is multiplication by g − 1 on the g-coordinate. Then S_{A}^{G} = ker(Δ_{A}^{(G)}) and we have an exact sequence of S_{A}^{G}-modules

0 → S_{A}^{G} → S_{A} →^{Δ_{A}^{(G)}} S_{A}^{(G)} → C → 0.

We know, by the Hilbert-Noether Theorem (see e.g. [1] 1.3.1, [8] 3.0.6), that S_{A}^{G} is noetherian and that S_{A} is finite over it. By Grothendieck’s Generic Freeness (or Flatness) Lemma (see e.g. [10] 14.4), there is a non-zero element f ∈ A such that C ⊗_{A} A[f^{−1}] is free as an A[f^{−1}]- module. Let M be a maximal ideal of A[f^{−1}] with residue field k and let B denote the localization of A[f^{−1}] at M.

Lemma 6.1. As a B-module, S_{B}^{G} is free and S_{B}^{G} ⊗_{B} k ≅ S_{k}^{G}.

Proof. As an A-module, B is flat, so if we apply − ⊗_{A} B to the exact sequence above, it remains exact and becomes

0 → S_{A}^{G} ⊗_{A} B → S_{B} →^{Δ_{B}^{(G)}} S_{B}^{(G)} → C ⊗_{A} B → 0.

All the terms but the first are certainly free over B, so the sequence splits over B and the first term must also be free over B. This first term is ker(Δ_{B}^{(G)}); thus it is isomorphic to S_{B}^{G}. If we now apply − ⊗_{B} k, we obtain

0 → S_{B}^{G} ⊗_{B} k → S_{k} →^{Δ_{k}^{(G)}} S_{k}^{(G)} → C ⊗_{A} B ⊗_{B} k → 0.

But ker(Δ_{k}^{(G)}) = S_{k}^{G}, by the discussion above. □

By the Hilbert-Noether Theorem again, S_{B}^{G} is finitely generated by homogeneous elements as a B-algebra, and thus it is finite over some polynomial ring B[d_{1}, …, d_{m}]. Consequently, S_{k}^{G} is finite over k[d_{1}, …, d_{m}].

The next lemma is standard.

Lemma 6.2. Any projective resolution P_{•} of S_{k}^{G} over k[d_{1}, …, d_{m}] can be lifted to a projective resolution Q_{•} of S_{B}^{G} over B[d_{1}, …, d_{m}] such that P_{•} ≅ Q_{•} ⊗_{B} k.

Proof. We lift the resolution step by step, starting at the 0-term. Clearly P_{0} can be lifted to a projective module Q_{0}, and the map P_{0} → S_{k}^{G} can be lifted to a map Q_{0} → S_{B}^{G}, by 6.1. Furthermore, this map is surjective, by Nakayama’s Lemma in each degree; hence it is split over B, since S_{B}^{G} is free over B, by 6.1 again.

Let Z_{0} denote the kernel of the lifted map. It is free of finite rank over B and, because of the splitting, Z_{0} ⊗_{B} k ≅ ker(P_{0} → S_{k}^{G}). We can now repeat the procedure at the 1-term and continue. □

The field k is finite, since any field that is finitely generated as a ring is finite (cf. [10] 4.19). Thus our previous results give us bounds on the degrees of the generators of the terms of the minimal projective resolution P_{•} of S_{k}^{G} over k[d_{1}, …, d_{m}]. These bounds are inherited by Q_{•}, and even by of Q_{•} ⊗_{B} K, which is a projective resolution of S_{K}^{G} over K[d_{1}, …, d_{m}].

It follows that the results 0.1, 0.2, 0.3, 0.4 are all valid for arbitrary fields. The same will be true for 7.2 and 8.2.

Remark. Experts will recognize that the argument can be summarized by saying that regularity is upper semicontinuous on flat families.

7. Horizontal Bounds

The proofs yield more precise information than can be stated in terms of regularity, although this is not useful for bounding the degrees of the generators of the invariants, which is why we only mention it here.

Instead of using the usual diagonal bound in the definition of regularity we can use a horizontal one: we set

hreg(R, M) = sup_{i}{a_{i}(R, M)},
Phreg(R, M) = sup_{i}{ρ_{i}(R, M)} − ∑_{i} |d_{i}|,
Exthreg(R, M) = sup_{i}{ϵ_{i}(R, M)} − ∑_{i} |d_{i}|.

It is still true that these numbers coincide. The same proof still works, the key point being that the proof of Lemma 1.1 is still valid; in fact, what we need is Lemma 3.3 in the case of the trivial group (see too [2] 5.7(i)). Also hreg is still clearly invariant under change of ring.

It follows from the definitions that hreg(R, M) ≤ reg(R, M).

Similarly we can define

Phreg(R, G; M) = sup_{i}{ρ_{i}(R, G; M)} − ∑_{i} |d_{i}|.

Lemma 5.2 now becomes:

Proposition 7.1. In the context of the Structure Theorem,

Phreg(k[c], G; S) ≤ −n.

From which we deduce, as before:

Theorem 7.2. We have

hreg(S^{G}) ≤ −n.

Our bound on ρ_{i} is thus improved by i. The statement of Proposition 2.1 now has both the reg(S) + σ(R) and the reg(S) + σ(R) + 1 terms replaced by hreg(S) + ∑ |d_{i}|.

Remark. All that we need for a version of Lemma 1.1 to hold is that the bound on ρ_{i+1} should not exceed the desired bound on ρ_{i} by more than 1 or, at any rate, not by more than min{i > 0 | R_{i} ≠ 0}. This allows the definition of many different well-behaved regularities between reg and hreg, cf. [2] §5.

8. Polynomial Tensor Exterior Algebras

One sometimes encounters invariants of algebras of the form k[V] ⊗_{k} Λ(V^{∗}), where V is a kG-module for some group G; V^{∗} is its contragredient (the dual module considered as a left kG-module); k[V] is the symmetric algebra on V^{∗}, but graded so that the elements of V^{∗} are in degree 2, and Λ(V^{∗}) is the exterior algebra on V^{∗}, graded with V^{∗} in degree 1.

More generally, let S be our usual polynomial ring with an action of G; for any positive integer r, let S^{⟨r⟩} denote the dilated ring with S_{ri}^{⟨r⟩} = S_{i} (0 in degrees not divisible by r). If S is a module over k[d_{1}, …, d_{m}], then S^{⟨r⟩} is a module over k[d_{1}^{⟨r⟩}, …, d_{m}^{⟨r⟩}], where |d_{i}^{⟨r⟩}| = r|d_{i}|.

Let X be a finite dimensional graded kG-module; we will write reg(X) for the top nonzero degree (this is consistent with the definition of reg(k, X), and we could just as well use hreg(X)).

Proposition 8.1. In the context of the Structure Theorem,

reg((S^{⟨r⟩} ⊗ X)^{G}) ≤ reg(X) − (r − 1)n,
hreg((S^{⟨r⟩} ⊗ X)^{G}) ≤ hreg(X) − rn.

Proof. We work over a finite field k and consider S as a module over the ring of Dickson invariants k[c]. It is easily verified that σ(k[c^{⟨r⟩}]) = rσ(k[c]) + (r − 1)n.

From Lemma 5.2, we know that ρ_{i}(k[c], G; S) ≤ σ(k[c]). By dilating the minimal relatively projective resolution of S we find that ρ_{i}(k[c^{⟨r⟩}], G; S^{⟨r⟩}) ≤ rσ(k[c]); by tensoring this dilated resolution with X we see that ρ_{i}(k[c^{⟨r⟩}], G; S^{⟨r⟩} ⊗ X) ≤ rσ(k[c]) + reg(X).

The proposition now follows in the usual way from the definition of regularity. □

Corollary 8.2. For a polynomial tensor exterior algebra of the type discussed above,

reg((k[V] ⊗ Λ(V^{∗}))^{G}) ≤ 0,
hreg((k[V] ⊗ Λ(V^{∗}))^{G}) ≤ − dim V.

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

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