Loci in Quotients by Finite Groups, Pointwise Stabilizers and the Buchsbaum Property

Gregor Kemper

IWR, Universität Heidelberg, Im Neuenheimer Feld 368 69 120 Heidelberg, Germany

                email Gregor.Kemper@iwr.uni-heidelberg.de

March 31, 2000

Abstract

Let K[V]^{G} be the invariant ring of a finite linear group G ≤ GL(V), and let G_{U} be the pointwise stabilizer of a subspace U ≤ V. We prove that the following numbers associated to the invariant ring decrease if one passes from K[V]^{G} to K[V]^{G_{U}}: the minimal number of homogeneous generators, the maximal degree of the generators, the number of syzygies and other Betti numbers, the complete intersection defect, the difference between depth and dimension, and the type. From this, theorems of Steinberg, Serre, Nakajima, Kac and Watanabe, Smith, and the author follow, which say that if K[V]^{G} is a polynomial ring, a hypersurface, a complete intersection, or Cohen-Macaulay, then the same is true for K[V]^{G_{U}}. Furthermore, K[V]^{G_{U}} inherits the Gorenstein property from K[V]^{G}. We give an algorithm which transforms generators of K[V]^{G} into generators of K[V]^{G_{U}}.

Let P be one of the properties mentioned above. We consider the locus of P in V//G := Spec (K[V]^{G}) and prove that for x ∈ Spec (K[V]) with image x′ in V//G, the local ring K[V]_{x′}^{G} has the property P if and only if P holds for the invariant ring K[V]^{G_{x}} of the point stabilizer. Using this, we prove that the non-Cohen-Macaulay locus in V//G is either empty, or it has dimension at least one and codimension at least 3. From this we deduce that K[V]^{G} is Buchsbaum if and only if it is Cohen-Macaulay. This proves a conjecture of Campbell et al..

Introduction

In a beautiful survey article, Stanley [28] described a number of structural properties which an invariant ring, or any other graded algebra R, may have. These properties are ordered hierarchically as follows:

R polynomial ring ⟹ R hypersurface ⟹ R complete intersection ⟹ R Gorenstein ⟹ R Cohen-Macaulay. (1)

Like Stanley, we restrict our attention to the case where G is a finite group acting linearly on a finite dimensional vector space V over a field K, but there is no restriction on K. We write K[V] for the symmetric algebra of the dual V^{∗}, and K[V]^{G} for the invariant ring. If K is infinite, K[V]^{G} can be viewed as the ring of polynomial functions on V which are constant an all G-orbits. For a subgroup H ≤ G, no general rules hold about the behavior of the above properties when one passes from the G-invariants to the H-invariants. However, the situation is much better if H arises as the pointwise stabilizer G_{U} of a linear subspace U ≤ V. In fact, this has been a recurrent subject of research, and far-reaching results are known. For example, if K[V]^{G} is a polynomial ring, then by a result due to Serre (see Bourbaki [3, Chapter V, § 6, Ex. 8] or Nakajima [23, Lemma 1.4]) the same is true for K[V]^{G_{U}}. In characteristic zero, this is the result of Steinberg [29], which says that the pointwise stabilizers of a reflection group are again reflection groups. Moreover, if K[V]^{G} is a hypersurface (resp. a complete intersection), then so is K[V]^{G_{U}} by Nakajima [26, Lemma 2.5] (resp. Kac and Watanabe [14]). Finally, Smith [27] and Kemper [17] independently proved that the Cohen-Macaulay property passes from K[V]^{G} to K[V]^{G_{U}}. It follows from the results of this paper that the same is true for the Gorenstein property (see Theorem B). Results of this type can be very useful for proving that a certain property does not hold for K[V]^{G}. For example, Nakajima [23] and Kemper and Malle [19] used Serre’s result to show that many invariant rings of modular reflection groups are not polynomial rings. See also Example 2.18.

Apart from structural properties, there are also interesting numerical invariants associated to a Noetherian graded algebra R = ⨁_{d=0}^{∞} R_{d} over a field K = R_{0}. The first is the minimal number k of homogeneous generators of R. R is a polynomial algebra if and only if k = dim(R), so we can interpret

pdef(R) := (minimal number of homogeneous generators of R) − dim(R)

as the polynomial defect of R. A further invariant which has been the subject of intensive research is β(R), the maximal degree of an element in a minimal homogeneous generating set, which is independent of the choice of the generators. In other words, β(R) is the minimal number d such that R can be generated by homogeneous elements of degree at most d.

If R is not a polynomial ring, there exist relations between minimal homogeneous generators of R. Let b_{1}(R) be the minimal number of relations which generate the ideal of relations. This number does not depend on the choice of the minimal generators. It follows from Krull’s principal ideal theorem that b_{1}(R) is bigger than or equal to pdef(R), and equality holds if and only if R is a complete intersection. Thus the number

cidef(R) := b_{1}(R) − pdef(R)

can be interpreted as the complete intersection defect. Giving homogeneous generators for R amounts to giving an epimorphism φ: S → R from a graded polynomial algebra S onto R, and giving generators for the ideal of relations is the same as giving an epimorphism from a free Smodule F_{1} onto ker(φ). The syzygies of the second kind are generators of the kernel of this latter epimorphism. Continuing this way, we obtain a free resolution

⋯ ⟶ F_{2} ⟶ F_{1} ⟶ F_{0} ⟶ R ⟶ 0  (2)

of R as an S-module. If in each step the generators of the previous kernel are chosen homogeneous and minimal, the ranks of the F_{i} only depend on R. We write

b_{i}(R) := rank(F_{i})

and call the b_{i}(R) the Betti numbers. They encode the homological complexity of R. For example, R is called Cohen-Macaulay if its depth (the maximal length of a homogeneous regular sequence) equals its dimension. So let us call

cmdef(R) := dim(R) − depth(R)

the Cohen-Macaulay defect. By the Auslander-Buchsbaum formula (see Benson [2, Theorem 4.4.4]), cmdef(R) + pdef(R) = hdim_{S}(R) := max{i | b_{i}(R) ≠ 0}, so we see how the Cohen-Macaulay defect is encoded by the Betti numbers, and what is says about the complexity of R. Finally R is Gorenstein if it is Cohen-Macaulay of type one. This brings a further invariant, the type r(R), into the play. We recall the definition on page 18.

In this paper we prove that all these numerical invariants become smaller if one passes from K[V]^{G} to K[V]^{G_{U}}. More precisely, we prove:

Theorem A. For a Noetherian graded algebra R over a field K = R_{0}, let f(R) be one of the functions pdef(R), β(R), cidef(R), cmdef(R), r(R), or b_{i}(R) introduced above. Let G ≤ GL(V) be a finite linear group on a finite dimensional vector space V over K, and let U ≤ V be a linear subspace. Then for the pointwise stabilizer

G_{U} := {σ ∈ G | σ(x) = x for all x ∈ U}

the following statements hold:

(a) f (K[V]^{G_{U}} ) ≤ f (K[V]^{G}).

(b) If U has a G_{U}-stable complement in V, then f (K[V]^{G_{U}} ) = f (K[V/U]^{G_{U}} ).

(c) With m := pdef (K[V]^{G}) − pdef (K[V]^{G_{U}} ), the stronger inequality

∑_{j=0}^{m} \binom{m}{j}b_{i−j} (K[V]^{G_{U}} ) ≤ b_{i} (K[V]^{G})

holds.

Note that by Maschke’s theorem the hypothesis of (b) is always satisfied if char(K) ∤ |G_{U}|.

For char(K) > 0, the result β (K[V]^{G_{U}} ) ≤ β (K[V]^{G}) contained in Theorem A was also proved by Smith [27]. As a side product from the proof we obtain an algorithm which converts generating invariants for K[V]^{G} into generators of K[V]^{G_{U}} (see Theorem 2.7). Theorem A has the following immediate consequence.

Theorem B. Let P be one of the properties occurring in (1). Then in the situation of Theorem A we have:

(a) P (K[V]^{G}) implies P (K[V]^{G_{U}} ).

(b) If U has a G_{U}-stable complement in V, then P (K[V]^{G_{U}} ) and P (K[V/U]^{G_{U}} ) are equivalent.

As mentioned above, Theorem B is already known for all properties except the Gorenstein property.

The properties P occurring in (1), with “polynomial ring” replaced by “regular”, are also applicable to a local ring. This raises the question of the locus of P: for which x ∈ V//G := Spec (K[V]^{G}) does P (K[V]_{x}^{G}) hold? We give a description of the loci in terms of the invariant rings of point stabilizers. Let 𝔮 ∈ Spec (K[V]) be a prime ideal. (We abbreviate elements from Spec (K[V]) or Spec (K[V]^{G}) by letters x, y or 𝔮, 𝔭, depending on whether we view them as points or as prime ideals.) Then we write

G_{𝔮} := {σ ∈ G | σ(f) − f ∈ 𝔮 for all f ∈ K[V]}  (3)

for the point stabilizer (or isotropy subgroup) of 𝔮. If 𝔮 is given by a point x ∈ V, G_{𝔮} is simply the set of elements in G which fix x. If 𝔮 is the ideal in K[V] generated by the linear forms vanishing on a subspace U ≤ V, then G_{U} = G_{𝔮} with G_{U} defined as in Theorem A. Conversely, for any 𝔮 ∈ Spec (K[V]), let U := (V^{∗} ∩ 𝔮)^{⊥} ≤ V be the subspace annulled by all linear forms lying in 𝔮. Then it is easy to see that G_{𝔮} = G_{U}. Thus the subgroups of G occurring as G_{𝔮}’s or G_{U}’s are the same, and we might as well have stated Theorems A and B for point stabilizers G_{𝔮} or even for pointwise stabilizers of arbitrary subsets S ⊆ Spec (K[V]). We write

π_{G}: Spec (K[V]) → V//G, 𝔮 ↦ K[V]^{G} ∩ 𝔮

for the categorical quotient. Note that since K[V] is integral over K[V]^{G}, π_{G} is surjective (see Eisenbud [7, Proposition 4.15]). We will prove:

Theorem C. Assume that K is a perfect field, and let P be a one of the properties occurring in (1). Then for x ∈ Spec(K[V]) and x′ := π_{G}(x) we have

P holds for K[V]_{x′}^{G} if and only if P holds for K[V]^{G_{x}}.

The hypothesis that K be perfect is a very mild one, and in fact it is only necessary for the properties of being a polynomial ring or a hypersurface. Since there are algorithms to decide which of the properties in (1) hold for the invariant ring of a finite group (see Kemper [15]), Theorem C enables us to explicitly determine the loci of these properties.

Using Theorem C, we prove that the non-Cohen-Macaulay locus (which is a closed subvariety in V//G) is either empty, or has dimension at least one and codimension at least 3. Stronger bounds hold for the dimension of the locus where the Cohen-Macaulay defect is greater than some given number (see Theorem 3.1). This result already appeared in Kemper [17, Korollar 5.15].

A property that was not mentioned in the hierarchy (1) is the Buchsbaum property, which is a weakening of the Cohen-Macaulay property. We recall the definition in Section 3.2. It was proved independently by Nakajima [25] and Campbell et al. [6] that if G is a p-group, then K[V]^{G} is Buchsbaum if and only if it is Cohen-Macaulay. Later Kemper [16, Theorem 1.7] proved the equivalence for a larger class of groups, which includes all groups with V^{G} ≠ 0 and all groups of order not divisible by char(K)^{2}. Campbell et al. [6, Conjecture 27] conjectured that the equivalence holds for all finite linear groups G. Using the result on the dimension of the non-Cohen-Macaulay locus, we prove this conjecture (see Theorem 3.4).

The plan of the paper is as follows. In Section 1 we prove that for x ∈ V the completions of the local rings K[V]_{π_{G}(x)}^{G} and K[V]_{π_{Gx} (x)}^{G_{x}} are isomorphic. For K algebraically closed, this is a consequence of Luna’s slice theorem, but we also give an elementary proof. Everything follows from this isomorphism. We formulate general conditions on a property P such that theorems on P which are of the same type as Theorems B and C can be deduced from this isomorphism. In the second section we show how the properties and numerical invariants that we are interested in can be expressed as properties (applicable to local and graded rings) for which the conditions given in Section 1 hold. Together with the results from Section 1, this yields proofs of Theorems A–C. Finally, Section 3 is devoted to the study of the non-Cohen-Macaulay locus and the Buchsbaum property.

Acknowledgments. I take this opportunity to thank Vladimir Popov for explaining Luna’s slice theorem to me, and Jürgen Herzog for bringing an important fact about Buchsbaum rings to my attention.

1 Invariants of pointwise stabilizers

In this section we set up the general framework to prove theorems of the type of Theorems A–C.

1.1 An isomorphism

We consider the following situation: Let X be an affine scheme of finite type over a field K and let G be a finite group of K-automorphisms of X. We write K[X] for the coordinate ring of X and K[X]^{G} for the invariant ring. For x ∈ X an ideal in K[X], we define the point stabilizer G_{x} as in (3). We denote the local ring of X at x ∈ X by K[X]_{x}, and its completion with respect to the maximal ideal by { \atop K[X]_{x}}.

Proposition 1.1. In the above situation, let x ∈ X be a K-geometric point, and write x′ := π_{G}(x) and x′′ := π_{G_{x}}(x). Then the inclusion K[X]^{G} ⊆ K[X]^{G_{x}} induces an isomorphism

{ \atop K[X]_{x′}^{G}} ⟶^{∼} { \atop K[X]_{x′′}^{G_{x}}}.

Proof. We give two proofs, one which relies on Luna’s slice theorem [21] and works for K algebraically closed and x smooth, and another which is elementary and works in the general situation.

So first assume that K is algebraically closed and x is a smooth point. From the finiteness of G it follows that the G-orbit G(x) of x is closed and separable (see Bardsley and Richardson [1] for the appropriate definitions). Clearly there exists a smooth, G_{x}-stable affine open subvariety Y ⊆ X containing x, and then Y and G(x) are transversal at x. Thus by Bardsley and Richardson [1, Proposition 7.3] there exists an affine open neighborhood S ⊆ Y of x which is an étale slice at x. In particular, the natural morphism S//G_{x} → X//G is étale. Therefore the induced homomorphism

{ \atop K[X]_{x′}^{G}} → { \atop K[S]_{x′′}^{G_{x}}} is an isomorphism. But K[S]_{x′′}^{G_{x}} ≅ K[X]_{x′′}^{G_{x}} since S is open in X, and the result follows.

Now we turn to the general proof. The inclusion K[X]^{G} ⊆ K[X]^{G_{x}} induces a monomorphism K[X]_{x′}^{G} ↪ K[X]_{x′′}^{G_{x}}. We wish to show that (a) and (b) from Lemma 1.2 hold for R = K[X]_{x′}^{G} and S = K[X]_{x′′}^{G_{x}}. Being a K-geometric point, x is given as the kernel 𝔪 of a homomorphism K[X] → K of K-algebras, and x′ and x′′ are the kernels 𝔭 and 𝔮 of the compositions K[X]^{G} ↪ K[X] → K and K[X]^{G_{x}} ↪ K[X] → K, respectively. Since the compositions are onto, we have K[X]^{G}/𝔭 ≅ K ≅ K[X]^{G_{x}}/𝔮, which implies (b) of Lemma 1.2.

By the prime avoidance lemma (see Eisenbud [7, Lemma 3.3]), there exists an f ∈ 𝔪 such that f ∉ σ(𝔪) for all σ ∈ G \ G_{x}. Then g := ∏_{τ∈G_{x}} τ(f) lies in 𝔮, but g ∉ σ(𝔪) for σ ∈ G \ G_{x}. Let J ⊆ 𝔮 be a maximal subideal generated by elements not lying in any σ(𝔪) for σ ∈ G \ G_{x}. Then

𝔮 ⊆ J ∪ ⋃_{σ∈G\G_{x}} (σ(𝔪) ∩ K[X]^{G_{x}}) .

By the above 𝔮 ⊆ σ(𝔪) ∩ K[X]^{G_{x}} is impossible for σ ∈ G \ G_{x}, hence we conclude by applying prime avoidance again that 𝔮 = J. Therefore 𝔮 is generated by elements f_{1}, …, f_{n} ∈ 𝔮 such that f_{i} ∉ σ(𝔪) for σ ∈ G\G_{x}. Choose representatives σ_{1}, …, σ_{r} for the left cosets of G_{x} in G, excluding G_{x} itself. Then g_{i} := ∏_{j=1}^{r} σ_{j}(f_{i}) lies in K[X]^{G_{x}} but not in 𝔪, so g_{i} is a unit in the localization K[X]_{𝔮}^{G_{x}} = K[X]_{x′′}^{G_{x}}. It follows that the maximal ideal 𝔮_{𝔮} of K[X]_{𝔮}^{G_{x}} is generated by the f_{i} · g_{i}. But f_{i} · g_{i} ∈ K[X]^{G} ∩ 𝔮 = 𝔭, which proves (a) of Lemma 1.2. This completes the proof.

The following lemma was used in the above proof.

Lemma 1.2. Let φ: R ↪ S be a monomorphism of Noetherian local rings with maximal ideals 𝔪 and 𝔫, respectively, such that

(a) φ(𝔪)S = 𝔫, and

(b) φ induces an isomorphism φ̄: R/𝔪 → S/𝔫.

Then the homomorphism φ̂: R̂ → Ŝ induced by φ is an isomorphism.

Proof. By Eisenbud [7, Theorem 7.2(a)], the assumption (a) implies that the natural map R̂⊗_{R}S → Ŝ is an isomorphism. Composing idR̂ ⊗φ: R̂ → R̂ ⊗_{R} S with this isomorphism yields φ̂. But by the flatness of completion (see Eisenbud [7, Theorem 7.2(b)]), idR̂ ⊗φ is injective, hence so is φ̂.

To show the surjectivity of φ̂, take ĝ ∈ Ŝ arbitrary. So ĝ is represented by a sequence (g_{i})_{i∈ℤ_{>0}} with g_{i} ∈ S such that g_{j} − g_{i} ∈ 𝔫^{i} for j ≥ i. We will construct f_{i} recursively such that f_{j} − f_{i} ∈ 𝔪^{i} for j ≥ i and φ(f_{i}) − g_{i} ∈ 𝔫^{i}. Then the f_{i} define an f̂ ∈ R̂ with φ̂(f̂) = ĝ. Suppose f_{i−1} has been constructed for some i > 0 (formally set f_{0} := 0). Choose generators x_{1}, …, x_{k} of 𝔪, then by the assumption (a) we have 𝔫 = (φ(x_{1}), …, φ(x_{k})) S. Therefore φ(f_{i−1}) − g_{i} ∈ 𝔫^{i−1} implies

φ(f_{i−1}) − g_{i} = ∑_{m} h_{m}φ(m)

with h_{m} ∈ S, where the sum ranges over the monomials m of degree i − 1 in the x_{j}. By (b) there exist r_{m} ∈ R such that φ(r_{m}) − h_{m} ∈ 𝔫. Set f_{i} := f_{i−1} − ∑_{m} r_{m}m. Then f_{i} − f_{i−1} ∈ 𝔪^{i−1} and

φ(f_{i}) − g_{i} = ∑_{m} (h_{m} − φ(r_{m})) φ(m) ∈ 𝔫^{i}.

These are the desired properties of f_{i}, hence the surjectivity of φ̂ is proved.

Remark. We adopt the notation of Proposition 1.1.

(a) In the case that X//G is irreducible, X//G_{x} is reduced and x′ is a normal point, Proposition 1.1 also follows from Grothendieck [10, Exposé I, Théorème 10.11].

(b) Let W be a finite dimensional KG-module and M := K[X] ⊗_{K} W. Then M^{G} is the module

of equivariants. It is not clear whether M̂_{x′}^{G} and { \atop M_{x′′}^{G_{x}}} are isomorphic. ◁

We specialize our situation by taking X to be a finite dimensional vector space V over K, and G ≤ GL(V) a finite linear group on V. K[V]^{G} is a graded algebra with unique maximal homogeneous ideal K[V]_{+}^{G} := π_{G}(0) consisting of the invariants vanishing at 0 ∈ V.

Proposition 1.3. Let G ≤ GL(V) be a finite linear group acting on a finite dimensional vector space V over a field K. Then for every point x ∈ V we have an isomorphism

{ \atop K[V]_{π_{G}(x)}^{G}} ≅ { \atop K[V]_{π_{Gx} (0)}^{G_{x}}}.

Proof. We define an automorphism φ of K[V] (as a K-algebra) by φ(f): v ↦ f(v + x) for v ∈ V. Then φ commutes with the G_{x}-action and maps the ideal 𝔭 ⊂ K[V] of polynomials vanishing at x to the ideal 𝔮 of polynomials vanishing at 0. This yields a G_{x}-equivariant isomorphism K[V]_{𝔭} ⟶^{∼} K[V]_{𝔮}, hence

(K[V]_{𝔭})^{G_{x}} ≅ (K[V]_{𝔮})^{G_{x}}.

Clearly (K[V]^{G_{x}})_{K[V]^{Gx} ∩𝔭} ⊆ (K[V]_{𝔭})^{G_{x}}, and f ∈ K[V] \ 𝔭 implies ∏_{σ∈G_{x}} σ(f) ∈ K[V]^{G_{x}} \ 𝔭, which yields the reverse inclusion. Correspondingly, (K[V]^{G_{x}})_{K[V]^{Gx} ∩𝔮} = (K[V]_{𝔮})^{G_{x}}, and hence K[V]_{π_{Gx} (x)}^{G_{x}} ≅ K[V]_{π_{Gx} (0)}^{G_{x}}. Combining this with Proposition 1.1 yields the result.

1.2 Local and geometric properties

We make the following definitions.

Definition 1.4. (a) We call a graded ring R = ⊕_{d=0}^{∞}R_{d} local if R_{0} is a field. In this case R_{+} := ⊕_{d>0}R_{d} is the unique homogeneous maximal ideal of R.

(b) Suppose that P is a property applicable to Noetherian local rings. Then for a Noetherian ring R we write

Loc_{R}(P) := {x ∈ Spec(R) | P(R_{x}) holds}

for the locus of P

(c) A property P applicable to Noetherian local rings and Noetherian local graded rings is called local if the following four conditions hold:

(L1) For a Noetherian local ring R, P(R) holds if and only if P(R̂) holds.

(L2) For a Noetherian local graded ring R, P(R) holds if and only if R_{+} ∈ Loc_{R}(P).

(L3) If P(R) holds for a Noetherian local graded domain R, then Loc_{R}(P) contains all R_{0}- geometric points in Spec(R).

(L4) Let R be a Noetherian local graded ring and S = R_{0}[x_{1}, …, x_{n}] a graded polynomial ring with all x_{i} homogeneous of degree 1. Then P (S ⊗_{R_{0}} R) and P(R) are equivalent.

(d) Let R be a Noetherian ring. A subset Y ⊆ X := Spec(R) is called constructible if it can be written as a finite union of subsets which are locally closed (i.e., intersections of open and closed sets) in X (see Hartshorne [11, Chapter II, Exercise 3.18]). If Y is constructible, R is an algebra over a field K and L is an extension field, we write Y_{L} := Spec(L) ×_{Spec(K)} Y ⊆ Spec(L ⊗_{K} R). Thus Y_{L} is formed by substituting every ideal I ⊆ R involved in the definition of Y by the ideal in L ⊗_{K} R generated by I.

(e) A property P applicable to Noetherian local rings and Noetherian local graded rings is called geometric (over a field K) if the conditions (L1) and (L2) hold, and for every Noetherian local graded domain R (with R_{0} = K) the following two conditions hold:

(G1) Loc_{R}(P) is constructible.

(G2) If K̄ is an algebraic closure of R_{0}, then Loc_{R⊗_{K} K̄}(P) = Loc_{R}(P)K̄.

Example 1.5. As we will see in Section 2, the properties occurring in (1) are all local and geometric (at least over a perfect field). ◁

The following theorem is a simple consequence of Proposition 1.3.

Theorem 1.6. Let G ≤ GL(V) be a finite linear group acting on a finite dimensional vector space V over a field K, and let P be a local property applicable to Noetherian local rings and Noetherian local graded rings. Then we have for every subspace U ≤ V:

(a) P(K[V]^{G}) implies P(K[V]^{G_{U}}).

(b) If U has a G_{U}-stable complement in V, then P (K[V]^{G_{U}} ) and P (K[V/U]^{G_{U}} ) are equivalent.

Proof. Let U be spanned by the vectors x_{1}, …, x_{m} ∈ V. Then G_{U} = G_{x_{1}} ∩ ⋯ ∩ G_{x_{m}}. Thus using induction on m reduces the proof of (a) to the case m = 1, i.e., G_{U} = G_{x}. But by (L3) from Definition 1.4, P(K[V]^{G}) implies P(K[V]_{π_{G}(x)}^{G}), which by (L1) is equivalent to P({ \atop K[V]_{π_{G}(x)}^{G}}). By

Proposition 1.3, this implies P({ \atop K[V]_{π_{Gx} (0)}^{G_{x}}}), which by (L1) and (L2) is equivalent to P(K[V]^{G_{x}}). This completes the proof of (a).

If U has a G_{U}-stable complement in V, then V ≅ V/U ⊕ U (as modules over the group ring KG_{U}) and hence K[V] ≅ K[V/U] ⊗_{K} K[U]. This implies K[V]^{G_{U}} ≅ K[V/U]^{G_{U}} ⊗_{K} K[U], and so the equivalence of P (K[V]^{G_{U}} ) and P (K[V/U]^{G_{U}} ) follows from (L4).

Our next goal is to obtain a description of loci in invariant rings. The following result gives this description for the K-geometric points.

Proposition 1.7. Let P be a property applicable to Noetherian local rings and Noetherian local graded rings, which satisfies the conditions (L1) and (L2) from Definition 1.4. Then in the situation of Proposition 1.3 we have for x ∈ V:

π_{G}(x) ∈ Loc_{K[V]^{G}}(P) if and only if P(K[V]^{G_{x}}) holds.

Proof. This is an immediate consequence of Proposition 1.3.

The next technical lemma is used to generalize Proposition 1.7 to non-geometric points in V//G.

Lemma 1.8. Let R be a finitely generated algebra over a field K and Y, Z ⊆ X := Spec(R) two constructible sets. Then the following statements are equivalent:

(a) Y = Z.

(b) Spec_{max}(R) ∩ Y = Spec_{max}(R) ∩ Z, where Spec_{max}(R) is the set of maximal ideals in R.

(c) YK̄ = ZK̄, where K̄ is an algebraic closure of K.

Proof. We can write

Y = ⋃_{i=1}^{n} (V_{X}(I_{i}) \ V_{X}(J_{i}))

with I_{i}, J_{i} ⊆ R ideals. Take any 𝔭 ∈ X. By renumbering the I_{i} and J_{i}, we can assume that there exists k ∈ {0, …, n} such that 𝔭 ∉ ⋃_{i=1}^{k} V_{X}(J_{i}) but 𝔭 ∈ V_{X}(J_{i}) for i > k. Thus

J := ⋂_{i=1}^{k} J_{i} ⊈ 𝔭, and J_{i} ⊆ 𝔭 for i > k.

With I := ⋂_{i=1}^{k} I_{i} we have 𝔭 ∈ Y if and only if I ⊆ 𝔭.

We first claim that 𝔭 lies in Y if and only if 𝔭 is the intersection of maximal ideals in Y. From this the equivalence of (a) and (b) follows. So assume that 𝔭 ∈ Y, and let M ⊆ Spec_{max}(R) be the set of maximal ideals containing 𝔭 but not J, and M′ ⊆ Spec_{max}(R) the set of maximal ideals containing 𝔭 and J. Since R is a Jacobson ring (see Eisenbud [7, Theorem 4.19]), we have

𝔭 = ⋂_{𝔮∈M} 𝔮 ∩ ⋂_{𝔮∈M′} 𝔮 = ⋂_{𝔮∈M} 𝔮 ∩ \sqrt{𝔭 + J}.

Since \sqrt{𝔭 + J} ⊈ 𝔭, it follows that ⋂_{𝔮∈M} 𝔮 = 𝔭. Since I ⊆ 𝔭, it follows that I ⊆ 𝔮 for 𝔮 ∈ M. Therefore M ⊆ Y. Conversely, assume 𝔭 = ⋂_{𝔮∈M} 𝔮 with M ⊆ Spec_{max}(R) ∩ Y. Then for every 𝔮 ∈ M and every i > k we have J_{i} ⊆ 𝔮, and therefore I ⊆ 𝔮 (otherwise, 𝔮 ∉ Y). This implies 𝔭 ∈ Y.

Now let 𝔭 be maximal. Then we claim that 𝔭 lies in Y if and only if there exists a 𝔮 ∈ YK̄ with 𝔭 = R ∩ 𝔮. Since the equivalence of (a) and (b) is already proved, this implies the equivalence of (a) and (c). So assume 𝔭 ∈ Y and choose a 𝔮 ∈ Spec(K̄ ⊗_{K} R) with 𝔭 ⊆ 𝔮. Then I ⊆ 𝔮 and therefore IR̄ ⊆ 𝔮, where we write R̄ := K̄ ⊗_{K} R. By the maximality of 𝔭 we get R ∩ 𝔮 = 𝔭, and therefore J ⊈ 𝔮. This implies JR̄ ⊈ 𝔮, hence 𝔮 ∈ YK̄. Conversely, assume that 𝔭 = R ∩ 𝔮 with 𝔮 ∈ YK̄. Then J_{i} ⊆ 𝔮 for i > k and therefore IR̄ ⊆ 𝔮 (otherwise 𝔮 ∉ YK̄). This implies I ⊆ 𝔭, i.e., 𝔭 ∈ Y. This completes the proof.

Theorem 1.9. Let G ≤ GL(V) be a finite linear group acting on a finite dimensional vector space V over a field K, and let P be a property applicable to Noetherian local rings and Noetherian local graded rings, which is geometric over K and satisfies the conditions (L1) and (L2) from Definition 1.4. Then for x ∈ Spec(K[V]) we have

π_{G}(x) ∈ Loc_{K[V]^{G}}(P) if and only if P(K[V]^{G_{x}}) holds.

Here G_{x} is defined by Equation (3).

Proof. Set X := Spec(K[V]). We have to show the equality of the sets

Y := π_{G}^{−1} (Loc_{K[V]^{G}}(P)) and Z := {x ∈ X | P(K[V]^{G_{x}}) holds} .

Y is constructible by Condition (G1) from Definition 1.4. Let M be the set of all subgroups H ≤ G such that P(K[V]^{H}) holds, and for H ≤ G set X^{H} := {x ∈ X | H ≤ G_{x}}. X^{H} is closed and thus

Z = ⋃_{H∈M} (X^{H} \ ⋃_{H≨H′≤G}X^{H′})

is constructible. By Lemma 1.8 we have to show that an x ∈ Spec_{max}(K̄ ⊗_{K} K[V]) lies in YK̄ if and only if it lies in ZK̄. We write V̄ := K̄ ⊗_{K} V. Then for H ≤ G we have K̄ ⊗_{K} K[V]^{H} ≅ K̄[V̄]^{H} and therefore

H ∈ M ⟺ K[V]_{+}^{H} ∈ Loc_{K[V]^{H}} (P) ⟺ K̄[V̄]_{+}^{H} ∈ Loc_{K[V]^{H}} (P)K̄

⟺ K̄[V̄]_{+}^{H} ∈ Loc_{K̄[V̄]^{H}} (P) ⟺ P(K̄[V̄]^{H}) holds,

where we used (L2) and (G2). Hence

ZK̄ = ⋃_{H∈M} (XH ; K̄ \ ⋃_{H≨H′≤G}XH′ ; K̄) = {x ∈ XK̄ | P(K̄[V̄]^{G_{x}}) holds} .

On the other hand,

YK̄ = (K̄ ⊗ π_{G})^{−1} (Loc_{K[V]^{G}}(P)K̄) = (K̄ ⊗ π_{G})^{−1} (Loc_{K̄[V̄]^{G}}(P)) ,

where we used (G2). Thus we have to show that for x ∈ Spec_{max}(K̄ ⊗_{K} K[V]) the property P (K̄[V̄]^{G_{x}}) holds if and only if (K̄ ⊗ π_{G})(x) lies in Loc_{K̄[V̄]^{G}}(P). But since K̄ is algebraically closed, such an x is a K̄-geometric point, and therefore the equivalence holds by Proposition 1.7.

Example 1.10. As an application, we get that for x ∈ Spec(K[V]), π_{G}(x) lies in the non-singular locus of V//G if and only if G_{x} has polynomial invariants.

Likewise, we obtain from Hochster and Eagon [13] that if the characteristic of K does not divide the group order |G_{x}|, then π_{G}(x) lies in the Cohen-Macaulay locus of V//G. This was also proved by Broer [4]. ◁

Corollary 1.11. In the situation of Theorem 1.9, let P be local and geometric over K. Then Loc_{K[V]^{G}}(P) is open in Spec(K[V]^{G}).

Proof. We write X := Spec(K[V]) and set

M := {H ≤ G | G_{V^{H}} = H} and F := {H ∈ M | P (K[V]^{H}) does not hold} .

We claim that

π_{G}^{−1} (Loc_{K[V]^{G}}(P)) = X \ ⋃_{H∈F} X^{H}.

This implies the corollary, since π_{G} is a finite morphism and hence closed (see Hartshorne [11, Chapter II, Exercise 3.5]), and π_{G} is surjective. To prove the claim, take x ∈ X with π_{G}(x) ∈ Loc_{K[V]^{G}}(P). By Theorem 1.9, P (K[V]^{G_{x}}) holds. Let H ∈ M be a subgroup with x ∈ X^{H}, so H ≤ G_{x}. Thus H ≤ (G_{x})_{V^{H}} ≤ G_{V^{H}} = H and therefore H = (G_{x})_{V^{H}}. Now by Theorem 1.6, P (K[V]^{H}) follows and hence H ∉ F. We conclude that x ∉ X^{H} for all H ∈ F. This yields the first inclusion.

To prove the reverse inclusion, take x ∈ X \ ⋃_{H∈F} X^{H}. Thus G_{x} ∉ F. But G_{x} = G_{U} for some subspace U ≤ V by the remark before Theorem C. Therefore G_{x} ∈ M, so P (K[V]^{G_{x}}) must hold. By Theorem 1.9, π_{G}(x) ∈ Loc_{K[V]^{G}}(P) follows.

2 Properties of invariant rings

With a few possible gaps, it can be picked out from various places in the literature that the properties occurring in (1) are local and geometric. In this section we will give a uniform proof for this fact, which also shows that the properties “f(R) ≤ m” for f one of the functions mentioned in Theorem A (except β(R)) are local and geometric. This is possible since all these functions can be expressed in terms of the embedding dimension and the Betti numbers.

2.1 The embedding dimension

Let R be a Noetherian local (graded) ring with maximal (homogeneous) ideal 𝔪. Then we define the embedding dimension of R as

edim(R) := dim_{R/𝔪} (𝔪/𝔪^{2}) .

Notice that edim(R) ≤ dim(R) for a Noetherian local ring R. It follows from the next proposition (which is folklore) that the same inequality holds for Noetherian local graded rings.

Proposition 2.1. Let R be a Noetherian local graded ring. Then edim(R) is the minimal number k such that R can be generated by k elements as an algebra over R_{0}. Moreover, there exist edim(R) homogeneous generators of R.

Proof. First suppose f_{1}, …, f_{k} generate R over K := R_{0}. The f_{i} still generate R if we subtract the components of degree 0, hence we may assume that f_{i} ∈ 𝔪. But the it is clear that the f_{i} + 𝔪^{2} generate 𝔪/𝔪^{2} as a vector space over K, so k ≥ edim(R).

Conversely, choose homogeneous f_{1}, …, f_{k} ∈ 𝔪 such that the f_{i} + 𝔪^{2} form a K-basis of (the graded vector space) 𝔪/𝔪^{2}, so k = edim(R). Let f ∈ R be homogeneous of positive degree. Then

f = ∑_{i=1}^{k} α_{i}f_{i} + ∑_{j=1}^{r} g_{i}h_{i}

with α_{i} ∈ K and g_{j}, h_{j} ∈ 𝔪. We can assume that the g_{i} are homogeneous with deg(g_{i}h_{i}) = deg(f). Therefore deg(g_{i}), deg(h_{i}) < deg(f), and by induction on deg(f) we conclude that g_{i}, h_{i} ∈ K[f_{1}, …, f_{k}]. Thus f ∈ K[f_{1}, …, f_{k}], which proves that f_{1}, …, f_{k} generate R.

Similarly, if R is a Noetherian local ring, then edim(R) is the minimal dimension of a regular local ring S which has R as an epimorphic image, provided such an S exists (see Bruns and Herzog [5, p. 72]).

Lemma 2.2. If R is a Noetherian local ring and R̂ its completion, then

edim(R̂) = edim(R).

Proof. This is clear since the associated graded rings of R and R̂ coincide (see Eisenbud [7, Theorem 7.1(b)]).

Lemma 2.3. Let R be a Noetherian local graded ring with maximal homogeneous ideal 𝔪 = R_{+}. Then

edim(R_{𝔪}) = edim(R).

Proof. The maximal ideal of R_{𝔪} is 𝔪_{𝔪}. Consider the natural homomorphism φ: 𝔪 → 𝔪_{𝔪}/𝔪_{𝔪}^{2}. Clearly 𝔪^{2} ⊆ ker(φ). Conversely, take f ∈ ker(φ). Then there exists g ∈ R \ 𝔪 such that gf ∈ 𝔪^{2}. Since 𝔪 is a maximal ideal, there exists h ∈ R such that hg − 1 ∈ 𝔪. We have hgf ∈ 𝔪^{2}, hence

f = (1 − hg)f + hgf ∈ 𝔪^{2}.

Therefore ker(φ) = 𝔪^{2}.

To prove that φ is surjective, take f/g ∈ 𝔪_{𝔪} with f ∈ 𝔪 and g ∈ R \ 𝔪. Again we have h ∈ R with hg − 1 ∈ 𝔪, so

\frac{hf}{1} − \frac{f}{g} = \frac{hg − 1}{1} · \frac{f}{g} ∈ 𝔪_{𝔪}^{2}.

Thus f/g + 𝔪_{𝔪}^{2} = φ(hf). It follows that 𝔪/𝔪^{2} and 𝔪_{𝔪}/𝔪_{𝔪}^{2} are isomorphic as R/𝔪-modules. But R/𝔪 ≅ R_{𝔪}/𝔪_{𝔪} since 𝔪 is maximal. From this the lemma follows.

Lemma 2.4. Let R be a Noetherian local graded ring and 𝔭 ∈ Spec(R). Then we have:

(a) edim(R_{𝔭}) ≤ edim(R).

(b) If R is a domain, then edim(R_{𝔭}) − dim(R_{𝔭}) ≤ edim(R) − dim(R).

Proof. By Proposition 2.1, R can be generated by k := edim(R) elements. Thus there exists a polynomial ring S over R_{0} in k indeterminates and an epimorphism π: S → R. With 𝔮 := π^{−1}(𝔭), this yields an epimorphism S_{𝔮} → R_{𝔭}. This epimorphism induces an isomorphism S_{𝔮}/𝔮_{𝔮} ⟶^{∼} R_{𝔭}/𝔭_{𝔭} and an epimorphism 𝔮_{𝔮}/𝔮_{𝔮}^{2} → 𝔭_{𝔭}/𝔭_{𝔭}^{2}. Therefore edim(R_{𝔭}) ≤ edim(S_{𝔮}). Since S_{𝔮} is regular (see Bruns and Herzog [5, Theorem 2.2.13 and Corollary 2.2.9]), we obtain

edim(R_{𝔭}) − dim(R_{𝔭}) ≤ dim(S_{𝔮}) − dim(R_{𝔭}) = ht(𝔮) − ht(𝔭) = = k − dim(S/𝔮) − ht(𝔭) = k − (dim(R/𝔭) + ht(𝔭)) .

This yields the inequality (a). Since dim(R/𝔭) + ht(𝔭) = dim(R) if R is a domain (see Eisenbud [7, Corollary 13.4]), (b) also follows.

Lemma 2.5. Let R be a Noetherian local graded ring and S = R_{0}[x_{1}, …, x_{n}] a graded polynomial ring with all x_{i} homogeneous of degree 1. Then

edim(S ⊗_{R_{0}} R) = edim(R) + n.

Proof. We have

(S ⊗ R)_{+} = ⨁_{i+j>0} S_{i} ⊗ R_{j} = S_{0} ⊗ R_{+} ⊕ S_{+} ⊗ R

and

(S ⊗ R)_{+}^{2} = S_{0} ⊗ R_{+}^{2} ⊕ S_{1} ⊗ R_{+} ⊕ S_{>1} ⊗ R,

since S_{>1} = S_{+}^{2}. Therefore the map S_{+}⊕R_{+} → (S⊗R)_{+}/(S⊗R)_{+}^{2}, (s, r) ↦ (s⊗1+1⊗r)+(S⊗R)_{+}^{2} is surjective and has kernel S_{+}^{2} ⊕ R_{+}^{2}. This yields the lemma.

Proposition 2.6. For m ≥ 0, define the property P_{m} by saying that P_{m} holds for a Noetherian local (graded) ring R if edim(R) − dim(R) ≤ m. Then P_{m} is local and geometric over every perfect field.

Proof. For a local ring R we have dim(R̂) = dim(R) (see Eisenbud [7, Corollary 10.12]), which together with Lemma 2.2 yields Condition (L1) from Definition 1.4. For a Noetherian local graded ring R with maximal homogeneous ideal 𝔪 we have dim(R) = ht(𝔪) = dim(R_{𝔪}) (see Eisenbud [7, Corollary 13.7]), which together with Lemma 2.3 yields (L2). (L3) follows from Lemma 2.4(b). Moreover, if S is a polynomial ring over a field K, then Noether’s normalization lemma yields dim(S ⊗_{K} R) = dim(S) + dim(R) for a finitely generated K-algebra R. Together with Lemma 2.5 this yields (L4). Thus P_{m} is local.

Now let R be a Noetherian local graded domain with K := R_{0} a perfect field, and take 𝔭 ∈ Spec(R). There exists a polynomial ring S := K[x_{1}, …, x_{k}] and an ideal I = (f_{1}, …, f_{r}) ⊆ S such that R ≅ S/I. Let 𝔍 := (∂f_{i}/∂x_{j}) be the Jacobian matrix. Set L := R_{𝔭}/𝔭_{𝔭}. From the proof of Theorem 16.19 in Eisenbud [7], we see that

dim_{L} (𝔭_{𝔭}/𝔭_{𝔭}^{2}) = k − rank(\overline{𝔍}) − dim(R/𝔭),

where \overline{𝔍} ∈ L^{m×r} denotes 𝔍 reduced modulo 𝔭. This yields

edim(R_{𝔭}) − dim(R_{𝔭}) = k − dim(R/𝔭) − ht(𝔭) − rank(\overline{𝔍}) = k − dim(R) − rank(\overline{𝔍}).

Therefore P_{m}(R_{𝔭}) is equivalent to rank(\overline{𝔍}) ≥ k − dim(R) − m =: s. But this is equivalent to the condition that the ideal generated by the s × s minors of 𝔍 is not contained in 𝔭. Hence Loc_{R}(P_{m}) is open, which proves (G1). (G2) follows from the observation that K̄ ⊗ R has the presentation K̄ ⊗ R = K̄[x_{1}, …, x_{k}]/(f_{1}, …, f_{r}), so Loc_{K̄⊗_{K}R}(P_{m}) is again determined by the s × s minors of 𝔍.

Thus Theorem 1.9 is applicable to the properties P_{m}, and from Theorem 1.6 we obtain the result that the invariant ring of a pointwise stabilizer G_{U} is generated by at most as many invariants as K[V]^{G}, which was already stated in Theorem A in the introduction. This result raises the question whether and how generators of K[V]^{G_{U}} can be constructed from generators of K[V]^{G}. Since all our proofs are explicit and constructive, this can indeed be done.

So suppose that f_{1}, …, f_{k} generate K[V]^{G} as a K-algebra. Pick a vector x from a basis of U and let 𝔭 ⊆ K[V]^{G} be the ideal of invariants vanishing at x. Then the images of the f_{i}^{(0)} := f_{i} − f_{i}(x) generate 𝔭/𝔭^{2} as a vector space over K. Let 𝔭̂ be the image of 𝔭 in the completion { \atop K[V]_{𝔭}^{G}}. The images of the f_{i}^{(0)} generate 𝔭̂/𝔭̂^{2}. If 𝔮 ⊆ K[V]^{G_{x}} is the ideal of G_{x}-invariants vanishing at x and 𝔮̂ its

image in { \atop K[V]_{𝔮}^{G_{x}}}, then by Proposition 1.1 the images of the f_{i}^{(0)} also generate 𝔮̂/𝔮̂^{2}. Now we apply the homomorphism φ: K[V] → K[V] of K-algebras given by φ(f)(v) = f(v+x) to the f_{i}^{(0)} and write

f̃_{i} := φ(f_{i}^{(0)}). By the proof of Proposition 1.3, φ induces an isomorphism { \atop K[V]_{𝔮}^{G_{x}}} ⟶^{∼} { \atop K[V]_{π_{Gx} (0)}^{G_{x}}}.

Therefore the images of the f̃_{i} generate 𝔪̂/𝔪̂^{2}, where 𝔪̂ is the image of 𝔪 := K[V]_{+}^{G_{x}}. By Lemmas 2.2 and 2.3 they also generate 𝔪/𝔪^{2} (as a K-vector space). Now a homogeneous generating system of 𝔪/𝔪^{2} of at most k elements can be chosen, and by the proof of Proposition 2.1 this system will also generate K[V]^{G_{x}}. If we allow for an increase in the number of generators, we can also decompose the f̃_{i} into their homogeneous components (which are all members in 𝔪). These components also generate 𝔪/𝔪^{2} as a K-vector space, and therefore K[V]^{G_{x}} as a K-algebra. We have proved:

Theorem 2.7. Let G ≤ GL(V) be a finite linear group acting on a finite dimensional vector space V over a field K, and let x ∈ V be a point. Apply the endomorphism φ of K[V] given by φ(f)(v) = f(v +x) to generators f_{1}, …, f_{k} ∈ K[V]^{G} of the invariant ring. Then the homogeneous components of the resulting polynomials generate the invariant ring K[V]^{G_{x}}.

To obtain generators of the invariant ring of a pointwise stabilizer G_{U}, we can iterate the process in Theorem 2.7 over a basis x_{1}, …, x_{r} of U. Although the polynomials φ(f_{i}) are usually not homogeneous, their maximal degree is bounded above by the maximal degree of the f_{i}. Thus for the maximal degree in a system of homogeneous generators we have the relation

β (K[V]^{G_{U}} ) ≤ β (K[V]^{G}) ,

which was stated in Theorem A. Example 2.8. Let G = S_{n} be the symmetric group acting on basis vectors e_{1}, …, e_{n} of V, and choose x := e_{n}, so G_{x} = S_{n−1}. K[V]^{G} is generated by the elementary symmetric polynomials f_{n,i}(x_{1}, …, x_{n}) (i = 1, …, n), where x_{1}, …, x_{n} ∈ V^{∗} is a dual basis. An easy calculation shows that applying φ from Theorem 2.7 yields

φ(f_{n,i}) := f_{n,i}(x_{1}, …, x_{n−1}, x_{n} + 1) = f_{n−1,i} + (x_{n} + 1)f_{n−1,i−1}

with f_{n−1,0} := 1. Thus φ(f_{n,i}) has homogeneous components f_{n−1,i−1} and f_{n−1,i}+x_{n}f_{n−1,i−1}. From these, the non-redundant generators f_{n−1,i−1} (1 < i ≤ n) and f_{n−1,1}+x_{n} can be chosen. Subtracting f_{n−1,1} from f_{n−1,1} + x_{n}, we obtain the expected generators x_{n} and f_{n−1,i} (i = 1, …, n − 1) of K[V]^{S_{n−1}}.

If we consider G = A_{n} the alternating group and assume char(K) ≠ 2, then K[V]^{G} has the additional generator g_{n} := ∏_{1≤i<j≤n}(x_{i} − x_{j}). Applying φ yields

φ(g_{n}) = ∏_{1≤i<j≤n−1}(x_{i} − x_{j}) · ∏_{1≤i≤n−1}(x_{i} − x_{n} − 1) = g_{n−1} · f

with f ∈ K[V]^{S_{n−1}}. Since the degree 0 component of f is (−1)^{n−1}, g_{n−1} occurs as a homogeneous component of (−1)^{n−1}φ(g_{n}). Therefore we obtain the expected generators of K[V]^{G_{x}} = K[V]^{A_{n−1}}. ◁

A Noetherian local graded ring is called a hypersurface if it is a quotient of a polynomial ring over a field by a principal ideal. The next lemma is folklore.

Lemma 2.9. Let R be a Noetherian local graded domain. Then

(a) R is (isomorphic to) a polynomial ring over a field if and only if P_{0} from Proposition 2.6 holds for R.

(b) R is a hypersurface if and only if P_{1}(R) holds.

Proof. First notice that K := R_{0} is the largest field contained in R.

(a) If R is a polynomial ring, then R = K[x_{1}, …, x_{k}] with k = dim(R), so k = edim(R) follows by Proposition 2.1. Conversely, if P_{0}(R) holds, then by Proposition 2.1 we have R ≅ K[x_{1}, …, x_{k}]/I with k = dim(R). Therefore I = 0 and R is polynomial.

(b) If R is a hypersurface, then R = K[x_{1}, …, x_{k}]/(f), so edim(R) ≤ k and dim(R) ≥ k − 1 by Krull’s principal ideal theorem (see Eisenbud [7, Theorem 10.2]). This implies P_{1}(R). Conversely, suppose edim(R) − dim(R) ≤ 1. By Proposition 2.1 we have R ≅ K[x_{1}, …, x_{k}]/I with k = edim(R). Hence ht(I) ≤ 1. Since I is a prime ideal, every non-zero polynomial in I has a prime factor f lying in I. But then (f) is a non-zero prime ideal contained in I and therefore I = (f).

Thus Theorem 1.9 can be used to compute the non-singular locus and the “hypersurface locus” of V//G, and Theorem 1.6 yields that if K[V]^{G} is a polynomial ring or a hypersurface, then the same holds for every K[V]^{G_{U}} with U ≤ V, as stated in Theorem B.

2.2 Betti numbers

For a Noetherian local (graded) ring R with maximal (homogeneous) ideal 𝔪, choose x_{1}, …, x_{k} ∈ 𝔪 whose images generate 𝔪/𝔪^{2} as a vector space over K := R/𝔪. Thus k = edim(R), and by Proposition 2.1 and Nakayama’s lemma (see Eisenbud [7, Corollary 4.8]), the x_{i} generate 𝔪. Let K_{•}(x_{1}, …, x_{k}) be the Koszul complex of x_{1}, …, x_{k} with coefficients in R, and H_{•}(x_{1}, …, x_{k}) := H_{•} (K_{•}(x_{1}, …, x_{k})) its homology (see Bruns and Herzog [5, Section 1.6]). By Bruns and Herzog [5, Proposition 1.6.5(b)], 𝔪 annihilates H_{•}(x_{1}, …, x_{k}). This makes H_{•}(x_{1}, …, x_{k}) into a vector space over K, and we define

b_{i}(R) := dim_{K} (H_{i}(x_{1}, …, x_{k}))

for i ∈ ℤ. This does not depend on the choice of the generators x_{1}, …, x_{k} (see Bruns and Herzog [5, p. 52]). We have b_{i}(R) = 0 for i < 0 and b_{0}(R) = 1. Let us call the b_{i}(R) the Betti numbers of R. The idea behind this term will become clear by Proposition 2.12.

Lemma 2.10. Let R be a Noetherian local ring and R̂ its completion. Then for all i ∈ ℤ we have

b_{i}(R̂) = b_{i}(R).

Proof. Let x_{1}, …, x_{k} be minimal generators of the maximal ideal 𝔪 ⊂ R. Then the images x̂_{1}, …, x̂_{k} minimally generate the maximal ideal 𝔪̂ ⊂ R̂, and the flatness of completion implies that H_{•}(x̂_{1}, …, x̂_{k}) ≅ R̂ ⊗_{R} H_{•}(x_{1}, …, x_{k}) by Bruns and Herzog [5, Proposition 1.6.7(b)]. The lemma follows.

Lemma 2.11. Let R be a Noetherian local graded ring with maximal homogeneous ideal 𝔪. Then for all i ∈ ℤ we have

b_{i}(R_{𝔪}) = b_{i}(R).

Proof. If x_{1}, …, x_{k} are minimal generators of 𝔪, then the x_{i}/1 minimally generate the maximal ideal 𝔪_{𝔪} ⊂ R_{𝔪}. The flatness of localization and Proposition 1.6.7(b) of Bruns and Herzog [5] now imply H_{•}(x_{1}/1, …, x_{k}/1) ≅ R_{𝔪} ⊗_{R} H_{•}(x_{1}, …, x_{k}), whence the result.

Proposition 2.12. Let π: S → R be an epimorphism of Noetherian local (graded) rings, where S is regular. Set K := R/𝔪 with 𝔪 ⊂ R the maximal (homogeneous) ideal, and m := edim(S)−edim(R). Then for i ∈ ℤ we have

∑_{j=0}^{m} \binom{m}{j}b_{i−j}(R) = dim_{K} (Tor_{i}^{S}(K, R)) .

Proof. The maximal (homogeneous) ideal of S is 𝔫 := π^{−1}(𝔪). Choose minimal generators y_{1}, …, y_{k} of 𝔫, k = edim(S). By Bruns and Herzog [5, Corollary 2.2.6], y_{1}, …, y_{k} form an Sregular sequence. Therefore K_{•}(y_{1}, …, y_{k}) provides a free resolution of K as an S-module (see Bruns and Herzog [5, Corollary 1.6.14]). It follows that

Tor_{i}^{S}(K, R) = H_{i} (K_{•}(y_{1}, …, y_{k}) ⊗_{S} R) .

Set x_{i} := π(y_{1}) = R ⊗_{S} y_{i}. Then by Bruns and Herzog [5, Proposition 1.6.7(a)] we obtain that K_{•}(x_{1}, …, x_{k}) ≅ R ⊗_{S} K_{•}(y_{1}, …, y_{k}), thus dim_{K} (Tor_{i}^{S}(K, R)) = dim_{K} (H_{i}(x_{1}, …, x_{k})). By renumbering the x_{i} we can assume that x_{1}, …, x_{k′} minimally generate 𝔪, so k − k′ = m. By Bruns and Herzog [5, Proposition 1.6.21] we have

H_{i}(x_{1}, …, x_{k}) = ⨁_{j=0}^{m} R^{\binom{m}{j}} ⊗_{R} H_{i−j}(x_{1}, …, x_{k′}).

This yields

dim_{K} (H_{i}(x_{1}, …, x_{k})) = ∑_{j=0}^{m} \binom{m}{j} dim_{K} (H_{i−j}(x_{1}, …, x_{k′})) = ∑_{j=0}^{m} \binom{m}{j}b_{i−j}(R).

The result now follows.

From Proposition 2.12 it follows that for a Noetherian local graded ring R the b_{i}(R) may be defined as on page 2 of the Introduction.

Lemma 2.13. Let R be a Noetherian local graded ring and 𝔭 ∈ Spec(R). Set m := edim(R) − edim(R_{𝔭}) − dim(R/𝔭). Then m ≥ 0 and for i ∈ ℤ we have

∑_{j=0}^{m} \binom{m}{j}b_{i−j}(R_{𝔭}) ≤ b_{i}(R).

Proof. By Proposition 2.1 we have an epimorphism π: S := K[x_{1}, …, x_{k}] → R with K := R_{0} and S a polynomial ring in k := edim(R) indeterminates. Let

⋯ ⟶ F_{2} ⟶ F_{1} ⟶ F_{0} ⟶ R ⟶ 0

be a minimal graded free resolution of R as an S-module. Then rank(F_{i}) = dim_{K} (Tor_{i}^{S}(K, R)) (see Eisenbud [7, Exercise A3.18]), hence by Proposition 2.12 we have b_{i}(R) = rank(F_{i}). With 𝔮 := π^{−1}(𝔭) we have R_{𝔭} ≅ S_{𝔮} ⊗_{S} R (see Eisenbud [7, Lemma 2.4]). By the flatness of localization the sequence

⋯ ⟶ S_{𝔮} ⊗_{S} F_{2} ⟶ S_{𝔮} ⊗_{S} F_{1} ⟶ S_{𝔮} ⊗_{S} F_{0} ⟶ R_{𝔭} ⟶ 0

provides a free resolution of R_{𝔭} over S_{𝔮}. Since S_{𝔮} ⊗_{S} F_{i} is free of the rank b_{i}(R), we obtain

dim_{L} (Tor_{i}^{S_{𝔮}}(L, R_{𝔭})) ≤ b_{i}(R),

where L := S_{𝔮}/𝔮_{𝔮} = R_{𝔭}/𝔭_{𝔭}. The lemma now follows from Proposition 2.12 with m replaced by edim(S_{𝔮}) − edim(R_{𝔭}). But S_{𝔮} is regular, so

edim(S_{𝔮}) = dim(S_{𝔮}) = ht(𝔮) = k − dim(S/𝔮) = edim(R) − dim(R/𝔭).

This completes the proof.

Lemma 2.14. Let R be a Noetherian local graded ring with K = R_{0} and S = K[x_{1}, …, x_{n}] a polynomial ring. Then for all i ∈ ℤ we have

b_{i}(S ⊗_{K} R) = b_{i}(R).

Proof. As in the proof of Lemma 2.13 we have a minimal graded free resolution

⋯ ⟶ F_{2} ⟶ F_{1} ⟶ F_{0} ⟶ R ⟶ 0

of R as a module over a polynomial ring S′ := K[y_{1}, …, y_{k}] with k = edim(R), and then rank(F_{i}) = b_{i}(R) holds. Tensoring this resolution over S′ by S ⊗_{K} S′ (which is free as an S′-module), we obtain the minimal graded free resolution

⋯ ⟶ (S ⊗_{K} S′)^{b_{2}(R)} ⟶ (S ⊗_{K} S′)^{b_{1}(R)} ⟶ (S ⊗_{K} S′)^{b_{0}(R)} ⟶ S ⊗_{K} R ⟶ 0.

Since edim (S ⊗_{K} R) = n + k by Lemma 2.5, this yields b_{i}(S ⊗_{K} R) = b_{i}(R).

Theorem 2.15. For m, r ≥ 0 and i ∈ ℤ define the property P_{m,i,r} by saying that P_{m,i,r} holds for a Noetherian (graded) local ring R if l := m − edim(R) + dim(R) ≥ 0 and

∑_{j=0}^{l} \binom{l}{j}b_{i−j}(R) ≤ r.  (4)

Then P_{m,i,r} is local and geometric over every perfect field.

Moreover, if edim(R) − dim(R) ≤ m for a Noetherian local graded ring R, then Loc_{R}(P_{m,i,r}) satisfies the conditions (G1) and (G2) from Definition 1.4 even if R_{0} is not perfect.

Proof. First observe that l ≥ 0 is equivalent to the property P_{m} defined in Proposition 2.6. Now the conditions (L1), (L2), and (L4) follow directly from Lemmas 2.10, 2.11, and 2.14. To prove (L3), let R be a Noetherian local graded domain such that P_{m,i,r}(R) holds, and let 𝔭 ∈ Spec(R) be a maximal ideal. Thus the inequality (4) holds and we have to show that

∑_{j′=0}^{l′} \binom{l′}{j′}b_{i−j′}(R_{𝔭}) ≤ r  (5)

with l′ := m − edim(R_{𝔭}) + dim(R_{𝔭}). From Lemma 2.13 we know that

∑_{j′′=0}^{l′′} \binom{l′′}{j′′}b_{i−j′′}(R_{𝔭}) ≤ b_{i}(R)

with l′′ := edim(R) − edim(R_{𝔭}) − dim(R/𝔭). Therefore

∑_{j=0}^{l} \binom{l}{j} ( ∑_{j′′=0}^{l′′}\binom{l′′}{j′′}b_{i−j−j′′}(R_{𝔭})) ≤ ∑_{j=0}^{l} \binom{l}{j}b_{i−j}(R) ≤ r.

We have l + l′′ = m − edim(R_{𝔭}) − dim(R/𝔭) + dim(R) = l′ since R is a domain. Hence ∑_{j+j′′=j′} \binom{l}{j}\binom{l′′}{j′′} = \binom{l′}{j′}, and (5) follows. We have proved that P_{m,i,r} is local.

To prove that P_{m,i,r} is geometric, let R be a Noetherian local graded domain with K := R_{0} and choose a polynomial ring S with dim(S) = edim(R) such that R is an epimorphic image of S. Let

⋯ ⟶^{φ_{2}} F_{2} ⟶^{φ_{1}} F_{1} ⟶^{φ_{0}} F_{0} ⟶ R ⟶ 0  (6)

be a minimal graded free resolution of R as an S-module, so b_{i}(R) = rank(F_{i}) by Proposition 2.12. Take 𝔭 ∈ Spec(R) and let 𝔮 be the preimage of 𝔭 in S. Tensoring (6) by S_{𝔮} yields a free resolution of R_{𝔭} as an S_{𝔮}-module. On the other hand, let

⋯ ⟶ F′_{2} ⟶ F′_{1} ⟶ F′_{0} ⟶ R_{𝔭} ⟶ 0  (7)

be a minimal free resolution of R_{𝔭} over S_{𝔮}. By Proposition 2.12 we have

rank(F′_{i}) = ∑_{j=0}^{l_{𝔭}} \binom{l_{𝔭}}{j}b_{i−j}(R_{𝔭})

with l_{𝔭} := edim(S_{𝔮}) − edim(R_{𝔭}). By Eisenbud [7, Theorem 20.2], the resolution (6) tensored with S_{𝔮} is isomorphic to the direct sum of (7) and a trivial complex. We write this direct sum as

⋯ ⟶ F′_{2} ⊕ S_{𝔮}^{m_{1}} ⊕ S_{𝔮}^{m_{2}} ⟶ F′_{1} ⊕ S_{𝔮}^{m_{0}} ⊕ S_{𝔮}^{m_{1}} ⟶ F′_{0} ⊕ S_{𝔮}^{m_{0}} ⟶ R_{𝔭} ⟶ 0

with m_{i} non-negative integers. For a finitely generated S_{𝔮}-module M we write μ(M) := dim_{S_{𝔮}/𝔮_{𝔮}}(M/𝔮_{𝔮}M) for the size of a minimal generating system. Since the above complex is isomorphic to (6) tensored with S_{𝔮}, we obtain μ (coker(S_{𝔮} ⊗ φ_{i})) = rank(F′_{i}) + m_{i−1} and rank(F′_{i}) + m_{i} + m_{i−1} = b_{i}(R). Thus we have

∑_{j=0}^{l_{𝔭}} \binom{l_{𝔭}}{j}b_{i−j}(R_{𝔭}) = μ (coker(S_{𝔮} ⊗ φ_{i})) + μ (coker(S_{𝔮} ⊗ φ_{i−1})) − b_{i−1}(R).  (8)

But for any value of j the locus of all 𝔮 ∈ Spec(S) such that μ (coker(S_{𝔮} ⊗ φ_{i})) ≤ j is open, given by some Fitting ideal of φ_{i} (see Bruns and Herzog [5, Lemma 1.4.7]). Thus from (8) we see that for any non-negative c the set of all 𝔭 ∈ Spec(R) such that ∑_{j=0}^{l_{𝔭}} \binom{l_{𝔭}}{j}b_{i−j}(R_{𝔭}) equals c is constructible. Since the Fitting ideals have the same generators when we replace R by K̄ ⊗_{K} R (see Eisenbud [7, Corollary 20.5]), this set is tensored by K̄ if we pass from R to K̄ ⊗_{K} R. Now P_{m,i,r} holds for R_{𝔭} if and only if l := m − edim(R_{𝔭}) + dim(R_{𝔭}) ≥ 0 and

∑_{j=0}^{l} \binom{l}{j}b_{i−j}(R_{𝔭}) ≤ r.

Since R and S are domains, we have

l − l_{𝔭} = m + dim(R_{𝔭}) − edim(S_{𝔮}) = m + dim(R) − edim(R),

which does not depend on 𝔭. The relation ∑_{j+j′=l} \binom{l−l_{𝔭}}{j′}\binom{l_{𝔭}}{j} = \binom{l}{l} (which also holds if l − l_{𝔭} is negative) leads to

∑_{j=0}^{l} \binom{l}{j}b_{i−j}(R_{𝔭}) = ∑_{j′=0}^{i} \binom{l − l_{𝔭}}{j′} (∑_{j=0}^{l_{𝔭}}\binom{l_{𝔭}}{j}b_{i−j−j′}(R_{𝔭})) .

Since the set of 𝔭 ∈ Spec(R) where the inner sum of the right hand side takes any specific value is constructible, the same is true for the set of 𝔭 where the whole sum is at most r. But the intersection of this set with Loc_{R}(P_{m}) is Loc_{R}(P_{m,i,r}). We have also seen by that Loc_{R⊗_{K} K̄}(P_{m,i,r}) = Loc_{R}(P_{m,i,r})K̄. Thus (G1) and (G2) from Definition 1.4 are satisfied.

As for the last statement, we have seen that the perfectness of R_{0} is only needed to ensure that the conditions (G1) and (G2) hold for Loc_{R}(P_{m}). But if edim(R)−dim(R) ≥ m, then 𝔭 ∈ Loc_{R}(P_{m}) holds for all 𝔭 ∈ Spec(R) by Lemma 2.4.

From Theorem 2.15, Theorem A(c) follows with Theorem 1.6. Since b_{i}(R) ≤ ∑_{j=0}^{l} \binom{l}{j}b_{i−j}(R) holds in the situation of Theorem 2.15, it also follows that b_{i} (K[V]^{G_{U}} ) ≤ b_{i}(K[V]^{G}) for all subspaces U ≤ V.

2.3 The complete intersection, Cohen-Macaulay, and Gorenstein properties

The properties of being a polynomial ring or a hypersurface were already dealt with in Section 2.1. Now we consider the properties of being a complete intersection, Gorenstein, or Cohen-Macaulay. We will show how these properties and the numbers measuring the deviation from them can be expressed in terms of Betti numbers and thus of the P_{m,i,r} from Theorem 2.15. As a corollary, it follows that the properties are local and geometric. The openness of the Gorenstein and complete intersection loci was already proved by Greco and Marinari [9]. For the Cohen-Macaulay locus, see also Matsumura [22, Exercise 24.2].

Recall that a Noetherian local (graded) ring R is called a complete intersection if it is the quotient of a regular (graded) ring modulo an ideal generated by a (homogeneous) regular sequence. This is equivalent to b_{1}(R) = edim(R) − dim(R) (see Bruns and Herzog [5, Theorem 2.3.3] for the local case; in the global case it is easy to see that R = K[x_{1}, …, x_{k}]/I is a complete intersection if and only if I is generated by k − dim(R) homogeneous relations.) Let us call

cidef(R) := b_{1}(R) − edim(R) + dim(R)

the complete intersection defect of R. By Krull’s principal ideal theorem, this number is nonnegative.

Proposition 2.16. For each r ≥ 0, the property “cidef(R) ≤ r” is local and geometric.

Proof. It follows from the definition of the b_{i}(R) that b_{0}(R) = 1 for all R. Thus if edim(R) − dim(R) ≤ m, then cidef(R) ≤ r is equivalent with the property P_{m,1,r+m}(R) from Theorem 2.15. Thus cidef(R) ≤ r holds if and only if P_{m,1,r+m}(R) holds for some m. This shows that “cidef(R) ≤ r” is local. For a Noetherian local graded domain R with m := edim(R) − dim(R), we have edim(R_{x}) − dim(R_{x}) ≤ m for x ∈ Spec(R) by Lemma 2.4, and hence Loc_{R}(cidef ≤ r) = Loc_{R}(P_{m,1,r+m}). By Theorem 2.15, this locus satisfies the conditions (G1) and (G2) from Definition 1.4 even if R_{0} is not perfect. This completes the proof.

For a Noetherian local (graded) ring R with maximal (homogeneous) ideal 𝔪 we write depth(R) for the maximal length of an R-regular sequence whose elements lie in 𝔪. R is Cohen-Macaulay if and only if depth(R) = dim(R), so we define

cmdef(R) := dim(R) − depth(R)

as the Cohen-Macaulay defect of R. The Cohen-Macaulay defect can be expressed in terms of the Betti numbers b_{i}(R) as follows: if hdim(R) := max{i ∈ ℤ | b_{i}(R) ≠ 0}, then

cmdef(R) = hdim(R) − edim(R) + dim(R)

(see Bruns and Herzog [5, Theorem 1.6.17]).

Proposition 2.17. For each r ≥ 0, the property “cmdef(R) ≤ r” is local and geometric.

Proof. If edim(R) − dim(R) ≤ m, then cmdef(R) ≤ r holds if and only if P_{m,i,0}(R) holds for all i > r + m. If R is an epimorphic image of a regular ring, then by Proposition 2.12 this is equivalent to P_{m,r+m+1,0}(R). But we can always assume that R is an epimorphic image of a regular ring, since in the local case P_{m,i,0}(R) is equivalent to the same condition on R̂. But by Cohen’s structure theorem (see Bruns and Herzog [5, Theorem A.21]), R̂ is an epimorphic image of a regular ring. Now the result follows as in the proof of Proposition 2.16.

Example 2.18. Consider the following linear groups given by Nakajima [24] as examples of groups generated by transvections whose invariant rings are not Cohen-Macaulay. Let m ≥ 3 be an integer, set n := 2m + 1 and take G to be the group of all n × n matrices over a finite field K = 𝔽_{q} of the form

σ_{α_{0},... ,α_{m}} := ( \begin{matrix}1 \\ & 1 & & & & 0 \\ . & . & . \\ & & & 1 \\ α_{0} & α_{1} & & & 1 \\ ⋮ & & ... & & & ... \\ α_{0} & & & α_{m} & & & 1 \end{matrix} )  (9)

with α_{0}, …, α_{m} ∈ K. For the point x = (−1, 1, …, 1) ∈ V := K^{n}, the point stabilizer is G_{x} = {σ_{α,... ,α} | α ∈ K}. Since rank (σ_{α,... ,α} − id) = m ≥ 3 for α ≠ 3, G_{x} is not generated by bireflections. Since G_{x} is a p-group, it follows by Kemper [18, Corollary 3.7] that K[V]^{G_{x}} is not Cohen-Macaulay. Therefore by Theorem B, K[V]^{G} is not Cohen-Macaulay. See Kemper [17, Beispiel 5.17] for a detailed computation of the non-Cohen-Macaulay locus. ◁

Now we turn to the Gorenstein property. The type of a Cohen-Macaulay local (graded) ring R with maximal (homogeneous) ideal 𝔪 and n := dim(R) is defined as

r(R) := dim_{R/𝔪} (Ext_{R}^{n}(R/𝔪, R)) .

If R is not Cohen-Macaulay, we formally set r(R) := ∞. A Noetherian local ring R is Gorenstein if it is of type 1 (see Bruns and Herzog [5, Theorem 3.20.10]). A Noetherian ring R is Gorenstein if all localizations R_{𝔭} at maximal ideals are Gorenstein (see Bruns and Herzog [5, Definition 3.1.18]). The following proposition tells us how the type and thus the Gorenstein property can be expressed in terms of the Betti numbers.

Lemma 2.19. Let R be a Noetherian local (graded) ring which is Cohen-Macaulay. Then with m := edim(R) − dim(R) we have

r(R) = b_{m}(R).

Proof. Let 𝔪 be the maximal (homogeneous) ideal of R and write K := R/𝔪. If R is local, then by Lemma 2.10 and Herzog and Kunz [12, Lemma 1.22] we have b_{m}(R̂) = b_{m}(R) and r(R̂) = r(R), so we may assume that R is complete. By Cohen’s structure theorem, R is an epimorphic image of a regular local ring S. On the other hand, if R is a local graded ring, it is the epimorphic image of a polynomial ring S. We may assume that dim(S) = edim(R) (see Proposition 2.1 and Bruns and Herzog [5, p. 72]). By Herzog and Kunz [12, Lemma 1.22(b)] (which holds for local graded Cohen-Macaulay rings as well as local Cohen-Macaulay rings), we have r(R) = dim_{K} (Ext_{S}^{n}(K, R)), where n := dim(R). Choose minimal generators x_{1}, …, x_{k} of the maximal (homogeneous) ideal of S, so k = edim(R). Then the Koszul complex K_{•}(x_{1}, …, x_{k}) provides a free resolution of K over S, so

Ext_{S}^{i}(K, R) = H_{i} (Hom_{S} (K_{•}(x_{1}, …, x_{k}), R)) .

From the self-duality of the Koszul complex we obtain H_{i} (Hom_{S} (K_{•}(x_{1}, …, x_{k}), R)) ≅ H_{k−i} (K_{•}(x_{1}, …, x_{k}) ⊗_{S} R) (see Bruns and Herzog [5, Proposition 1.6.10(d)]). In particular, r(R) = b_{k−n}(R). But k − n = m.

Proposition 2.20. For all positive integers k, the property of having type at most k is local and geometric. In particular, the Gorenstein property is local and geometric.

Proof. By Lemma 2.19 and the proof of Proposition 2.17, a Noetherian local (graded) ring R with m := edim(R) − dim(R) is of type r(R) ≤ k if and only if b_{m}(R) ≤ k and b_{m+1}(R) = 0. It is easy to check that this is equivalent to the condition that for some (and then for all) m ≥ edim(R)−dim(R) the properties P_{m,m,k}(R) and P_{m,m+1,0}(R) hold. This yields the first statement.

For the second statement, we only need to show that a Noetherian local graded ring R with maximal homogeneous ideal 𝔪 is Gorenstein if and only if R is of type 1. But by the locality, the latter condition is equivalent to saying that R_{𝔪} is of type 1, i.e., R_{𝔪} is Gorenstein. Moreover, by Bruns and Herzog [5, Exercise 3.6.20(c)] R_{𝔪} is Gorenstein if and only if R is Gorenstein. This completes the proof.

3 The Cohen-Macaulay and Buchsbaum properties

In this section we use Theorem C to obtain bounds on the dimension of the non-Cohen-Macaulay locus in K[V]^{G}. From these, we deduce that K[V]^{G} is Buchsbaum if and only if it is Cohen-Macaulay.

3.1 The non-Cohen-Macaulay locus

By Proposition 2.17, the property “cmdef(R) ≤ m” is local and geometric. It follows by Corollary 1.11 that the locus Loc_{K[V]^{G}}(cmdef > m) is closed. The next theorem gives an upper and a lower bound for the dimension of this locus, which for m = 0 is the non-Cohen-Macaulay locus.

Theorem 3.1. Let G ≤ GL(V) be a finite linear group acting on a finite dimensional vector space V over a field K. Let m be a non-negative integer such that cmdef (K[V]^{G}) > m. Then

0 < dim ( Loc_{K[V]^{G}}(cmdef > m)) < dim_{K}(V) − m − 2.

In particular, the non-Cohen-Macaulay locus of K[V]^{G} is either empty, or it has dimension at least one and codimension at least 3.

Proof. Write X := Spec (K[V]) and Y := {x ∈ X | cmdef (K[V]^{G_{x}}) > m}. Theorem 1.9 and Proposition 2.17 yield

π_{G}^{−1} (Loc_{K[V]^{G}}(cmdef > m)) = Y,

and the finiteness of π_{G} yields that dim (Loc_{K[V]^{G}}(cmdef > m)) = dim(Y) (see Eisenbud [7, Corollary 9.3]). Let P ≤ G be a Sylow p-subgroup of G with p := char(K). The P -orbit of a vector 0 ≠ v ∈ V spans a finite-dimensional vector space over 𝔽_{p}. Since P is a p-group, there exists a vector 0 ≠ x from this space which is fixed by P. Hence P ⊆ G_{x}. Since the index of G_{x} is not divisible by p, it follows that depth (K[V]^{G_{x}}) ≤ depth (K[V]) (see Kemper [17, Proposition 1.21]), so cmdef (K[V]^{G_{x}}) > m. It follows that x lies in Y and therefore also K · x ⊆ Y. This yields the lower bound for the dimension.

Y is a union of subspaces X^{H} with H ≤ G subgroups such that cmdef (K[V]^{G_{x}}) > m. Let

P′ ≤ H be a Sylow p-subgroup of such an H. Then again cmdef (K[V]^{P′}) > m. But by Ellingsrud and Skjelbred [8] we have

depth (K[V]^{P′}) ≥ min {dim_{K}(V^{P′}) + 2, dim_{K}(V)} ,

so cmdef (K[V]^{P′}) ≤ max {dim_{K}(V) − dim_{K}(V^{P′}) − 2, 0}. Therefore cmdef (K[V]^{P′}) > m im-

plies m < dim_{K}(V) − dim_{K}(V^{P′}) − 2 and hence

dim_{K}(V^{H}) ≤ dim_{K}(V^{P′}) < dim_{K}(V) − m − 2.

This yields the upper bound.

Remark. (a) Using the same arguments as in the proof of Theorem 3.1, with Theorem 5 from Landweber and Stong [20] instead of Ellingsrud and Skjelbred’s result, one obtains that the singular locus in K[V]^{G} has codimension at least 2. This is well-known and also follows from the fact that K[V]^{G} is a normal domain.

(b) The following example shows that it may happen that the non-Gorenstein locus of an invariant ring consists of only one point and has codimension 2. Consider the group generated by ( \begin{matrix}ζ_{3} & 0 \\ 0 & ζ_{3} \end{matrix} ), where ζ_{3} ∈ K is a primitive third root of unity. K[V]^{G} is generated by the four monomials of degree 3 and has the Hilbert series

H (K[V]^{G}, t) = \frac{1 + 2t^{3}}{(1 − t^{3})^{2}} .

Therefore K[V]^{G} is not Gorenstein by Stanley [28, p. 503]. Since all non-zero vectors have the trivial group as point stabilizer, we obtain π_{G}(0) as the non-Gorenstein locus. Thus none of the bounds from Theorem 3.1 can be extended to the locus where a stronger property occurring in (1) on page 1 is violated.

(c) The following example shows that the bounds from Theorem 3.1 cannot be sharpened. Let G = Z_{p} be a cyclic group of order p := char(K) and let V_{n} be the n-dimensional indecomposable KG-module with 4 ≤ n ≤ p. Then by Ellingsrud and Skjelbred [8] we have depth (K[V]^{G}) = 3, so cmdef (K[V]^{G}) = n − 3. Thus

dim (Loc_{K[V]^{G}} (cmdef > n − 4)) = dim_{K}(V^{G}) = 1 = n − (n − 4) − 3.

Thus both bounds from Theorem 3.1 are simultaneously reached. ◁

3.2 Buchsbaum invariant rings are Cohen-Macaulay

Let R be a Noetherian local ring with maximal ideal 𝔪. A sequence x_{1}, …, x_{r} ∈ 𝔪 is called weakly regular if for each i = 1, …, r

(x_{1}, …, x_{i−1}) : (x_{i}) = (x_{1}, …, x_{i−1}) : 𝔪,

i.e., if for every y ∈ R such that yx_{i} lies in the ideal (x_{1}, …, x_{i−1}) generated by x_{1}, …, x_{i−1} and for every x ∈ 𝔪, xy lies in (x_{1}, …, x_{i−1}). Clearly x_{1}, …, x_{r} is weakly regular if it is regular. A sequence x_{1}, …, x_{r} ∈ 𝔪 is called a system of parameters if r = dim(R) and dim (R/(x_{1}, …, x_{r})) = 0. R is called Buchsbaum if every system of parameters is weakly regular (see Stückrad and Vogel [30, Definition 1.5]). It follows from the unmixedness theorem (see Bruns and Herzog [5, Theorem 2.1.6]) that in a Cohen-Macaulay ring every system of parameters is a regular sequence. Hence if R is Cohen-Macaulay, it is also Buchsbaum. A Noetherian local graded ring R with maximal homogeneous ideal 𝔪 is called Buchsbaum if R_{𝔪} is Buchsbaum (see Stückrad and Vogel [30, Definition 3.1]). In the graded situation it is also clear that Cohen-Macaulay implies Buchsbaum.

One reason why the Buchsbaum property is useful is that if R is a Noetherian local (graded) Buchsbaum ring with maximal (homogeneous) ideal 𝔪, then every system of parameters x_{1}, …, x_{r} ∈ 𝔪 “measures the depth”, i.e., depth(R) is the maximal integer k such that x_{1}, …, x_{k} is regular (see Campbell et al. [6, Proposition 25]). This follows easily from the definition and from the fact that all maximal regular sequences have equal length. This property of Buchsbaum rings was used by Campbell et al. [6] and Kemper [16] to prove that in many cases the Buchsbaum property and the Cohen-Macaulay property of K[V]^{G} are equivalent. We will use the following lemma to prove that this is in fact always the case.

Lemma 3.2 (Stückrad and Vogel [30, Chapter I, Corollary 1.11]). Let R be a Noetherian local ring with maximal ideal 𝔪. If R is Buchsbaum, then for all 𝔭 ∈ Spec(R) with 𝔭 ≠ 𝔪, the localization R_{𝔭} is Cohen-Macaulay.

I thank Jürgen Herzog for bringing this lemma to my attention. We obtain the following consequence.

Corollary 3.3. Let R be a Noetherian local graded domain with maximal homogeneous ideal 𝔪. If R is Buchsbaum, the R_{𝔭} is Cohen-Macaulay for all 𝔭 ∈ Spec(R) \ {𝔪}.

Proof. By way of contradiction, assume that R_{𝔭} is not Cohen-Macaulay for a 𝔭 ∈ Spec(R) \ {𝔪}. Since the non-Cohen-Macaulay locus is a closed cone (see Matsumura [22, Exercise 24.2] or Kemper [17, Satz 5.4]), it is given by a homogeneous radical ideal I ⊆ R. Since I ⊆ 𝔭, I is a proper subideal of 𝔪. I is the intersection of the minimal primes lying above it and all minimal primes are homogeneous (see Eisenbud [7, Proposition 3.12]), hence there exists a homogeneous ideal 𝔮 ⊂ R such that I ⊆ 𝔮 ⫋ 𝔪. Thus R_{𝔮} is not Cohen-Macaulay. But R_{𝔮} = (R_{𝔪})_{𝔮_{𝔪}} and 𝔮_{𝔪} ≠ 𝔪_{𝔪}. Thus Lemma 3.2 yields that R_{𝔪} is not Buchsbaum, a contradiction.

The condition (L3) from Definition 1.4 follows for the Buchsbaum property from Corollary 3.3. Moreover, (L2) is satisfied by definition, and (L1) is Lemma 1.13 in Stückrad and Vogel [30]. It is probably true that the Buchsbaum property is local, but we do not need this result here.

Theorem 3.4. Let G ≤ GL(V) be a finite linear group acting on a finite dimensional vector space V over a field K. Then the invariant ring K[V]^{G} is Buchsbaum if and only if it is Cohen-Macaulay.

Proof. If K[V]^{G} is Cohen-Macaulay, it is certainly Buchsbaum. Conversely, if K[V]^{G} is not Cohen-Macaulay, then by Theorem 3.1 the non-Cohen-Macaulay locus of K[V]^{G} has positive dimension. Therefore there exists 𝔭 ∈ Spec (K[V]^{G}), 𝔭 ≠ K[V]_{+}^{G}, such that K[V]_{𝔭}^{G} is not Cohen-Macaulay. By Corollary 3.3, this implies that K[V]^{G} is not Buchsbaum.

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