Non-commutative resolutions of linearly reductive quotient singularities

We prove existence of non-commutative crepant resolutions (in the sense of van den Bergh) of quotient singularities by finite and linearly reductive group schemes in positive characteristic. In dimension two, we relate these to resolutions of singularities provided by G-Hilbert schemes and F-blowups. As an application, we establish and recover results concerning resolutions for toric singularities, as well as canonical, log terminal, and F-regular singularities in dimension 2.

Definition 1.1.Let R be a Noetherian, local, complete, and Cohen-Macaulay commutative ring and let M be a nonzero reflexive R-module.The endomorphism ring End R (M ) is called (1) a non-commutative resolution (NCR for short) if End R (M ) has finite global dimension, and (2) a non-commutative crepant resolution (NCCR for short) if End R (M ) has global dimension equal to dim R and End R (M ) is a Cohen-Macaulay R-module.
1.2.Linearly reductive quotient singularities.In this article, we study NCCRs in the following situation.
Setup 1.2.Let k be an algebraically closed field of characteristic p > 0. Let S := k[[x 1 , ..., x n ]] be a formal power series ring over k.Let G be a finite group scheme over k that acts on Spec S, such that the action is free in codimension one.We let R := S G ⊆ S be the invariant subring and set X := Spec R.
Definition 1.3.A finite group scheme G over k is called linearly reductive if every k-linear and finite-dimensional representation of G is semi-simple.
Definition 1.4.A scheme X = Spec R as in Setup 1.2 is called a quotient singularity by the group scheme G.If G is linearly reductive, then X is called a linearly reductive quotient singularity (LRQ singularity for short).
For background and details on quotient singularities by finite group schemes and especially linearly reductive ones, we refer to [AOV08, LMM21a, LMM21b, Li22, LS14].1.3.NCCRs for LRQ singularities.In Section 2, we first establish the following result, which extends a classical proposition of Auslander [Au62].Our extension was essentially already obtained by Faber, Ingalls, Okawa, and Satriano [FIOS22, Proposition 2.26], although they work in a slightly different setup.Moreover, a similar result was proven in [BHZ18,CKWZ19], again in somewhat different setups.
Theorem 1.5.There exists a skew group scheme ring where H is the dual Hopf algebra of H 0 (G, O G ).Moreover, there exists a natural isomorphism S * G → End R (S) of usually non-commutative R-algebras.
Next, we establish the following equivalences, which generalise results of Broer and the second-named author [Br05,Ya12b].
Theorem 1.6.The following are equivalent: (1) R is a pure subring of S.
(3) G is linearly reductive.As an application, we obtain that LRQ singularities admit NCCRs.
Theorem 1.7.If G is linearly reductive, then End R (S) and thus, also an NCCR of R.
This result was already known in characteristic zero, as well as for finite groups of order prime to the characteristic of the ground field: in these cases, the first assertion follows, for example, from combining results of Auslander [Au62] and Yi [Yi94], and the second assertion was established by Toda and the second-named author [TY09].

Auslander's results and dimension two.
There are some classical results of Auslander [Au86] that carry over to the situation of this article.To state them, we introduce the following categories: In Section 3, we will show that these categories are related as follows.
(1) The functors induce equivalences of categories.Simple representations correspond to indecomposable modules under these equivalences.
This can be viewed as a non-commutative McKay correspondence.
1.5.F-blowups and G-Hilbert schemes.For a variety X over a perfect field k of positive characteristic p > 0 and an integer e ≥ 1, the secondnamed author introduced in [Ya12a] the e.th F-blowup FB e (X) → X, which is a proper and birational morphism and which is an isomorphism over the smooth locus of X.In Section 4, we show the following result.
Theorem 1.9.If G is linearly reductive, then there exist natural, proper, and birational morphisms Moreover, ψ e is an isomorphism if e is sufficiently large. (1) This extends results of Toda and the second-named author [TY09, Ya12a] from the case where G is a finite group of order prime to p, the characteristic of the ground field.Assertion (1) was already shown by the first-named author [Li22] and extends results of Ishii, Ito, and Nakamura [Is02, IN19,IN99] from the case where G is a finite group of order prime to p. Assertion (2) in characteristic zero was established by Bridgeland, King, and Reid, as well as Nakamura [BKR01,Na01].1.6.Examples.In Section 5, we apply these results to some classes of singularities: (1) We establish the existence of NCCRs for normal and Q-factorial toric singularities.This recovers some results of Faber, Mullen, and Smith [FMS19] and extends some results of Špenko and van den Bergh [SvB17] to positive characteristic.(2) We establish the existence of NCCRs for F-regular surface singularities via their description as LRQ singularities.This includes all canonical and log terminal singularities in dimension 2 and characteristic p ≥ 7. We then recover results of Hara [Ha12] and the first-named author [Li22] that G-Hilbert schemes and sufficiently high F-blowups yield the minimal resolution of such singularities.Although some of these results were previously known, it is interesting that our approach gives a natural and uniform approach via the description of these singularities as quotients by finite and linearly reductive group schemes.
After finishing the first draft of this paper, we noticed that Hashimoto and Kobayashi [HK23, Lemma 3.9] independently obtained a result similar to our Theorems 1.6 and 2.4.

Skew group scheme rings
For G a finite group scheme over a field k of characteristic p ≥ 0 that acts on a k-algebra S, we construct in this section the skew group scheme ring 2.2.Invariant rings.With assumptions and notations from the previous paragraph, we let R := S G ⊆ S be the ring of invariants with respect to the G-action on S. In the language of Hopf algebras, these are the H-invariants of S with H = H 0 (G, O G ) * as defined, for example, in [Mo93, Definition 1.7.1].
The multiplication S × S → S and the G-action on S are R-linear and thus, we obtain morphisms S → End R (S) and H → End R (S).It is easy to see that these combine to a natural homomorphism of usually non-commutative R-algebras.
We now specialise to the case where . Moreover, we will also assume that the action ρ is free in codimension one, that is, there exists a Zariski-closed subset Z ⊂ Spec S of codimension at least two, such that there is an induced action G×V → V where V := SpecS\Z and where action is free in the scheme sense.More precisely, if π : Spec S → Spec R denotes the quotient morphism by the G-action, then we set U := Spec R\π(Z) and then, freeness means that the morphism ) is an isomorphism.In the language of Hopf algebras, this means that if Spec S ′ is an affine open subset of V which is stable under the G-action, then the H-action on S with The following generalises a classical result of Auslander [Au62, page 118].A similar statement in a slightly different setup was already shown [FIOS22, Proposition 2.26], but since our proof does not proceed via quotient stacks, we decided to give it nevertheless.
Proposition 2.3.With notations and assumptions as in Setup 1.2, R is normal and the natural morphism is an isomorphism of R-algebras.
Proof.We let H := H 0 (G, O G ) * be the dual Hopf algebra and then R is the ring of invariants of S with respect to the H-action.Let K be the field of fractions of R. Since taking invariants is compatible with localisation [Sk04, Lemma 1.1], we conclude Therefore, S is finite and reflexive as R-module [StProj, Lemma 15.23.20].It then follows from [StProj, Lemma 15.23.8] that End R (S) is a reflexive Rmodule.Since S * G is a finite and free as an S-module, we conclude that S * G is a reflexive R-module.
Since S * G and End R (S) both are reflexive R-modules, it suffices to show that the natural homomorphism S * G → End R (S) is an isomorphism at each prime of height one of R. When localising at primes of height one, the Haction is Galois because ρ is free in codimension one and then, the statement follows from [KT81, Theorem 1.7], see also [Mo93,Theorem 8.3.3].
The following generalises a result from the second-namend author [Ya12b, Corollaries 3.3 and 6.18] from finite groups to finite group schemes.
Theorem 2.4.We keep the notations and assumptions of Setup 1.2.Let m ⊆ S be the maximal ideal and let j ⊆ S * G be the Jacobson radical.Then, the following are equivalent: (1) R is a pure subring of S.
(7) ⇒ (4) : We consider the pullback functor F * from the module category of S * G to the module category of S 1/p * G that is induced by the inclusion S * G ֒→ S 1/p * G.Note that the same proof as the one of [Ya12b, Proposition 6.19] shows that this functor is identical to the functor Hom R (S, S 1/p ) ⊗ − : End R (S)-mod → End R 1/p (S 1/p )-mod.
This completes the proof of the claimed equivalences.The last assertion of the theorem follows, for example, from [HH89, Theorem 2.6].
Remark 2.5.If R is strongly F-regular, then it is log terminal [HW02].Thus, in the situation of the above theorem, if G is linearly reductive, then the quotient singularity Spec R G is log terminal.If a Noetherian local ring R of equicharacteristic zero admits an NCCR, then X = Spec R is log terminal, see [SvB08,IY23], as well as [DITW20].On the other hand, there are quotient singularities in positive characteristic that are not log terminal, see, for example [Ya19].
Remark 2.6.In the situation of Theorem 2.4, even if R is not strongly Fregular, then it can be F-pure [HSY13, Section 3b] (see also [Ar77,Section 4] and [Ar75, page 64]) or F-rational [Ha17].There is also the case where R is neither log-canonical nor Cohen-Macaulay [Ya19] -in particular, it is neither F-pure nor F-rational.
Corollary 2.7.With notations and assumptions as in Theorem 2.4, assume that the equivalent statements hold.Then, (1) End R (S) is an NCCR of R.
(2) For sufficiently large e, is Morita equivalent to End R (S) and thus, it is also an NCCR of R.

Auslander's results and dimension two
We keep the notations and assumptions of Setup 1.2 and assume moreover that G is linearly reductive.In this section, we show that the category P of projective S * G-modules is equivalent to the category Rep k (G) of G-representations, as well as to the category add R (S) generated by direct summands of the R-module S. If moreover n = 2, then these are equivalent to the category L of reflexive R-modules.This generalises classical results of Auslander [Au86].
3.1.Auslander's results.In [Au86, Section 1], the following proposition is shown in the case where n = 2 and where G is a finite group, whose order is prime to the characteristic of k.
Proposition 3.1.We keep the notations and assumptions of Setup 1.2, let m ⊆ S be the maximal ideal, and assume that G is linearly reductive.We define the two categories We continue with the following result, which in similar contexts is sometimes called Auslander's projectivisation.Proposition 3.2.We keep the assumptions and notations of Proposition 3.1 and define the category add R (S) : summands of finite sums of S.
Then, the functor P → add R (S) P → P G induces equivalences of categories.
Proof.See, for example, [Le12, Proposition 2.4].Note that this proposition relies on [ARS95, Proposition II.2.1], which is stated for Artinian rings only.However, the elementary proof there also works for Noetherian rings, which is sufficient for our purposes.
Under this equivalence, the regular (resp.trivial) representation of G in Rep k (G) gets mapped to S (resp.R).Let ρ i : G → GL(V i ) be the set of finite-dimensional, k-linear, and simple representations of G up to isomorphism.We have the well-known decomposition of the regular representation (1) is the decomposition of S into indecomposable and reflexive R-modules.Applying Proposition 2.3 to this, we conclude the following.
Corollary 3.3.Under the assumptions of Proposition 3.2, there exists an isomorphism of R-algebras 3.2.Dimension 2. Now, we specialise further to the case n = 2.Then, R is normal and two-dimensional, and thus, a finite R-module is reflexive if and only if it is Cohen-Macaulay, see [BH88, Proposition 1.4.1].In particular, S is a reflexive R-module and thus, the objects of add R (S) are reflexive R-modules.
Proposition 3.4.Under the assumptions and notations of Proposition 3.4, assume n = 2 and define the category L : finite and reflexive R-modules.
Then, (1) up to isomorphism, the indecomposable reflexive R-modules are precisely the indecomposable R-summands of S.
(2) The inclusion add R (S) ⊆ L. is an equivalence of categories.In particular, there exist only a finite number of nonisomorphic indecomposable reflexive R-modules.
Proof.In [Au86, Proposition 2.1], this is shown in the case where G is a finite group, whose order is prime to the characteristic of k.The same arguments also work for G a finite and linearly reductive group scheme over k.

F-blowups
In this section, we study two-dimensional LRQ singularities and prove that they can be resolved by sufficiently high F-blowups.We show this by relating F-blowups to G-Hilbert schemes.
4.1.F-blowups.We start by recalling F-blowups that were introduced by the second-named author in [Ya12a] and which are characteristic p variants of higher Nash blowups.More precisely, let X be an n-dimensional variety over a perfect field k of characteristic p > 0, let X sm ⊆ X be the smooth locus, and let F e : X e → X be the e.th iterated Frobenius.For a K-rational point x ∈ X sm (K), the fibre (F e ) −1 (x) is a zero-dimensional subscheme of length p en of X e ⊗ k K and thus, corresponds to a K-rational point of the Hilbert scheme Hilb p en (X e ).
Definition 4.1.The e.th F-blowup, denoted FB e (X), is the closure of By [Ya12a, Corollary 2.6], there exists a natural morphism π e : FB e (X) → X, which is projective, birational, and an isomorphism over X sm .If we set M e := (F e ) * O Xe , then (π e ) * M e /(tors) is locally free.Moreover, π e is the universal proper birational morphism having this property.In other words, the e.th F-blowup of X is the blowup at the module M e , see [OZ93,Vi06].4.2.G-Hilbert schemes.Now, we assume that we are in the situation of Setup 1.2.If G is linearly reductive, then there exists a Hilbert-Chow morphism Hilb G (Spec S) → Spec R, as shown by the first-named author in [Li22, Section 4.3], which extends the classical results of Ito and Nakamura [IN99].Existence of the G-Hilbert scheme for G a finite and linearly reductive group scheme is due to Blume [Bl11].
Lemma 4.2.The G-Hilbert scheme is the blowup at the R-module S.
Proof.Let Y → Spec R be the blowup at the R-module S. We will show that the birational correspondence between Y and Hilb G (Spec S) extends to morphisms in both directions.If U denotes the universal family over Hilb G (Spec S), then we have inclusions Thus, Hilb G (Spec S) → Spec R is a flattening of the R-module S. Using the universality of Y , we obtain a morphism Hilb G (Spec S) → Y .On the other hand, the flat Y -scheme (Y × Spec R Spec S) red ⊆ Y × Spec k Spec S induces a morphism Y → Hilb G (Spec S).
Theorem 4.3.We keep the notations and assumptions of Setup 1.2 and assume moreover that G is linearly reductive.Then, for each e ≥ 1, we have a natural morphism ψ e : Hilb G (Spec S) → FB e (Spec R).
Moreover, it is an isomorphism if e is sufficiently large.
Proof.If Y 1 and Y 2 are the blowups at R-modules M 1 and M 2 respectively, then the blowup at M 1 ⊕ M 2 is the unique irreducible component of Y 1 × Spec R Y 2 that surjects onto Spec R.This shows that the blowup at a module M depends only on the set of isomorphism classes of indecomposable modules that appear as direct summands of M .It follows also that if every indecomposable summand of M also appears as a direct summand of M ′ , then there exists a natural morphism from the blowup at M ′ to the blowup at M .This gives the existence of the natural morphism ψ e , as desired.
For e ≫ 0, the indecomposable R-modules that appear as summands of R 1/p e are the same as direct summands of S [Ya12b, Proposition 4.1], which shows that ψ e is an isomorphism.
Remark 4.4.The fact that ψ e is an isomorphism for sufficiently large e can be viewed as a commutative version of the Morita equivalence between the two NCCRs End R (S) = S * G and End R (R 1/p e ), which we established in Corollary 2.7.We refer the interested reader to [TY09] for a discussion in the case where G is a finite group of order prime to p.
If n = 2, then Hilb G (Spec S) → Spec R is the minimal resolution of singularities, which is due to Ishii, Ito, and Nakamura [Is02, IN99,IN19] if G is a finite group of order prime to p and which is due to the first-named author [Li22, Theorem 4.5] if G is a finite and linearly reductive group scheme.Together with the previous results, we conclude the following.
Corollary 4.5.Under the assumptions of Theorem 4.3 assume moreover n = 2.If e is sufficiently large, then is the minimal resolution of singularities of Spec R.
Corollary 4.6.Under the assumptions of Theorem 4.3 assume moreover that n = 3 and R is Gorenstein.If e is sufficiently large, then Proof.This follows from Thereom 6.3.1 of [vB04], in the same way as Corollary 3.6 of [TY09] does.
We end this section by the following lemma, which gives an easy criterion for when an LRQ singularity is Gorenstein.In characteristic zero or if G is a finite group of order prime to p, then this is due to Watanabe [Wa74].The extension to the linear reductive case should be known to the experts, but we could not find a proper reference.See also the survey in [DK15, Section 3.9.5].
Proposition 4.7.In Setup 1.2 assume moreover that G is linearly reductive.After a change of coordinates, we may assume that the G-action is linear, that is, that G is a subgroup scheme of GL n,k and that the G-action on S = k[[x 1 , ..., x n ]] is compatible with the embedding of G into GL n,k and the usual GL n,k -action on S. The following are equivalent: (1) G is a subgroup scheme of SL n,k .
Proof.The assertion on linearisation follows from [Sa12, proof of Corollary 1.8], see also the discussion in [LMM21a, Section 6.2].
(2) ⇒ (1): Since G is linearly reductive, there exists a lift of G and the linear G-action on From this, we obtain the canonical lift X can → SpecW (k) of the LRQ singularity X = Spec S G over the ring W (k) of Witt vectors.We let K be the field of fractions of W (k) and let K be an algebraic closure of K.
Seeking a contradiction, suppose that G (which is naturally a subgroup scheme of GL n,k ) is not a subgroup scheme of SL n,k and that X is Gorenstein.There exists a finite field extension L ⊇ K, such that the generic fibre of the lift of G over L is a constant group scheme associated to some finite group G abs .Since G is not a subgroup scheme of SL n,k , we have that G abs is not a subgroup of SL n,K and thus, a fortiori, not of SL n,L , see also the discussion in [Li22,Section 4.4].Let X L := X ⊗ Spec W (k) Spec L be the generic fibre of X base-changed to L. Since X L is isomorphic to Spec L[[x 1 , ..., x n ]] G abs , it is not Gorenstein by the characteristic zero results already mentioned above.On the other hand, X is Gorenstein since X is [Ma86, Theorem 23.4], which implies that the geometric generic fibre of X L over L is also Gorenstein [Ma86,Theorem 18.2 and Theorem 23.6].This is a contradiction.

Examples
In this section, we apply the results of the previous sections to a couple of classes of singularities, such as Q-factorial toric singularities, F-regular surface singularities, and canonical surface singularities.
By a singularity X, we mean in this section the spectrum X = Spec R where R is a local and complete k-algebra with k an algebraically closed field.
5.1.Toric singularities.Let us say that a normal singularity X is toric if arises as the completion of a normal toric variety at a closed point.The following result should be well-known to the experts, see [CLS11,Theorem 11.4.8] in characteristic zero.
Proposition 5.1.Let X be a normal n-dimensional singularity over an algebraically closed field k.Set S := k[[x 1 , ..., x n ]] and let T n := (G m,k ) n be the n-dimensional torus together with its usual k-linear action on S.Then, the following are equivalent: (1) X is toric and Q-factorial.
(2) X is isomorphic to SpecR with R ∼ = S G , where G is a finite subgroup scheme of T n , and where G acts via the T n -action on S with an action that is free in codimension one.
Proof.We follow the proof of [LMM21a, Proposition 7.3] and generalise it to our situation: (1) ⇒ (2) : If X is toric, then it is analytically isomorphic to Speck[M ] for some affine semi-group M .Clearly, we may assume that X is not smooth and then, Speck[M ] has no torus factors.In this situation, the Cox construction (see, for example [GS15, Section 3.1]) realises Spec k[M ] as a quotient and where the G-action is linear and diagonal (see, for example, [GS15, Proposition 5.8 and Corollary 5.9]).Since X is Q-factorial, Cl(X) is finite and thus, G is a finite group scheme.Moreover, G is a subgroup scheme of T n .By the linearly reductive version of the Chevalley-Todd theorem [Sa12], the G-action is small, that is, free in codimension one.
(2) ⇒ (1) : We have that X is toric by [GS15, Theorem 5.2].Let G ′ be the associated reduced subscheme of G, which is a finite group and let R ′ := S G ′ .Then, as is well-known, R ′ is Q-factorial.For e ≫ 0, we have Now, if D is a Weil divisor of SpecR and if D ′ is its pullback to SpecR ′ , then for some n > 0, nD ′ is Cartier and thus, defined by some element f ∈ R ′ .Thus, p e nD is a Cartier divisor, which is defined by f p e ∈ R.This shows that X is Q-factorial.
(2) Finite subgroup schemes of G m,k are kernels of multiplication-by-N for some N ≥ 0 and thus, isomorphic to µ N .Similarly, finite subgroup schemes of T n = (G m,k ) n are of the form n i=1 µ N i for some N i 's with N i ≥ 0. In particular, they are diagonalisable.
(3) If G is a subgroup scheme of T N for some N and it acts on Spec S freely in codimension one, then we may assume that the G-action is linear because G is linearly reductive.Since G is diagonalisable, simultaneous diagonalisation implies that G is a subgroup scheme of T n and that the G-action on S is factors through the usual T n -action on S.
Corollary 5.3.Let X = Spec R be a normal n-dimensional, toric, and Qfactorial singularity over an algebraically closed field of characteristic p > 0.
( Theorem 5.5.Let X be a normal two-dimensional singularity over an algebraically closed field of characteristic p > 0.Then, the following are equivalent (1) X is F-regular (resp.Gorenstein and F-regular).
(2) X is the quotient singularity by a finite and linearly reductive subgroup scheme G of GL 2,k (resp.SL 2,k ).Moreover, if p ≥ 7, then this is equivalent to (3) X is a log terminal (resp.canonical) singularity.
By the results of the previous sections, we thus obtain the following.
Corollary 5.6.Let X = SpecR be a normal two-dimensional and F-regular singularity over an algebraically closed field of characteristic p > 0.
(1) If R = S G with S = k[[x 1 , x 2 ]] and G as in Theorem 5.5.(2), then End R (S) is an NCCR of R.
(2) If e is sufficiently large, then End R (R 1/p e ) is an NCCR of R.
(3) If e is sufficiently large, then FB e (X) is the minimal resolution of singularities of X.
Remark 5.7.Assertion (3) is a theorem of Hara [Ha12], which we recover here in the context of LRQ singularities and G-Hilbert schemes.
5.3.Canonical surface singularities.If X is a canonical surface singularity over an algebraically closed field k of characteristic p ≥ 0, then it is a rational double point.If p > 0, then these have been classified by Artin [Ar77] and they are all of the form X = Spec R with for a suitable polynomial f = f (x 1 , x 2 , x 3 ).If p ≥ 7, then the results recalled in Section 5.2 show that all canonical surface singularities are F-regular and LRQ singularities and thus, the results about NCCRs and F-blowups of the previous sections apply.However, if p < 7, then not all canonical surface singularities are F-regular.The following result is more or less well-known.
Proposition 5.8.Every canonical surface singularity over an algebraically closed field admits an NCCR.
Proof.Let Spec R be a canonical surface singularity over an algebraically closed field.By [AV85], there are only finitely many indecomposable maximal Cohen-Macaulay modules over R up to isomorphism.Let M be the direct sum of all of them.By [Le07, Theorem 6], End R (M ) is an NCCR.
(1) If p = 5, then the singularities E 0 8 and E 1 8 (notation as in [Ar77]) are quotient singularities with G isomorphic to α 5 and C 5 = Z/5Z, respectively.This is in contrast to the group that is usually assigned to E 8 -singularities if p = 0 or p ≥ 7, namely the binary icosahedral group, which is a non-abelian group of order 120.We have S * G ∼ = End R (S) by Proposition 2.3, but this ring does not have finite global dimension by Theorem 2.4.Thus, although these singularities admit NCCRs, they are not given by End R (S).
(3) If X is a canonical surface singularity, then FB e (X) has only rational singularities and it is dominated by the minimal resolution of X by [HSY13, Proposition 3.2].However, it is not necessarily true that FB e (X) is a resolution of singularities even if e is sufficiently large, see [HSY13, Theorem 1.1] for a couple of examples, which include the E 0 8 -singularity in characteristic p = 5.This has to do with the fact that for every e ≥ 1, there exists an indecomposable maximal Cohen-Macaulay module of R that is not a summand of R 1/p e .Thus, an NCCR End R (M ) of such a singularity, which exists by Proposition 5.8, is not of the form End R (R 1/p e ).
(4) S * G has finite global dimension.Moreover, if S * G has finite global dimension, then gl.dim(S * G) = n.
for example, [Li22, Proposition A.2].By [Ya02, Corollary 4.2], we have that gl.dim(S * G) = gl.dim(S#H) is finite.(4) ⇒ (3) : From the previous proposition, the natural map S * G → End R (S) is an isomorphism.This implies that S is a faithful S * G-module.Let H := H 0 (G, O G ) * be the dual Hopf algebra.Let 0 = t ∈ ℓ H be a left integral.By [Ya02, Corollary 2.3], there exists a c ∈ S with tc = 1.If m S = (x 1 , ..., x n ) denotes the maximal ideal of S, we let c ′ := c ∈ S/m S = k.Since tc = 1 in S we also have tc ′ = 1 in k.Applying [Ya02, Corollary 2.3] to (S/m S )#H = H, we conclude gl.dim(H) = 0.This implies that H is semi-simple (see, for example, [We94, Theorem 4.2.2]) and using [Li22, Proposition A.2], we conclude that G is linearly reductive.(3) ⇒ (1) : By [Sa12, proof of Corollary 1.8], we may assume that the G-action on S is linear.The statement then follows from [BH88, Remark 6.5.3(b)].(1) ⇒ (4) : We have S * G ∼ = End R (S) by Proposition 2.3.In particular, End R (S) is free and hence, Cohen-Macaulay as an S-module.By [Yo90, Proposition (1,8)], it is also Cohen-Macaulay as an R-module.By [Ya12b, Corollary 2.11] and Assumption (1), it is an NCCR and in particular, it has finite global dimension.(4) ⇒ (5) : See [Le12, Proposition 12.7].(5) ⇒ (4) : Obvious.(6) ⇒ (7) : Obvious.(3) ⇒ (6) : We have J(S * G) ⊇ m(S * G) by [La01, (5.9)].On the other hand, since k * G is semisimple, we have Rep k (G) : finite-dimensional k-linear G-representations P : finite and projective S * G-modules, which are related as follows.(1) If W ∈ Rep k (G), then there is a natural S * G-module structure on S ⊗ k W that extends the S-action on S and the G-action on W .Moreover, S ⊗ k W is a finite and projective S * G-module, that is, lies in P. (2) If P is a finite S * G-module, then P/mP is a finite-dimensional and k-linear G-representation, that is, lies in Rep k (G).(3) If P ∈ P, then there exists an isomorphism of S * G-modules P ∼ = S ⊗ k (P/mP ).(4) The functor Rep k (G) → P W → S ⊗ k W induces an equivalence of categories.Simple G-representations correspond to indecomposable S * G-modules under this equivalence.Proof.We have that S * G is semiperfect by [La01, (23.3)].The Jacobson radical of S * G is equal to m * G by Theorem 2.4.The map that sends P to P/mP defines a bijection from isomorphism classes of indecomposable projective S * G-modules to isomorphism classes of simple k * G-modules by [AF92, Proposition 27.10].Since (S ⊗ k (P/mP )) /m (S ⊗ k (P/mP )) = P/mP, the map W → S ⊗ k W is the inverse of the above bijection.
is the invariant subring, then we study a natural homomorphism of usually non-commutative R-algebrasS * G → End R (S)and show that it is an isomorphism if the action of G on S is free in codimension one.If p > 0, then we show that R ⊆ S is a pure subring if and only if R is strongly F-regular if and only if G is linearly reductive if and only if S * G has finite global dimension.2.1.Skew group scheme rings.Let k be a field, let S be a commutative k-algebra, and let G → Spec k be a finite group scheme.Assume that we have an action ρ : G → Aut Spec S/Spec k .Let G abs be a finite group that acts on a k-algebra S. Let G → Spec k be the constant group scheme associated to G abs .Then, H := H 0 (G, O G ) * is isomorphic to the group algebra k[G abs ] with its usual Hopf algebra structure.From this and [Mo93, Example 4.1.6],we conclude that S * G as just defined coincides with the classical skew group ring S * G abs .
[SvB17]and G as in Theorem 5.5.(2), then End R (S) is an NCCR of R.(2) For e sufficiently large, End R (R 1/p e ) is an NCCR of R.(3) If n = 2 and if e is sufficiently large, then FB e (X) is the minimal resolution of singularities of X. (4) If n = 3, if e is sufficiently large, and if R is Gorenstein, then FB e (X) is a crepant resolution of singularities of X.Remark 5.4.Faber, Muller, and Smith[FMS19]proved that if Spec R is a normal toric singularity in characteristic p > 0 and if e is sufficiently large, then End R (R 1/p e ) is an NCR.Recall that this means that the latter ring has finite global dimension, but that it is not necessarily Cohen-Macaulay.A similar result in characteristic zero was established by Špenko and van den Bergh in[SvB17].