A McKay Correspondence in Positive Characteristic

Abstract

We establish a McKay correspondence for finite and linearly reductive subgroup schemes of ${\mathbf {SL}}_2$ in positive characteristic. As an application, we obtain a McKay correspondence for all rational double point singularities in characteristic $p\geq 7$. We discuss linearly reductive quotient singularities and canonical lifts over the ring of Witt vectors. In dimension 2, we establish simultaneous resolutions of singularities of these canonical lifts via G-Hilbert schemes. In the appendix, we discuss several approaches towards the notion of conjugacy classes for finite group schemes: This is an ingredient in McKay correspondences, but also of independent interest.

1 Introduction

1.1 Klein’s classification and McKay’s correspondence

Felix Klein [Kl84] classified finite subgroups G of ${\mathbf {SL}}_2({\mathbb {C}})$ : up to conjugation, there are two infinite series and three isolated cases.

1. 1. The associated quotient singularity ${\mathbb {C}}^2/G$ is called a Klein singularity and the singularities arising this way are precisely the rational double point singularities. Its minimal resolution of singularities is a union of ${\mathbb {P}}^1$ ’s, whose dual intersection graph $\Gamma $ is a simply-laced Dynkin diagram of finite type, that is, of type $A_n$ , $D_n$ , $E_6$ , $E_7$ or $E_8$ .

2. 2. John McKay [Mc80] associated a finite graph $\widehat {\Gamma }$ to $G \subset {\mathbf {SL}}_2({\mathbb {C}})$ , whose vertices correspond to the isomorphism classes of the simple representations of G. This graph is a Dynkin diagram of affine type $\widehat {A}_n$ , $\widehat {D}_n$ , $\widehat {E}_6$ , $\widehat {E}_7$ or $\widehat {E}_8$ .

3. 3. After these preparations, the classical McKay correspondence consists of the following observations:

   1. (a) The graph $\Gamma $ is obtained from $\widehat {\Gamma }$ by removing the vertex corresponding to the trivial representation.

   2. (b) There exists a bijection between conjugacy classes of G, vertices of $\widehat {\Gamma }$ and isomorphism classes of simple representations of G.

   3. (c) There exists a bijection between finite subgroups of ${\mathbf {SL}}_2({\mathbb {C}})$ up to conjugacy, the above Dynkin diagrams of affine type, Klein singularities and the above Dynkin diagrams of finite type.

By now, there are various approaches to and a vast literature on this subject, such as [Kn85, Ko85, Mc80, Mc81, St85] and many more. Also, there are now generalisations into very different directions: higher dimensional algebraic geometry [Re02], K-theory [GV83], derived categories of coherent sheaves [BKR01, KV00], representations of quivers [Ki16], non-commutative geometry and Hopf algebras [CKWZ16] and string theory [Aetal09] - just to mention a few.

1.2 Positive characteristic

Now, let k be an algebraically closed field of characteristic $p>0$ .

1.2.1 Wild McKay correspondence

The classical McKay correspondence as sketched above is still partially available over k if G is assumed to be a finite subgroup of ${\mathbf {SL}}_2(k)$ of order prime to p - this is the tame case. If p divides the order of G - this is the modular or wild case - then this correspondence breaks down. We refer to Yasuda’s surveys [Ya22, Ya23] about conjectures and partial results concerning such wild McKay correspondences.

1.2.2 Linearly reductive McKay correspondence

In this article, we show that if G is a finite and linearly reductive subgroup scheme of ${\mathbf {SL}}_{2,k}$ , then there is a reasonable version of the classical McKay correspondence. For example, we obtain a McKay correspondence for all rational double point singularities if $p\geq 7$ . Instead of considering groups, we allow non-reduced group schemes over k, but we require their categories of k-linear and finite-dimensional representations to be semi-simple. We refer to [TY20] about conjectures and partial results concerning a McKay correspondence for the group scheme $\boldsymbol {\alpha }_p$ , which is not linearly reductive.

Thus, let G be a finite and linearly reductive subgroup scheme of ${\mathbf {SL}}_{2,k}$ with $p\geq 7$ . (In this introduction, we will exclude small characteristics whenever this is makes our discussion easier.) Let $x\in X:=U/G$ with $U={\mathbb {A}}_k^2$ or $U=\widehat {{\mathbb {A}}}_k^2$ be the associated Klein singularity, which is a rational double point. One goal of this article is to define a notion of conjugacy class for G, to construct graphs $\Gamma $ and $\widehat {\Gamma }$ and to establish bijections as above.

Let us make three comments:

1. 1. It is interesting in its own that a McKay correspondence can be extended from finite group schemes of length prime to p, that is, the tame case, to linearly reductive group schemes.

2. 2. What makes this linearly reductive McKay correspondence really interesting is that the bijection in Theorem 1.1 is not true when considering finite groups of order prime to p only, see Example 1.2.

3. 3. Probably, many more aspects of the classical McKay correspondence can be carried over to the linearly reductive setting, but rather than writing a whole monograph, we decided to establish only some basic bijections.

1.3 Linearly reductive group schemes

Let G be a finite and linearly reductive group scheme over an algebraically closed field k of characteristic $p\geq 0$ . By definition, this means that the category $\mathrm {Rep}_k(G)$ of k-linear and finite-dimensional representations of G is semi-simple.

If $p=0$ , then every finite group scheme over k is étale and linearly reductive. and in fact, it is the constant group scheme associated to a finite group. In fact, the functor $G\mapsto G_{\mathrm {abs}}:=G(k)$ induces an equivalence of categories between finite group schemes over k and finite groups.

If $p>0$ , then every linearly reductive group scheme admits a canonical semi-direct product decomposition

$$ \begin{align*}G\,\cong\,G^\circ\rtimes G^{\mathrm{\acute{e}t}}, \end{align*} $$

where $G^{\mathrm {\acute {e}t}}$ is a group scheme of length prime to p (and thus, the constant group scheme associated to a finite group of order prime to p) and where $G^\circ $ is infinitesimal and diagonalisable. The latter implies that $G^\circ $ is a product of group schemes of the form $\boldsymbol {\mu }_{p^n}$ . Conversely, every such semi-direct product of a diagonalisable group scheme with the constant group scheme associated to a finite group of order prime to p is linearly reductive. This structure result is usually attributed to Nagata [Na61], but see also [AOV08], [Ch92] and [Ha15]. Linearly reductive group schemes over k strictly contains the class of constant group schemes associated to finite groups of order prime to p.

1.3.1 Abstract groups

Associated to G, there is an abstract finite group $G_{\mathrm {abs}}$ and we refer for the slightly technical definition to Section 2.2. The order of $G_{\mathrm {abs}}$ is equal to the length of G. For example, if G is étale over k, then we have $G_{\mathrm {abs}}\cong G(k)$ . The assignment $G\mapsto G_{\mathrm {abs}}$ establishes an equivalence of categories

(1.1) $$ \begin{align} \left\{ \begin{array}{l} \mbox{finite linearly reductive}\\ \mbox{group schemes over }k \end{array}\right\} \quad\leftrightarrow\quad \left\{ \begin{array}{l} \mbox{finite groups with a}\\ \mbox{unique }p\mbox{-Sylow subgroup} \end{array}\right\}, \end{align} $$

see [LMM21] and Lemma 2.2.

1.3.2 Canonical lifts

In [LMM21], we showed that if G is a finite and linearly reductive group scheme, then there exists a lift of G over the ring of Witt vectors $W(k)$ . We note that $G^\circ $ and $G^{\mathrm {\acute {e}t}}$ even lift uniquely to $W(k)$ and we define the canonical lift $G_{\mathrm {can}}\to \mathrm {Spec}\: K$ of G to be the unique lift that is a semi-direct product of the lifts of $G^\circ $ and $G^{\mathrm {\acute {e}t}}$ . Any lift of G to some extension field of K becomes isomorphic to $G_{\mathrm {can}}$ after possibly passing to some further field extension. Moreover, the finite group $G_{\mathrm {can}}(\overline {K})$ is isomorphic to $G_{\mathrm {abs}}$ .

1.3.3 Representation theory

By [LMM21] and Proposition 2.6, there exist canonical equivalences of representation categories

(1.2) $$ \begin{align} \mathrm{Rep}_{k}(G) \,\to\, \mathrm{Rep}_{\overline{K}}(G_{\mathrm{can},\overline{K}}) \,\to\, \mathrm{Rep}_{\mathbb{C}}(G_{\mathrm{abs}}) \end{align} $$

that are compatible with degrees, direct sums, tensor products, duals and simplicity. These equivalences induce isomorphisms of rings

$$ \begin{align*}K_k(G) \,\to\, K_{\overline{K}}(G_{\mathrm{can},\overline{K}}) \,\to\, K_{{\mathbb{C}}}(G_{\mathrm{abs}}), \end{align*} $$

see Corollary 2.8. Here, $K_F(G)$ denotes the K-group associated to F-linear and finite-dimensional G-representations.

1.3.4 Hopf algebras

If G is a finite group scheme over k, then the multiplication map turns $H^0(G,{\mathcal O}_G)$ into a finite-dimensional Hopf algebra over k. We discuss finite group schemes and among them the linearly reductive ones from the point of Hopf algebras in Appendix A.

1.4 Linearly reductive subgroup schemes of ${\mathbf {SL}}_{2,k}$

Let k be an algebraically closed field of characteristic $p\geq 0$ . Hashimoto [Ha15] extended Klein’s classification [Kl84] of finite subgroups of ${\mathbf {SL}}_2({\mathbb {C}})$ up to conjugation to the setting of finite and linearly reductive subgroup schemes of ${\mathbf {SL}}_{2,k}$ . If $p\geq 7$ , then one obtains a list analogous to Klein’s classical list. If $p\in \{2,3,5\}$ , then some classical cases are missing, but there are no new cases.

1.5 McKay graph and McKay correspondence

Let G be a finite and linearly reductive subgroup scheme of ${\mathbf {SL}}_{2,k}$ . As in McKay’s original construction [Mc80], we associate an affine Dynkin diagram $\widehat {\Gamma }$ to G, its embedding into ${\mathbf {SL}}_{2,k}$ and the set of isomorphism classes of simple representations of G. This is the McKay graph associated to this data. In fact, we will see that it is compatible with the equivalences induced by (1.1) and (1.2) and we refer to Section 3.1 for details. We establish the following version of McKay’s theorem [Mc80] in positive characteristic.

Theorem 1.1 (Theorem 3.6)

Let k be an algebraically closed field of characteristic $p\geq 0$ . There exists a bijection between non-trivial, finite and linearly reductive subgroup schemes of ${\mathbf {SL}}_{2,k}$ up to conjugation and affine Dynkin graphs of type

$$ \begin{align*}\begin{array}{lcl} \widehat{A}_n, \widehat{D}_n, \widehat{E}_6, \widehat{E}_7, \widehat{E}_8 &\quad \mbox{if} \quad & p=0\quad \mbox{or} \quad p\geq7, \\ \widehat{A}_n, \widehat{D}_n, \widehat{E}_6, \widehat{E}_7 &\quad \mbox{if} \quad & p=5,\\ \widehat{A}_n, \widehat{D}_n &\quad \mbox{if} \quad & p=3, \\ \widehat{A}_n &\quad \mbox{if} \quad &p=2. \end{array} \end{align*} $$

By construction, this bijection is compatible with the classical McKay correspondence via the lifting results and the equivalences (1.1), (1.2).

Example 1.2. The linearly reductive group scheme corresponding to $\widehat {A}_n$ is $\boldsymbol {\mu }_{n+1}$ . This group scheme is reduced, that is, étale, if and only if p does not divide $n+1$ . In particular, it is crucial to allow non-reduced group schemes in order to obtain a bijection as in characteristic zero.

1.6 Linearly reductive quotient singularities

Consider ${\mathbf {GL}}_{2,k}$ with its usual linear action on $U={\mathbb {A}}^2_k$ or $U=\widehat {{\mathbb {A}}}_k^2$ . If G is a finite, linearly reductive and very small (see Definition 4.1) subgroup scheme of ${\mathbf {GL}}_{2,k}$ , then the associated quotient singularity $x\in X:=U/G$ is a two-dimensional linearly reductive quotient singularity in the sense of [LMM21]. By loc.cit., such a singularity determines G together with its embedding $G\to {\mathbf {GL}}_{2,k}$ up to isomorphism and conjugation, respectively.

In Section 4.3, we will see that a minimal resolution of singularities of a two-dimensional linearly reductive quotient singularity $x\in X=U/G$ is provided by the G-Hilbert scheme

(1.3) $$ \begin{align} \pi \,:\, G\text{-}\mathrm{Hilb}(U) \,\to\, U/G, \end{align} $$

which generalises work of Ishii, Ito and Nakamura [IN19, IN99].

Remark 1.3. In dimension two, a linearly reductive quotient singularity is the same as an F-regular singularity [LMM21]. Thus, every two-dimensional F-regular singularity can be resolved by a suitable G-Hilbert scheme, see also Remark 4.6.

If moreover G is a subgroup scheme of ${\mathbf {SL}}_{2,k}$ , then $x\in X=U/G$ is called a Klein singularity. Klein singularities are rational double point singularities. If $p=0$ or $p\geq 7$ , then conversely every rational double point is a Klein singularity by Hashimoto [Ha15] and, independently, by [LS14]. If $p\in \{2,3,5\}$ , then not every rational double point is a Klein singularity.

1.7 Canonical lifts and simultaneous resolutions

Let G be a very small, finite and linearly reductive subgroup scheme of ${\mathbf {GL}}_{2,k}$ and let $x\in X=U/G$ be the associated linearly reductive quotient singularity. Assume $p>0$ and let $W(k)$ be the ring of Witt vectors of k. In Section 4.4, we will establish the existence of a canonical lift

$$ \begin{align*}\mathcal{X}_{\mathrm{can}}\,\to\,\mathrm{Spec}\: W(k) \end{align*} $$

of $x\in X=U/G$ . Using G-Hilbert schemes in families, we will see in Section 4.5 that it admits a simultaneous and minimal resolution of singularities

$$ \begin{align*}\widetilde{\pi} \,:\, \mathcal{Y}\,\to\,\mathcal{X}_{\mathrm{can}}\,\to\,\mathrm{Spec}\: W(k). \end{align*} $$

We will prove this resolution to be unique, see Theorem 4.10.

1.8 McKay correspondence for Klein singularities

Let G be a finite and linearly reductive subgroup scheme of ${\mathbf {SL}}_{2,k}$ and let $x\in X=U/G$ be the associated Klein singularity.

As discussed in Section 1.5, we have the McKay graph $\widehat {\Gamma }$ associated to G, its embedding into ${\mathbf {SL}}_{2,k}$ and the set of isomorphism classes of simple representations of G.

Let $\pi :Y\to X$ be a minimal resolution of singularities. Since $x\in X$ is a rational double point, the exceptional divisor $\mathrm {Exc}(\pi )$ of $\pi $ is a configuration of ${\mathbb {P}}^1$ ’s, whose dual intersection graph $\Gamma $ is a simply-laced Dynkin diagram.

Theorem 1.4 (Theorem 5.3)

Let k be an algebraically closed field of characteristic $p\geq 0$ . Let G be a finite and linearly reductive subgroup scheme of ${\mathbf {SL}}_{2,k}$ and let $x\in X=U/G$ be the associated Klein singularity. Then, there exists a natural bijection of the graph $\Gamma $ with the graph obtained from $\widehat {\Gamma }$ by removing the vertex corresponding to the trivial representation.

Since every rational double point in characteristic $p\geq 7$ is a Klein singularity, we obtain the following.

Corollary 1.5. There exists a linearly reductive McKay correspondence for rational double point singularities in every characteristic $p\geq 7$ .

To establish this theorem, we use the Ishii-Ito-Nakamura resolution of singularities (1.3), as well as Hecke correspondences as in the work of Ito and Nakamura [IN99] and Nakajima [Na96, Na01].

1.9 Generalisations and variants

Let G be a very small, finite and linearly reductive subgroup scheme of ${\mathbf {GL}}_{2,k}$ and let $x\in X=U/G$ be the associated two-dimensional linearly reductive quotient singularity. Let $\pi :Y\to X$ be the minimal resolution of singularities and let $\mathrm {Exc}(\pi )$ be the exceptional divisor of $\pi $ .

1. 1. In Theorem 5.4, we associate a representation of G to each point of $\mathrm {Exc}(\pi )$ . This generalises results of Ishii and Nakamura [Is02, IN19].

2. 2. In Theorem 5.6, we establish a bijection between the components of $\textrm {Exc}(\pi )$ and reflexive ${\mathcal O}_X$ -modules. Probably, this result should be viewed as a theorem on two-dimensional F-regular singularities, see Remark 5.7. It generalises work of Artin and Verdier [AV85], Wunram [Wu88] and Ishii and Nakamura [IN19].

3. 3. In Theorem 7.1, we establish an equivalence of derived categories of coherent sheaves ${\mathcal {D}}(Y)$ and ${\mathcal {D}}^G(U)$ (G-equivariant sheaves on U). This generalises work of Kapranov and Vasserot [KV00], Bridgeland, King and Reid [BKR01] and Ishii, Nakamura and Ueda [Is02, IN19, IU15]. Of course, these articles themselves generalise work of Gonzalez-Sprinberg and Verdier [GV83] from K-theory to derived categories of coherent sheaves.

1.10 Ito-Reid correspondence

If G is a finite subgroup of ${\mathbf {SL}}_2({\mathbb {C}})$ , then Ito and Reid [IR96] found a natural bijection, the Ito-Reid correspondence, between the conjugacy classes of G and the vertices of the McKay graph $\widehat {\Gamma }$ .

Theorem 1.6. Let k be an algebraically closed field of characteristic $p\geq 0$ . There exists an Ito-Reid correspondence for finite and linearly reductive subgroup schemes of ${\mathbf {SL}}_{2,k}$ .

This can be proven using lifting results and the Ito-Reid correspondence over ${\mathbb {C}}$ , see Section 6.3 for details.

The main difficulty is to define a notion of conjugacy class for finite and linearly reductive group schemes that makes such a correspondence work: Let us recall the ring $K_k(G)$ from Section 1.3.3. If G is a finite and linearly reductive group scheme over k, then we define the set of conjugacy classes of G to be

$$ \begin{align*}\mathrm{Spec}\: \left({\mathbb{C}}\otimes K_k(G)\right). \end{align*} $$

At first sight, this might look rather artificial. One should think of it as defining conjugacy classes to be “dual” to simple representations. Moreover, our definition is compatible with lifting over $W(k)$ and it induces a bijection of the conjugacy classes of G with the conjugacy classes of $G_{\mathrm {abs}}$ . We refer to Section 6.1 and Appendix B.2 for details.

1.11 Conjugacy classes

In Theorem 1.6, we had to find a definition of conjugacy classes that makes this theorem true. This begs for the question whether there are other definitions or approaches, which is a question that is interesting in its own.

Let G be a finite group scheme (not necessarily linearly reductive) over an algebraically closed field k. In Appendix B, we study the following approaches to the notion of the set of conjugacy classes of G:

1. 1. The set of conjugacy classes $G(k)/\sim $ of the group of k-rational points of G.

2. 2. The spectrum of $F\otimes K_k(G)$ , where F is a field of characteristic zero that contains “sufficiently many” roots of unity.

3. 3. The scheme that represents the functor of conjugacy classes from schemes over k to sets defined by $S\mapsto G(S)/\sim $ ,

4. 4. the isotypical component of the trivial representation of the adjoint representation of G.

5. 5. The simple subrepresentations of the extended adjoint representations $^{\mathrm {Ad}}A$ and $^{\mathrm {Ad}}(A^*)$ , which are defined using the quantum doubles of the Hopf algebra $A:=H^0(G,{\mathcal O}_G)$ and its dual $A^*$ .

All approaches lead essentially to the “same answer” in characteristic zero. However, they usually lead to very different notions in positive characteristic. On the other hand, all approaches have their merits and drawbacks. For linearly reductive group schemes, the approach (2) leads to a definition that is compatible with lifting and that leads to an Ito-Reid correspondence.

1.12 Organisation of this article

1. - In Section 2, we recall basic facts about finite and linearly reductive group schemes over algebraically closed fields.

2. - In Section 3, we construct the McKay graph $\widehat {\Gamma }$ . The main result is Theorem 3.6, a McKay correspondence.

3. - In Section 4, we recall basic facts about linearly reductive quotient singularities. We establish an Ishii-Ito-Nakamura-type resolution of singularities, the canonical lift of such a singularity and a unique simultaneous resolution of singularities of the canonical lift.

4. - In Section 5, we revisit the Ishii-Ito-Nakamura-type resolution $\pi $ of singularities of $x\in X=U/G$ and we introduce Hecke correspondences, which leads to Theorem 5.3, a bijection between simple and non-trivial representations of G and components of $\pi $ . We also study generalisations of this result to two-dimensional linearly reductive quotient singularities.

5. - In Section 6, we establish an Ito-Reid correspondence between conjugacy classes of G and exceptional divisors in the minimal resolution of singularities of $x\in X=U/G$ .

6. - In Section 7, we study derived categories of G-equivariant sheaves on U and on the minimal resolution $\pi :Y\to X=U/G$ .

7. - In Appendix A, we recall results on finite group schemes from the point of view of Hopf algebras. We recall the adjoint representation, quantum doubles and the extended adjoint representation.

8. - In Appendix B, we study several approaches toward the notion of a conjugacy class for finite group schemes.

2 Linearly reductive group schemes

In this section, we recall a couple of general facts about finite and linearly reductive group schemes over algebraically closed fields. We discuss the close relationship between such a group scheme G and a certain abstracted group $G_{\mathrm {abs}}$ associated to it. For the relationship to Hopf algebras, we refer to Appendix A.

2.1 Group schemes

Let G be a finite group scheme over an algebraically closed field k of characteristic $p\geq 0$ . Since k is perfect, there is a short exact sequence of finite group schemes over k

(2.1) $$ \begin{align} 1\,\to\,G^\circ\,\to\,G\,\to\, G^{\mathrm{\acute{e}t}} \,\to\,1, \end{align} $$

where $G^\circ $ is the connected component of the identity and where $G^{\mathrm {\acute {e}t}}$ is an étale group scheme over k. The reduction $G_{\mathrm {red}}\to G$ provides a canonical splitting of (2.1) and we obtain a canonical semi-direct product decomposition $G\cong G^\circ \rtimes G^{\mathrm {\acute {e}t}}$ . Since k is algebraically closed, $G^{\mathrm {\acute {e}t}}$ is the constant group scheme associated to the finite group $G(k)=G^{\mathrm {\acute {e}t}}(k)$ of k-rational points. Moreover, $G^\circ $ is an infinitesimal group scheme of length equal to some power of p. In particular, if $p=0$ or if the length of G is prime to p, then $G^\circ $ is trivial and then, G is étale.

If M is a finitely generated abelian group, then the group algebra $k[M]$ carries a Hopf algebra structure and the associated commutative group scheme is denoted $D(M):=\mathrm {Spec}\: k[M]$ . By definition, such group schemes are called diagonalisable. For example, we have $D({\mathbf {C}}_n)\cong \boldsymbol {\mu }_n$ , where ${\mathbf {C}}_n$ denotes the cyclic group of order n. We have that $\boldsymbol {\mu }_n$ is étale over k if and only if $p\nmid n$ .

A finite group scheme G over k is said to be linearly reductive if every k-linear and finite-dimensional representation of G is semi-simple. If $p=0$ , then all finite group schemes over k are étale and linearly reductive. If $p>0$ , then, by a theorem that is often attributed to Nagata [Na61, Theorem 2] (but see also [AOV08, Proposition 2.10], [Ch92] and [Ha15, Section 2]), a finite group scheme over k is linearly reductive if and only if it is an extension of a finite and étale group scheme, whose length is prime to p, by a diagonalisable group scheme.

2.2 Abstract groups and canonical lifts

Let G be a finite and linearly reductive group scheme over an algebraically closed field k of characteristic $p>0$ . Following [LMM21, Section 2], we study the finite group

$$ \begin{align*}G_{\mathrm{abs}} \,:=\, \left(\underline{((G^\circ)^{D}(k))}_{{\mathbb{C}}}\right)^{D}({\mathbb{C}}) \,\rtimes\, G^{\mathrm{\acute{e}t}}(k). \end{align*} $$

Here, $-^D$ denotes $\underline {\mbox {Hom}}(-,{\mathbb {G}}_m)$ , the Cartier dual, of a commutative and finite group scheme. If G is étale, then $G_{\mathrm {abs}}=G(k)$ and if $G=\boldsymbol {\mu }_{p^n}$ , then $G_{\mathrm {abs}}={\mathbf {C}}_{p^n}$ . In any case, the order of $G_{\mathrm {abs}}$ is equal to the length of G.

Definition 2.1. The finite group $G_{\mathrm {abs}}$ is called the abstract finite group associated to G.

Lemma 2.2. The functor

$$ \begin{align*}G \,\mapsto\, G_{\mathrm{abs}} \end{align*} $$

establishes an equivalence of categories between the category of finite and linearly reductive group schemes over k and the category of finite groups with a normal and abelian p-Sylow subgroup.

Proof. [LMM21, Lemma 2.1].

Remark 2.3. A finite group G with a normal and abelian p-Sylow subgroup P is the same as a finite group with a unique and abelian p-Sylow subgroup. In this case, the Schur-Zassenhaus theorem implies that G is isomorphic to a semi-direct product $P\rtimes G/P$ .

Next, we study lifts to characteristic zero: let $W(k)$ be the ring of Witt vectors of k, let K be its field of fractions and let $\overline {K}$ be an algebraic closure of K. By [LMM21, Proposition 2.4], there exist lifts of G as finite and flat group scheme over $W(k)$ . More precisely, $G^\circ $ and $G^{\mathrm {\acute {e}t}}$ even lift uniquely to $W(k)$ , but their extension class usually does not, see also [LMM21, Example 2.6]. However, there is a unique lift ${\mathcal G}_{\mathrm {can}}$ of G to $W(k)$ that is characterised by being a semi-direct product of the unique lift of $G^\circ $ with the unique lift of $G^{\mathrm {\acute {e}t}}$ .

Definition 2.4. The lift $\mathcal {G}_{\mathrm {can}}\to \mathrm {Spec}\: W(k)$ is called the canonical lift of G. We set $G_{\mathrm {can}}:=\mathcal {G}_K\to \mathrm {Spec}\: K$ and also call it canonical lift.

Every other lift of G to some extension of $R\supseteq W(k)$ differs from ${\mathcal G}_{\mathrm {can},R}$ by a twist and thus, there is only one geometric lift of G to $\overline {K}$ up to isomorphism, namely $G_{\mathrm {can},\overline {K}}$ , see [LMM21, Section 2.2].

Since $\overline {K}$ is algebraically closed and of characteristic zero, $G_{\mathrm {can},\overline {K}}$ is the constant group scheme associated to the finite group $G_{\mathrm {can}}(\overline {K})$ . In fact, we have $G^\circ \cong \prod _i \boldsymbol {\mu }_{p^{n_i}}$ for some $n_i$ ’s and if we set $N:=\max \{n_i\}_i$ , fix a primitive $p^N$ .th root of unity $\zeta _{p^N}$ and set $K_N:=K(\zeta _{p^N})$ , then $G_{\mathrm {can},K_N}$ is the constant group scheme associated to the finite group $G_{\mathrm {can}}(K_N)$ and we have $G_{\mathrm {can}}(K_N)=G_{\mathrm {can}}(\overline {K})$ .

Lemma 2.5. There exist isomorphisms of finite groups

$$ \begin{align*}G_{\mathrm{abs}} \,\cong\, G_{\mathrm{can}}(K_N) \,=\,G_{\mathrm{can}}(\overline{K}). \end{align*} $$

In particular, there exist isomorphisms

$$ \begin{align*}G_{\mathrm{can},K_N} \,\cong\, \left(\underline{G_{\mathrm{abs}}}\right)_{K_N} \quad \mbox{and} \quad G_{\mathrm{can},\overline{K}} \,\cong\, \left(\underline{G_{\mathrm{abs}}}\right)_{\overline{K}} \end{align*} $$

of finite group schemes over $K_N$ and $\overline {K}$ , respectively.

2.3 Representation theory

Let k be an algebraically closed field of characteristic $p>0$ , and let $W(k)$ , K and $\overline {K}$ be as in the previous section. The equivalence of Lemma 2.2 can be extended to representations and K-theory. If G is a finite group or a finite group scheme over k, we denote by $\mathrm {Rep}_k(G)$ the category of its k-linear and finite-dimensional representations. Let us first recall [LMM21, Corollary 2.11].

Proposition 2.6. Let G be a finite and linearly reductive group scheme over k. Then, there exist canonical equivalences of categories

$$ \begin{align*}\mathrm{Rep}_{k}(G) \,\to\, \mathrm{Rep}_{K_N}(G_{\mathrm{can},K_N}) \,\to\, \mathrm{Rep}_{\overline{K}}(G_{\mathrm{can},\overline{K}}) \,\to\, \mathrm{Rep}_{\mathbb{C}}(G_{\mathrm{abs}}), \end{align*} $$

which are compatible with degrees, direct sums, tensor products, duals and simplicity.

Remark 2.7. In particular, this allows us to define characters or even a character table of G via Proposition 2.6 and $G_{\mathrm {abs}}$ .

Let $K_k(G)$ be the K-group associated to $\mathrm {Rep}_k(G)$ . In fact, $K_k(G)$ has a natural structure of a commutative ring with one, where the sum (resp. product) structure comes from direct sums (resp. tensor products) of representations. A straightforward application of Proposition 2.6 is the following.

Corollary 2.8. There exist isomorphisms of rings

$$ \begin{align*}K_k(G) \,\to\, K_{K_N}(G_{\mathrm{can},K_N}) \,\to\, K_{\overline{K}}(G_{\mathrm{can},\overline{K}}) \,\to\, K_{{\mathbb{C}}}(G_{\mathrm{abs}}). \end{align*} $$

3 McKay graph and McKay correspondence

In this section, we introduce the McKay graph associated to a finite and linearly reductive subgroup scheme G over an algebraically closed field k of characteristic $p\geq 0$ and a representation $\rho :G\to {\mathbf {GL}}_{n,k}$ . This induces a bijection between certain affine Dynkin diagrams and finite and linearly reductive subgroup schemes of ${\mathbf {SL}}_{2,k}$ . As an application, we establish a linearly reductive McKay correspondence.

3.1 McKay graph

Let G be a finite and linearly reductive group scheme over an algebraically closed field k of characteristic $p\geq 0$ . Let $\{\rho _i\}_i$ be the finite set of isomorphism classes of k-linear and simple representations of G. Following tradition, we assume that $\rho _0$ is the trivial representation. We fix a representation $\rho :G\to {\mathbf {GL}}(V)$ . If G is a subgroup scheme of ${\mathbf {SL}}_{n,k}$ or ${\mathbf {GL}}_{n,k}$ , then $\rho $ is usually the linear representation corresponding to the embedding of G into this linear algebraic group. By assumption, $\mathrm {Rep}_k(G)$ is semi-simple. Therefore, there exist unique integers $a_{ij}\in {\mathbb {Z}}_{\geq 0}$ for each i, such that we have isomorphisms of k-linear representations

$$ \begin{align*}\rho\otimes\rho_i \,\cong\, \bigoplus_{j}\, \rho_j^{\oplus a_{ij}}. \end{align*} $$

Associated to this data, we define the McKay graph, denoted $\Gamma (G,\{\rho _i\},\rho )$ :

1. - The vertices are the $\{\rho _{i}\}_i$ . (Some sources exclude the the trivial representation $\rho _0$ .)

2. - There are $a_{ij}$ edges from the vertex corresponding to $\rho _i$ to the vertex corresponding to $\rho _j$ .

We now establish a couple of elementary properties of this graph, which are well-known in the classical case and which immediately carry over to the linearly reductive situation. We leave the proof of the first lemma to the reader.

Lemma 3.1. We have

$$ \begin{align*}a_{ij} \,=\, \dim_k\, \mathrm{Hom}(\rho_i,\rho_j\otimes\rho). \end{align*} $$

In particular, if $\rho $ is self-dual, that is, $\rho \cong \rho ^\vee $ , then $a_{ij}=a_{ji}$ for all $i,j$ . In this case, we can consider $\Gamma (G,\{\rho _i\},\rho )$ as an undirected graph.

Lemma 3.2. Let $\rho :G\to {\mathbf {SL}}_{2,k}$ be a homomorphism of group schemes over k, considered as a 2-dimensional representation. Then, $\rho $ is self-dual.

Proof. Being a 2-dimensional representation, $\rho ^\vee $ is isomorphic to $\rho \otimes \det (\rho )$ and the lemma follows.

Lemma 3.3. Let $\rho :G\to {\mathbf {GL}}_{n,k}$ be a faithful representation. Then, every irreducible representation of G occurs as subrepresentation of $\rho ^{\otimes m}$ for some suitable m.

Proof. This is well-known for finite groups. Using Proposition 2.6, it carries over to finite and linearly reductive group schemes.

Corollary 3.4. If $\rho $ is a faithful representation, then the graph $\Gamma (G,\{\rho _i\},\rho )$ is connected.

Proof. It suffices to note that the number of paths in $\Gamma $ of length m that connect the vertices corresponding to $\rho _i$ and $\rho _j$ is equal to the multiplicity of $\rho _i$ in $\rho _j\otimes \rho ^{\otimes m}$ , see, for example, the proof of [Ki16, Theorem 8.13].

Let k be an algebraically closed field of characteristic $p>0$ , and let $W(k)$ , K and $\overline {K}$ be as in Section 2.2. There, we also discussed the canonical lift $G_{\mathrm {can}}$ of G over K and we saw that there exists an isomorphism of finite groups $G_{\mathrm {abs}}\cong G_{\mathrm {can}}(\overline {K})$ . Proposition 2.6 implies the following result.

Proposition 3.5. Let G be a finite and linearly reductive group scheme over k. Let $\{\rho _i\}_i$ be the set of isomorphism classes of simple representations of G. Let $\rho $ be a finite-dimensional representation of G.

1. 1. Proposition 2.6 yields sets of representations $\{\rho _{\mathrm {can},i}\}_i$ and $\{\rho _{\mathrm {abs},i}\}_i$ of $G_{\mathrm {can}}$ and $G_{\mathrm {abs}}$ , respectively, which are the sets of isomorphism classes of simple representations of $G_{\mathrm {can}}$ and $G_{\mathrm {abs}}$ , respectively.

2. 2. Proposition 2.6 yields representations $\rho _{\mathrm {can}}$ and $\rho _{\mathrm {abs}}$ of $G_{\mathrm {can}}$ and $G_{\mathrm {abs}}$ , respectively.

This data leads to a bijection of the three McKay graphs

$$ \begin{align*}\Gamma(G,\{\rho_i\}_i,\rho),\quad \Gamma(G_{\mathrm{can}},\{\rho_{\mathrm{can},i}\}_i,\rho_{\mathrm{can}}),\quad \Gamma(G_{\mathrm{abs}},\{\rho_{\mathrm{abs},i}\}_i,\rho_{\mathrm{abs}}). \end{align*} $$

3.2 McKay correspondence

We now run through the classical McKay correspondence [Mc80] in our setting, where we follow [Ki16, Section 8.3]. Let k be an algebraically closed field of characteristic $p\geq 0$ and let G be a finite and linearly reductive subgroup scheme of ${\mathbf {SL}}_{2,k}$ .

1. 1. The K-group $K_k(G)$ carries a symmetric bilinear form

   $$ \begin{align*}([V],[W])_0 \,:=\, \dim_k\,\mathrm{Hom}_G(V,W). \end{align*} $$

2. 2. We consider the closed embedding $G\to {\mathbf {SL}}_{2,k}$ as a 2-dimensional representation $\rho :G\to {\mathbf {SL}}_{2,k}\to {\mathbf {GL}}_{2,k}$ and define an operator

   $$ \begin{align*}A\,:\,K_k(G)\,\to\, K_k(G),\qquad [V]\mapsto [V]\otimes\rho\,. \end{align*} $$

   Since $\rho $ is self-dual by Lemma 3.2, it follows that A is symmetric with respect to $(-,-)_0$ , see [Ki16, Lemma 8.12].

3. 3. Using $\rho $ , we define a symmetric bilinear form on $K_k(G)\otimes _{\mathbb {Z}}{\mathbb {R}}$ by

   $$ \begin{align*}([V],[W]) \,:=\, \left([V],\, (2-A)[W]\right)_0, \end{align*} $$
   which is positive semi-definite. The class $\delta \in K_k(G)$ of the regular representation of G generates the radical of $(-,-)$ .

4. 4. Let $\{\rho _i\}$ be the set of isomorphism classes of simple representations of G.

   1. (a) The McKay graph $\widehat {\Gamma }=\Gamma (G,\{\rho _i\}_i,\rho )$ is an affine Dynkin diagram of type $\widehat {A}_n$ , $\widehat {D}_n$ with $n\geq 4$ , $\widehat {E}_6$ , $\widehat {E}_7$ or $\widehat {E}_8$ .

   2. (b) After removing the vertex corresponding to the trivial representation $\rho _0$ from $\widehat {\Gamma }$ , we obtain a finite Dynkin diagram of type $A_n$ , $D_n$ with $n\geq 4$ , $E_6$ , $E_7$ or $E_8$ , respectively.

If G is a finite subgroup of ${\mathbf {SL}}_2({\mathbb {C}})$ , then these statements are part of the classical McKay correspondence, see [Mc80] or [Ki16, Section 8.3]. In our setting of finite and linearly reductive subgroup schemes of ${\mathbf {SL}}_{2,k}$ , the above claims immediately follow from the classical McKay correspondence together with the lifting results Proposition 2.6 and Proposition 3.5. We then obtain the following analog of McKay’s theorem [Mc80].

Theorem 3.6. Let k be an algebraically closed field of characteristic $p\geq 0$ . There exists a bijection between finite, non-trivial and linearly reductive subgroup schemes of ${\mathbf {SL}}_{2,k}$ up to conjugation and affine Dynkin diagrams of type

$$ \begin{align*}\begin{array}{lcl} \widehat{A}_n, \widehat{D}_n, \widehat{E}_6, \widehat{E}_7, \widehat{E}_8 &\quad \mbox{if} \quad & p=0 \quad \mbox{or} \quad p\geq7, \\ \widehat{A}_n, \widehat{D}_n, \widehat{E}_6, \widehat{E}_7 &\quad \mbox{if} \quad & p=5,\\ \widehat{A}_n, \widehat{D}_n &\quad \mbox{if} \quad & p=3 \mbox{ and} \\ \widehat{A}_n &\quad \mbox{if} \quad &p=2. \end{array} \end{align*} $$

Proof. If $p=0$ , then this is part of the classical McKay correspondence. If $p>0$ , then this follows from the linearly reductive McKay correspondence just discussed together with Hashimoto’s classification [Ha15, Theorem 3.8] of finite and linearly reductive subgroup schemes of ${\mathbf {SL}}_{2,k}$ .

Remark 3.7. The linearly reductive group schemes corresponding to $\widehat {E}_6$ , $\widehat {E}_7$ and $\widehat {E}_8$ are étale and correspond to finite groups of order prime to p. The linearly reductive group scheme corresponding to $\widehat {A}_n$ (resp. $\widehat {D}_n$ ) is étale if and only if $p\nmid (n+1)$ (resp. $p\nmid (n-2)$ ). We refer to [LS14, Proposition 4.2] for details. Thus, even if p is sufficiently large, then one does not obtain a bijection in Theorem 3.6 with finite groups of order to prime to p only.

Remark 3.8. Steinberg [St85] established many properties of the McKay graph and the McKay correspondence for finite subgroups of ${\mathbf {SU}}_2({\mathbb {C}})$ without using classification lists. It seems plausible to obtain a proof of Theorem 3.6 along these lines without using lifting results or classification lists.

4 Linearly reductive quotient singularities

In this section, we recall linearly reductive quotient singularities in the sense of [LMM21] and some general results, including the canonical lift of such a singularity over the ring of Witt vectors. In dimension two, we establish a minimal resolution of singularities using G-Hilbert schemes as in the work of Ishii, Ito and Nakamura [Is02, IN19, IN99]. We also show that the canonical lift admits a unique minimal and simultaneous resolution of singularities. Finally, we discuss Klein singularities and their relation to rational double point singularities.

4.1 Quotient singularities

Let k be an algebraically closed field of characteristic $p\geq 0$ . If G is a finite and linearly reductive group scheme over k, if V is a finite-dimensional k-vector space and if $\rho :G\to {\mathbf {GL}}(V)$ is a linear representation, then we define the $\lambda $ -invariant of $\rho $ as in [LMM21, Definition 2.7] to be

$$ \begin{align*}\lambda(\rho) \,:=\, \max_{\{{\mathrm{id}}\}\neq\boldsymbol{\mu}_n\subseteq G} \dim V^{\boldsymbol{\mu}_n}, \end{align*} $$

where $V^{\boldsymbol {\mu }_n}$ denotes the subspace of $\boldsymbol {\mu }_n$ -invariants. As explained in [LMM21, Remark 2.8], the representation $\rho $ is faithful if and only if $\lambda (\rho )\neq \dim V$ . Moreover, $\rho $ contains no pseudo-reflections if and only if $\lambda (\rho )\leq \dim V-2$ and in this case, the representation $\rho $ is said to be small.

Definition 4.1. The representation $\rho $ is called very small if $\lambda (\rho )=0$ .

Note that in dimension two the notions of small and very small coincide. We refer to [LMM21, Section 3] for the classification of finite and linearly reductive group schemes that admit a very small representation. By [LMM21, Proposition 6.5], $\lambda (\rho )$ is equal to the dimension of the non-free locus of the induced G-action on the spectrum of the formal power series ring $k[[V^\vee ]]=(k[V^\vee ])^\wedge $ . Following [LMM21, Definition 6.4] and using the linearisation result [LMM21, Proposition 6.3], we define:

Definition 4.2. A linearly reductive quotient singularity over k is an isolated singularity that is analytically isomorphic to $\mathrm {Spec}\: k[[V^\vee ]]^G$ , where G is a finite and linearly reductive group scheme over k and where $\rho :G\to {\mathbf {GL}}(V)$ is a very small representation.

Remark 4.3. We set $U:=\mathrm {Spec}\: k[V^\vee ]$ or $U:=\mathrm {Spec}\: k[[V^\vee ]]$ and simply write $x\in X=U/G$ with the assumptions of Definition 4.2 implicitly understood.

Properties of these singularities have been studied in [LMM21]: for their local étale fundamental groups, class groups and F-signatures, we refer to [LMM21, Section 7].

By [LMM21, Theorem 8.1], a linearly reductive quotient singularity determines the finite and linearly reductive group scheme G together with the very small representation $\rho :G\to {\mathbf {GL}}(V)$ uniquely up to isomorphism and conjugacy, respectively. It thus makes sense to refer to $\rho (G)\subset {\mathbf {GL}}(V)$ as the finite and linearly reductive subgroup scheme of ${\mathbf {GL}}(V)$ associated to the linearly reductive quotient singularity $x\in X=U/G$ . In particular, the classification of linearly reductive quotient singularities in dimension d is “the same” as the classification of very small, finite and linearly reductive subgroup schemes of ${\mathbf {GL}}_{d,k}$ up to conjugacy. We refer to [LMM21, Section 3] for details and this classification.

4.2 Minimal resolution of singularities

If $x\in X$ is a normal surface singularity, then it admits a unique minimal resolution of singularities

$$ \begin{align*}\pi\,:\, Y\,\to\,X. \end{align*} $$

If $x\in X$ is moreover a rational singularity, then the exceptional locus of $\pi $ is a union of ${\mathbb {P}}^1$ ’s, whose dual intersection graph $\Gamma $ contains no cycles. The graph $\Gamma $ is called the type of $x\in X$ . If the type determines the singularity up to analytic isomorphism, then the singularity is said to be taut.

Over ${\mathbb {C}}$ , taut singularities have been classified by Laufer [La73]. For example, two-dimensional finite quotient singularities over ${\mathbb {C}}$ are taut. Since two-dimensional linearly reductive quotient singularities over algebraically closed fields of positive characteristic are F-regular (see [LMM21, Proposition 7.1]), they are taut by Tanaka’s theorem [Ta15].

Remark 4.4. Very small, finite and linearly reductive subgroup schemes of ${\mathbf {GL}}_{2,k}$ have been classified in [LMM21, Theorem 3.4], extending Brieskorn’s classification [Br67, Satz 2.9] of small subgroups of ${\mathbf {GL}}_2({\mathbb {C}})$ . The types of the associated quotient singularities in terms of this classification are given by [Br67, Satz 2.11].

Finite and linearly reductive subgroup schemes of ${\mathbf {SL}}_{2,k}$ are automatically very small and they have been classified by Hashimoto [Ha15, Theorem 3.8], extending Klein’s classification [Kl84] of finite subgroups of ${\mathbf {SL}}_2({\mathbb {C}})$ and we refer to [Ha15, LS14] for the types of the associated quotient singularities (see also Theorem 4.12 below).

4.3 The Ishii-Ito-Nakamura resolution

In [IN99], Ito and Nakamura showed that if G is a finite subgroup of ${\mathbf {SL}}_2({\mathbb {C}})$ , then a minimal resolution of singularities of ${\mathbb {C}}^2/G$ is provided by the G-Hilbert scheme $G\text{-}\mathrm {Hilb}({\mathbb {C}}^2)$ . Ishii [Is02] extended this to quotient singularities ${\mathbb {C}}^2/G$ for G a finite and very small subgroup of ${\mathbf {GL}}_2({\mathbb {C}})$ and Ishii and Nakamura [IN19] extended this to quotient singularities $U/G$ for G a finite and very small subgroup of ${\mathbf {GL}}_2(k)$ of order prime to p.

Let k be an algebraically closed field of characteristic $p\geq 0$ . Let G be a very small, finite and linearly reductive subgroup scheme of ${\mathbf {GL}}_{2,k}$ . Set $U:={\mathbb {A}}^2_k$ or $\widehat {{\mathbb {A}}}_k^2$ and let $x\in X:=U/G$ be the associated linearly reductive quotient singularity. By [Bl11], there exists a G-Hilbert scheme $G\text{-}\mathrm {Hilb}(U)$ over k that parametrises zero-dimensional G-invariant subschemes $Z\subset U$ (so-called clusters), such that the G-representation on $H^0(Z,{\mathcal O}_Z)$ is the regular representation. Taking a cluster Z to its G-orbit (see, for example, [Bl11, Remark 3.3]) induces a morphism

$$ \begin{align*}\pi\,:\,Y\,:=\,G\text{-}\mathrm{Hilb}(U) \,\to\, U/G\,=\,X \end{align*} $$

over k.

Theorem 4.5. The morphism $\pi $ is a minimal resolution of singularities.

Proof. If $p=0$ , then this is shown in [IN19], extending [IN99]. If $p>0$ and G is of length prime to p, then this is shown in [IN19]. However, this proof also works if G is linearly reductive and p divides the length of G.

Remark 4.6. By [LMM21, Theorem 11.5], every two-dimensional F-regular singularity $x\in X$ is a linearly reductive quotient singularity, say $x\in X=U/G$ for some finite and linearly reductive subgroup scheme G of ${\mathbf {GL}}_{2,k}$ . By [LMM21, Theorem 8.1], G and its embedding into ${\mathbf {GL}}_{2,k}$ are unique up to isomorphism and conjugacy, respectively. By Theorem 4.5, $G\text{-}\mathrm {Hilb}(U)\to U/G=X\ni x$ is the minimal resolution of singularities. In particular, every two-dimensional F-regular singularity can be resolved by a suitable G-Hilbert scheme.

4.4 Canonical lifts

Let k be an algebraically closed field of characteristic $p>0$ , let $W(k)$ be the ring of Witt vectors, let K be its field of fractions and let $\overline {K}$ be an algebraic closure. Let $x\in X=U/G$ be a d-dimensional linearly reductive singularity, where $U={\mathbb {A}}^d_k$ or $U=\widehat {{\mathbb {A}}}_k^d$ . Next, we set ${\mathcal U}:={\mathbb {A}}^d_{W(k)}$ or $\widehat {{\mathbb {A}}}^d_{W(k)}$ , respectively. In Section 2.2, we recalled the canonical lift ${\mathcal G}_{\mathrm {can}}$ of G over $W(k)$ . By [LMM21, Proposition 2.9], there exists a unique lift of the G-action from U to ${\mathcal U}$ . From this, we obtain a flat family

(4.1) $$ \begin{align} \mathcal{X}_{\mathrm{can}}\,:=\,{\mathcal U}/{\mathcal G}_{\mathrm{can}} \,\to\, \mathrm{Spec}\: W(k)\,. \end{align} $$

of linearly reductive quotient singularities over $W(k)$ , whose special fibre over k is $x\in X=U/G$ .

Definition 4.7. The family (4.1) is called the canonical lift of the linearly reductive quotient singularity $x\in X=U/G$ .

By the Lefschetz principle, the geometric generic fibre of $\mathcal {X}_{\mathrm {can}}$ can be identified with a finite quotient singularity of the form ${\mathbb {C}}^d/G_{\mathrm {abs}}$ , where $G_{\mathrm {abs}}$ is the abstract group associated to G and the embedding of $G_{\mathrm {abs}}\to {\mathbf {GL}}_d({\mathbb {C}})$ corresponds to the embedding $G\to {\mathbf {GL}}_{d,k}$ provided by Proposition 2.6. The canonical lift is unique in the following sense:

Proposition 4.8. We keep the notations and assumptions. Let $W(k)\subseteq R$ be a finite extension of complete DVRs and let $\mathcal {X}\to \mathrm {Spec}\: R$ be a lift of $x\in X$ that is of the form ${\mathcal V}/{\mathcal G}\to \mathrm {Spec}\: R$ for some flat lift ${\mathcal G}$ of G to R and ${\mathcal V}\cong {\mathcal U}\times _RS$ . Then, there exists a finite extension $R\subseteq S$ of complete DVRs, such that

1. 1. There exists an isomorphism

   $$ \begin{align*}{\mathcal G}_{\mathrm{can}} \times_{W(k)}S \,\cong\, {\mathcal G}\times_RS \end{align*} $$
   of group schemes over S.

2. 2. The very small representation $\rho :G\to {\mathbf {GL}}_{d,k}$ corresponding to the G-action on U lifts uniquely to ${\mathcal G}_{\mathrm {can}}$ and ${\mathcal G}$ , respectively, and they become conjugate over S.

3. 3. There is an isomorphism

   $$ \begin{align*}\mathcal{X}_{\mathrm{can}}\times_{W(k)}S \,\cong\, \mathcal{X}\times_RS \end{align*} $$
   of deformations of $x\in X$ over S.

Proof. Since ${\mathcal G}$ and ${\mathcal G}_{\mathrm {can},R}$ are lifts of G to R, they become isomorphic after passing to a finite extension $R\subseteq S$ , see also the discussion in Section 2.2. There, we also saw that the linear representation $\rho :G\to {\mathbf {GL}}_{d,k}$ lifts uniquely to ${\mathcal G}$ and ${\mathcal G}_{\mathrm {can}}$ , respectively, and that they become conjugate over S. From this, we deduce an isomorphism $\mathcal {X}_{\mathrm {can}}\times _{W(k)}S\cong \mathcal {X}\times _RS$ .

Remark 4.9. If $d\geq 3$ , then [LMM21, Corollary 10.10] shows that a d-dimensional linearly reductive quotient singularity $x\in X$ admits precisely one lift over $W(k)$ , namely the canonical lift. If $d=2$ , then linearly reductive quotient singularities usually have positive dimensional deformation spaces and admit many non-isomorphic lifts to $W(k)$ , see [LMM21, Section 12].

4.5 Simultaneous resolution of singularities

If $\mathcal {X}\to S$ is a deformation of a rational double point singularity $x\in X$ , then it admits a simultaneous resolution of singularities, but usually only after some finite base-change $S'\to S$ , see [Ar74]. In the case where $\mathcal {X}\to S$ is a family of rational surface singularities, then such a finite $S'\to S$ exists if S maps to the so-called Artin component inside the versal deformation space of $x\in X$ . Moreover, due to the existence of flops, these simultaneous resolutions (if they exist) are not unique in general.

In the special case where $x\in X$ is a two-dimensional linearly reductive quotient singularity over some algebraically closed field k of characteristic $p>0$ and where $\mathcal {X}_{\mathrm {can}}\to \mathrm {Spec}\: W(k)$ is the canonical lift of $x\in X$ , we will now show that there exists a simultaneous and minimal resolution of singularities over $W(k)$ and that it is unique. This simultaneous resolution can be most elegantly constructed using the Ishii-Ito-Nakamura resolution from Section 4.3 in families.

Let k be an algebraically closed field of characteristic $p>0$ . Let G be a finite and linearly reductive group scheme over k and let $\rho :G\to {\mathbf {GL}}_{2,k}$ be a very small representation. Let ${\mathcal G}_{\mathrm {can}}\to \mathrm {Spec}\: W(k)$ be the canonical lift of G and let $\widetilde {\rho }:{\mathcal G}_{\mathrm {can}}\to {\mathbf {GL}}_{2,W(k)}$ be the lift of $\rho $ to $W(k)$ . Let ${\mathcal U}:={\mathbb {A}}^2_{W(k)}$ and let

$$ \begin{align*}\mathcal{X}_{\mathrm{can}} \,:=\, {\mathcal U}/{\mathcal G}_{\mathrm{can}} \,\to\, \mathrm{Spec}\: W(k) \end{align*} $$

be the canonical lift of $x\in X$ . By [Bl11], there exists a ${\mathcal G}_{\mathrm {can}}$ -Hilbert scheme

$$ \begin{align*}{\mathcal G}_{\mathrm{can}}\text{-}\mathrm{Hilb}({\mathcal U}) \,\to\,\mathrm{Spec}\: W(k) \end{align*} $$

that parametrises ${\mathcal G}_{\mathrm {can}}$ -invariant subschemes $Z\subset {\mathcal U}$ that are finite and flat over $W(k)$ (so-called ${\mathcal G}_{\mathrm {can}}-$ clusters) and such that the ${\mathcal G}_{\mathrm {can}}$ -representation on $H^0(Z,{\mathcal O}_Z)$ is the regular representation. Taking such a cluster Z to its ${\mathcal G}_{\mathrm {can}}$ -orbit (see, for example, [Bl11, Remark 3.3]) induces a morphism

(4.2) $$ \begin{align} \widetilde{\pi}\,:\, \mathcal{Y}\,:=\,{\mathcal G}_{\mathrm{can}}\text{-}\mathrm{Hilb}({\mathcal U}) \,\to\, {\mathcal U}/{\mathcal G}_{\mathrm{can}}\,=\,\mathcal{X}_{\mathrm{can}} \end{align} $$

over $W(k)$ .

Theorem 4.10. Keeping the assumptions and notations

$$ \begin{align*}\widetilde{\pi} \,:\, \mathcal{Y} \,\to\, \mathcal{X}\,\to\,\mathrm{Spec}\: W(k) \end{align*} $$

is a simultaneous minimal resolution of singularities of the canonical lift $\mathcal {X}\to \mathrm {Spec}\: W(k)$ of the linearly reductive quotient singularity $x\in X=U/G$ .

1. 1. The simultaneous resolution $\widetilde {\pi }$ is unique up to isomorphism.

2. 2. The exceptional locus of $\widetilde {\pi }$ is a union of ${\mathbb {P}}^1_{W(k)}$ ’s meeting transversally and we denote by $\Gamma $ its dual intersection graph.

3. 3. The special fibre and the generic fibre are linearly reductive quotient singularities of type $\Gamma $ and $\widetilde {\pi }$ identifies the components of the exceptional loci of $\widetilde {\pi }_k$ and $\widetilde {\pi }_K$ .

Proof. We recall that we defined $G_{\mathrm {can}}:={\mathcal G}_{\mathrm {can},K}$ , which will simplify the notation in the following. The generic and special fibre of (4.2) over K and k are isomorphic to

$$ \begin{align*}\begin{array}{cccccc} &G_{\mathrm{can}}\text{-}\mathrm{Hilb}({\mathcal U}_K) &\to& {\mathcal U}_K/G_{\mathrm{can}} &\to&\mathrm{Spec}\: K\\ \mbox{and} \quad& G\text{-}\mathrm{Hilb}(U) &\to& U/G &\to&\mathrm{Spec}\: k, \end{array} \end{align*} $$

respectively, by [Bl11, Remark 3.1]. Now, $G_{\mathrm {can},\overline {K}}\text{-}\mathrm {Hilb}({\mathcal U}_{\overline {K}}) \to {\mathcal U}_{\overline {K}}/G_{\mathrm {can},\overline {K}}$ and $G\text{-}\mathrm {Hilb}(U) \to U/G$ , are minimal resolutions of singularities by Theorem 4.5. Thus, $G_{\mathrm {can}}\text{-}\mathrm {Hilb}({\mathcal U}_K) \to {\mathcal U}_K/G_{\mathrm {can}}$ is a resolution of singularities, which is minimal since it is minimal over $\overline {K}$ . Thus, $\mathcal {Y}\to \mathcal {X}\to \mathrm {Spec}\: W(k)$ is a simultaneous minimal resolution of singularities.

The exceptional fibres $\widetilde {\pi }_K$ and $\widetilde {\pi }_k$ of $\widetilde {\pi }$ over K and k are unions of ${\mathbb {P}}^1$ ’s that intersect transversally. The types, that is dual resolution graphs, associated to $\widetilde {\pi }_K$ and $\widetilde {\pi }_k$ are the same (see also Remark 4.4) and we denote this graph by $\Gamma $ . In particular, the exceptional fibres of $\widetilde {\pi }_K$ and $\widetilde {\pi }_k$ have the same numbers of irreducible components. In particular, the specialisation maps of Néron-Severi lattices

$$ \begin{align*}\mathrm{NS}(\mathcal{Y}_K) \,\leftarrow\, \mathrm{NS}(\mathcal{Y}) \,\to\, \mathrm{NS}(\mathcal{Y}_k) \end{align*} $$

are isometries of lattices. These identify the components of the exceptional loci of $\widetilde {\pi }_K$ and $\widetilde {\pi }_k$ . Moreover, given such a component of the exceptional locus of $\widetilde {\pi }_k$ , it is isomorphic to ${\mathbb {P}}^1_k$ and it extends uniquely to a ${\mathbb {P}}^1_{W(k)}$ in the exceptional locus of $\widetilde {\pi }$ . This establishes claims (2) and (3).

It remains to prove claim (1): Let $\mathcal {Y}'\to \mathcal {X}\to \mathrm {Spec}\: W(k)$ be a simultaneous resolution of singularities that coincides with the minimal resolution on special and generic fibres, respectively. Let $\alpha :\mathcal {Y}^{\prime }_K\,\to \,\mathcal {Y}_K$ be an isomorphism over $\mathcal {X}_K$ and choose a relatively (to $\mathcal {X}$ ) ample invertible sheaf $\mathcal {L}$ on $\mathcal {Y}_K$ . Then, $\alpha ^*\mathcal {L}$ is relatively ample on $\mathcal {Y}_K$ . Since the types of the singularities of $\mathcal {X}_K$ and $\mathcal {X}_k$ are the same, the fibres of $\mathcal {Y}^{\prime }_K\to \mathcal {X}_K$ and $\mathcal {Y}^{\prime }_k\to \mathcal {Y}_k$ contain the same number of exceptional divisors. Thus, the specialisation map of Néron-Severi lattices $\mathrm {NS}(\mathcal {Y}^{\prime }_K)\to \mathrm {NS}(\mathcal {Y}^{\prime }_k)$ is an injective map between two lattices of the same rank and the same discriminant, whence an isometry. Similarly, the specialisation map of Néron-Severi lattices $\mathrm {NS}(\mathcal {Y}_K)\to \mathrm {NS}(\mathcal {Y}_k)$ is an isometry. In particular, we can identify the components of the exceptional loci of $\mathcal {Y}_K$ and $\mathcal {Y}_k$ . (If the types of $\mathcal {X}_k$ and $\mathcal {X}_K$ differ, then this specialisation map is usually only injective, there are usually more $(-2)$ -curves in the special fibre and this is also the place where the difficulties with flops begin.) In particular, $\mathcal {L}$ and $\alpha ^*\mathcal {L}$ extend to relative ample invertible sheaves on $\mathcal {Y}$ and $\mathcal {Y}'$ , respectively. By [Ko09, Theorem 5.14], the isomorphism $\alpha $ extends to an isomorphism $\mathcal {Y}'\to \mathcal {Y}$ over $\mathcal {X}$ and the claimed uniqueness follows.

4.6 Rational double point singularities and Klein singularities

We specialise Definition 4.2 to the following case.

Definition 4.11. A Klein singularity is a linearly reductive quotient singularity as in Definition 4.2 with $\dim V=2$ and $\det \rho =1$ , that is, $\rho $ is a homomorphism of G to ${\mathbf {SL}}_{2,k}$ .

In particular, a Klein singularity is a two-dimensional and linearly reductive quotient singularity, it is a rational surface singularity, and since $\det (\rho )=1$ , it is Gorenstein. Quite generally, rational and Gorenstein surface singularities are precisely the rational double point singularities [Ar66]. We refer to [Du79] or [SSS] for more background on surface singularities and rational double point singularities.

If $x\in X$ is a rational double point, then its type $\Gamma $ is a simply-laced finite Dynkin graph. In characteristic zero, every rational double point singularity is taut. In positive characteristic, a Klein singularity is F-regular and thus, taut. In the case of rational double points, this also follows from Artin’s explicit classification [Ar77]. On the other hand, rational double point singularities that are not F-regular need not be taut. We have the following relation between rational double point singularities and Klein singularities in positive characteristic, see [LMM21, Theorem 11.2].

Theorem 4.12. Let k be an algebraically closed field of characteristic $p\geq 0$ . Let $x\in X$ be a normal surface singularity over k.

1. 1. If $p=0$ or $p\geq 7$ , then $x\in X$ is a Klein singularity if and only if $x\in X$ is a rational double point singularity.

2. 2. If $p>0$ , then $x\in X$ is a Klein singularity if and only if $x\in X$ is a rational double point singularity and F-regular. The finite Dynkin graphs of these singularities are of type

   $$ \begin{align*}\begin{array}{lcl} A_n, D_n, E_6, E_7, E_8 &\quad \mbox{if} \quad & p\geq7, \\ A_n, D_n, E_6, E_7 &\quad \mbox{if} \quad & p=5,\\ A_n, D_n &\quad \mbox{if} \quad & p=3, \\ A_n &\quad \mbox{if} \quad &p=2. \end{array} \end{align*} $$

A Klein singularity $x\in X=U/G$ is a linearly reductive quotient singularity and thus, determines G and $\rho :G\to {\mathbf {SL}}_{2,k}$ up to isomorphism and conjugacy, respectively. Thus, the classification of Klein singularities boils down to the classification of finite and linearly reductive subgroup schemes of ${\mathbf {SL}}_{2,k}$ and we refer to Remark 4.4 for this classification.

Remark 4.13. Theorem 3.6 and Theorem 4.12 rely on the classification of finite and linearly reductive subgroup schemes of ${\mathbf {SL}}_{2,k}$ . It is therefore no surprise that the classification lists coincide.

5 Hecke correspondences

Let $x\in X=U/G$ be a Klein singularity over an algebraically closed field k of characteristic $p>0$ . In Section 2.2, we discussed the canonical lift $\mathcal {X}\to \mathrm {Spec}\: W(k)$ of $x\in X$ and in Section 4.5 we established a simultaneous minimal resolution of singularities of the canonical lift. More precisely, this simultaneous resolution was constructed using G-Hilbert schemes extending previous work of Ishii, Ito and Nakamura [Is02, IN19, IN99].

In this section, we refine this resolution of singularities as in the work of Ito and Nakamura [IN99] and Nakajima [Na96, Na01]: we eventually obtain a bijection between the components of the minimal resolution of singularities of the Klein singularity $x\in X=U/G$ and the simple and non-trivial representations of G using special Hecke correspondences.

5.1 The Ito-Nakamura resolution revisited

We first slightly extend the setup of Section 4.3. Let k be an algebraically closed field of characteristic $p>0$ and let $x\in X=U/G$ be a Klein singularity as in Definition 4.11, that is, we have $U={\mathbb {A}}^2_k$ and G a very small, finite and linearly reductive subgroup scheme of ${\mathbf {SL}}_{2,k}$ .

Let $\{\rho _i\}_{i\in I}$ be the set of isomorphism classes of simple representations of G. Given a finite-dimensional representation $\rho $ of G, there exist non-negative integers $\nu _i\in {\mathbb {Z}}_+$ , such that $\rho $ is isomorphic to $\bigoplus _i\rho _i^{\oplus \nu _i}$ and we combine these into a multi-index $\nu =(\{\nu _i\})\in {\mathbb {Z}}_{\geq 0}^I$ . We set $\dim (\nu ):=\sum _i\nu _i\dim \rho _i$ , which is the dimension of the representation associated to $\nu $ .

For any integer $n\geq 1$ , the G-action on U induces a G-action on the Hilbert scheme $\mathrm {Hilb}^n(U)$ , which parametrises zero-dimensional subschemes of length n of U. We consider the fixed point scheme

$$ \begin{align*}\mathrm{Hilb}^{n,G}(U) \,:=\, \left( \mathrm{Hilb}^n(U) \right)^G, \end{align*} $$

that is, the largest subscheme of $\mathrm {Hilb}^n(U)$ on which G acts trivially. It parametrises G-invariant and zero-dimensional subschemes of length n of U. Given $\nu =(\{\nu _i\})_i\in {\mathbb {Z}}_{\geq 0}^I$ with $\dim (\nu )=n$ , we define

$$ \begin{align*}H^\nu\,=\,\left\{ Z\,\in\,\mathrm{Hilb}^{n,G}(U)\,|\, H^0(Z,{\mathcal O}_Z)\,\cong\, \bigoplus \rho_i^{\oplus \nu_i}\mbox{ as }G\mbox{-representation} \right\}, \end{align*} $$

which defines a subscheme of $\mathrm {Hilb}^{n,G}(U)$ . Adapting [Ki16, Lemma 12.4], which follows [IN99, Section 9], to our situation, we have the following.

Lemma 5.1. We keep the assumptions and notations and let $n\geq 1$ be an integer.

1. 1. The scheme $\mathrm {Hilb}^{n,G}(U)$ is smooth over k.

2. 2. We have a decomposition

   $$ \begin{align*}\mathrm{Hilb}^{n,G}(U) \,=\, \bigsqcup_{\nu}\, H^\nu, \end{align*} $$
   where the disjoint union runs over all multi-indices $\nu \in {\mathbb {Z}}_{\geq 0}^I$ of dimension n. Each $H^\nu $ is a smooth subscheme of $\mathrm {Hilb}^{n,G}(U)$ .

Proof. Since U is two-dimensional, $\mathrm {Hilb}^n(U)$ is smooth over k by [Fo68, Theorem 2.4]. To show Claim (1), it remains to show that the fixed point scheme for the G-action is also smooth, which follows from Lemma 5.2 below. Then, Claim (2) is obvious.

Lemma 5.2. Let k be an algebraically closed field, let X be a scheme that is smooth over k, let G be a finite and linearly reductive group scheme over k and assume that G acts on X. Then, the fixed point scheme $X^G\subseteq X$ is smooth over k.

Proof. Let $x\in X^G$ . Passing to the completion of the local ring ${\mathcal O}_{X,x}$ , using that X is smooth over k and passing to coordinates such that the G-action is linear (this is always possible by [Sa12, proof of Corollary 1.8]), we may assume that

$$ \begin{align*}\widehat{{\mathcal O}}_{X,x} \,\cong\, k[[u_1,...,u_d]] \end{align*} $$

and that the G-action is linear. In this description it is easy to see that the G-invariant subscheme of $\mathrm {Spec}\:\widehat {{\mathcal O}}_{X,x}$ is smooth (see also the proof of [IN99, Lemma 9.1]), which implies that $X^G$ is smooth near x.

An important special case is the regular representation of G, where we have $\nu _i=\dim \rho _i$ for all i and in this case, we will write $\delta $ for the corresponding multi-index. The dimension of $\delta $ is equal to the length of G. In this case, we have

$$ \begin{align*}\pi\,:\, H^\delta \,=\, G\text{-}\mathrm{Hilb}(U) \,\to\, U/G \,=\,X, \end{align*} $$

which we already studied in Section 4.3. There, we saw that $\pi $ is a minimal resolution of singularities of the Klein singularity $x\in X=U/G$ .

5.2 Hecke correspondences

We keep the assumptions and notations of the previous section. We let n be the length of G and let $\delta $ be the multi-index corresponding to the regular representation. For $i\in I$ , we set $\alpha _i:=(0,...,0,1,0...)$ with the non-zero entry in the i.th position. We define

$$ \begin{align*}B_i \,:=\, \left\{ J_1 \,\in\, H^{\delta-\alpha_i},\quad J_2\,\in\, H^\delta\quad |\quad J_2\subseteq J_1 \right\} \,\subseteq\, H^{\delta-\alpha_i}\times H^\delta, \end{align*} $$

which is a Hecke correspondence, see [Na96], [Na01, Section 6.1] or [Ki16, Section 12.4]. We let

$$ \begin{align*}E_i \,:=\, \mathrm{pr}_2(B_i)\,\subseteq\, H^\delta \,=\, G\text{-}\mathrm{Hilb}(U) \end{align*} $$

be the image under the projection $\mathrm {pr}_2$ onto the second factor.

In characteristic zero, the following result is due to Nakajima [Na96] and independently to Ito and Nakamura [IN99], see also [Ki16, Section 12.4].

Theorem 5.3. Let k be an algebraically closed field of characteristic $p\geq 0$ . Let G be a finite and linearly reductive subgroup scheme of ${\mathbf {SL}}_{2,k}$ , let $x\in X:=U/G$ be the associated Klein singularity, and let $\pi :H^\delta \to X$ be the Ito-Nakamura resolution.

1. 1. The assignment $i\mapsto E_i$ defines a bijection between the set $\{\rho _i\}_i$ of isomorphism classes of non-trivial simple representations of G and the set of irreducible components of the exceptional divisor $\mathrm {Exc}(\pi )$ of $\pi $ .

2. 2. If $i\neq j$ , then

   $$ \begin{align*}B_i\cdot B_j \,=\, A_{ij} \end{align*} $$
   where the numbering of the $\{\rho _i\}_i$ is as in [IN99, Section 10].

We thus obtain a bijection between the dual resolution graph of $\pi $ and the graph obtained by removing the vertex $\rho _0$ from the McKay graph of G with respect to the embedding $G\to {\mathbf {SL}}_{2,k}\to {\mathbf {GL}}_{2,k}$ .

Proof. The case-by-case analysis of [IN99, Section 12] carries directly over from finite subgroups of ${\mathbf {SL}}_2({\mathbb {C}})$ to finite and linearly reductive subgroup schemes of ${\mathbf {SL}}_{2,k}$ .

Here is a second proof of this theorem that uses our lifting results:

Proof. If $p=0$ , then the Lefschetz principle allows to reduce to $k={\mathbb {C}}$ , where the theorem is due to Ito, Nakajima and Nakamura [IN99, Na96].

We now assume $p>0$ . Let $\mathcal {X}_{\mathrm {can}}={\mathcal U}/{\mathcal G}_{\mathrm {can}}\to \mathrm {Spec}\: W(k)$ be the canonical lift of $x\in X=U/G$ (see Section 4.4) and let $\widetilde {\pi }:\mathcal {Y}\to \mathcal {X}_{\mathrm {can}}\to \mathrm {Spec}\: W(k)$ be the simultaneous resolution of singularities (see Section 4.5). Moreover, we have the abstract group $G_{\mathrm {abs}}$ and the canonical lift $G_{\mathrm {can}}$ associated to G (see Section 2.2) and we have a bijection of simple representations of $G_{\mathrm {abs}}$ , $G_{\mathrm {can}}$ and G by Proposition 2.6.

The generic fibre $\mathcal {X}_K$ is isomorphic to ${\mathcal U}_K/G_{\mathrm {can}}$ and using the Lefschetz principle, the geometric generic fibre can be identified with ${\mathbb {C}}^2/G_{\mathrm {abs}}$ . Over ${\mathbb {C}}$ , we have Theorem 5.3 for $G_{\mathrm {abs}}$ ${\mathbb {C}}^2/G_{\mathrm {abs}}$ and its minimal resolution of singularities by [IN99, Na96].

By the Lefschetz principle, we have it for $G_{\mathrm {can}}$ , $\mathcal {X}_K={\mathcal U}_{K}/G_{\mathrm {can}}$ and $\widetilde {\pi }_K$ . Using the comparison results Proposition 2.6 and Theorem 4.10, we obtain it for $\mathcal {X}_k={\mathcal U}_k/{\mathcal G}_{\mathrm {can},k}$ and $\widetilde {\pi }_k$ , that is, for $X=U/G$ , G and $\pi $ .

5.3 Local McKay correspondence

We now extend work of Ishii and Nakamura [Is02, IN19] to our linearly reductive setting. Let G be a very small, finite and linearly reductive subgroup scheme of ${\mathbf {GL}}_{2,k}$ and let $x\in X=U/G$ be the associated two-dimensional linearly reductive quotient singularity. Until the end of this section, we do not require it to be a Klein singularity. Note that in dimension two, linearly reductive quotient singularities are the same as F-regular singularities, see Remark 4.6.

Let $\{\rho _i\}_i$ be the set of simple representations of G. Let $\rho _0$ be the trivial representation and we choose our numbering of the $\rho _i$ ’s to be the one of [IN19, Theorem 3.6]. Let $\pi :Y\to X$ be its minimal resolution of singularities and let $\{E_i\}$ be the irreducible components of the exceptional divisor $\textrm {Exc}(\pi )$ of $\pi $ . Then, there is exists a connection between the $\{\rho _i\}$ and the exceptional divisors of $\pi $ as follows - this is yet another version of the McKay correspondence.

Theorem 5.4. Keeping assumptions and notations, let ${\mathfrak m}\subset {\mathcal O}_U$ be the maximal ideal corresponding to the origin, let $y\in Y$ be a closed point, let $Z_y$ be the G-invariant cluster of U corresponding to y and let ${\mathcal I}_{Z_y}\subset {\mathcal O}_U$ be its ideal sheaf. Then, the G-representation on ${\mathcal I}_{Z_y}/{\mathfrak m}{\mathcal I}_{Z_y}$ is given by

$$ \begin{align*}\left\{ \begin{array}{ll} \rho_i\,\oplus\,\rho_0 &\quad \mbox{if }y\in E_i\,\backslash\,\bigcup_{j\neq i}E_j\\ \rho_i\,\oplus\,\rho_j\,\oplus\,\rho_0 & \quad \mbox{if }y\in E_i\cap E_j \quad \mbox{with} \quad i\neq j. \end{array} \right. \end{align*} $$

Proof. For $k={\mathbb {C}}$ , this is [Is02, Theorem 7.1]. For k algebraically closed of arbitrary characteristic and G a very small and finite subgroup of ${\mathbf {GL}}_{2}(k)$ of order prime to p, this is [IN19, Theorem 3.6]. However, these proofs also work if G is a very small and finite and linearly reductive subgroup scheme of ${\mathbf {GL}}_{2,k}$ .

Remark 5.5. In [IN19], the G-quiver structure of $G\text{-}\mathrm {Hilb}(U)$ was studied in the case where G is a very small and finite subgroup of ${\mathbf {GL}}_{2}(k)$ . We leave the extension of these results to the linearly reductive case to the reader.

5.4 Reflexive sheaves on the minimal resolution

We end this section by shortly digressing on work of Wunram [Wu88], Ishii and Nakamura [IN19], which generalises work of Artin and Verdier [AV85]. We keep the assumptions and notations from Section 5.3. Let $F\subset Y$ be the fundamental divisor of $\pi $ , see [Ar66].

Theorem 5.6. Keeping assumptions and notations, there exists a bijection between

1. 1. the set of irreducible components $\{E_i\}$ of $\mathrm {Exc}(\pi )$ and

2. 2. the set of non-trivial indecomposable full ${\mathcal O}_Y$ -modules $\{{\mathcal M}_i\}$ , special in the sense that $H^1(Y,{\mathcal M}_i^{\vee })=0$

This correspondence ${\mathcal M}_i\mapsto E_i$ is defined by

$$ \begin{align*}c_1({\mathcal M}_i)\cdot E_j \,=\, \delta_{i,j}\,. \end{align*} $$

The rank of ${\mathcal M}_i$ is equal to $c_1({\mathcal M}_i)\cdot F$ , the multiplicity of $E_i$ in F.

Proof. For $k={\mathbb {C}}$ , this is [Wu88]. For k algebraically closed of arbitrary characteristic and G a very small and finite subgroup of ${\mathbf {GL}}_{2,k}$ of order prime to p, this is [IN19, Theorem 2.8]. However, these proofs also work if G is a very small, finite and linearly reductive subgroup scheme of ${\mathbf {GL}}_{2,k}$ .

Here, a full ${\mathcal O}_Y$ -module is as defined in [IN19, Definition 2.4]. By [IN19, Corollary 2.5], the assignment ${\mathcal M}\mapsto \pi _*{\mathcal M}$ sets up a bijection between the set of (indecomposable) full ${\mathcal O}_Y$ -modules with the set of (indecomposable) reflexive ${\mathcal O}_X$ -modules. In particular, the above theorem yields a bijection between the set of irreducible components of $\mathrm {Exc}(\pi )$ and the set of non-trivial, indecomposable and reflexive ${\mathcal O}_X$ -modules.

Remark 5.7. The group scheme G plays no rôle in this theorem and the discussion thereafter. Probably, these results should be viewed as results on two-dimensional F-regular singularities (see Remark 4.6).

6 Conjugacy classes and Ito-Reid correspondence

For a finite group, the number of isomorphism classes of complex simple representations is equal to the number of conjugacy classes. Thus, one can choose a bijection between these two sets. In particular, for a finite subgroup of ${\mathbf {SL}}_2({\mathbb {C}})$ one can choose a bijection between the vertices of its McKay graph and the conjugacy classes of G. In [IR96], Ito and Reid gave a canonical bijection between these two sets. In this section, we extend this Ito-Reid correspondence to finite and linearly reductive subgroup schemes of ${\mathbf {SL}}_{2,k}$ . The main difficulty is to define a suitable notion of conjugacy classes for finite and linearly reductive group schemes.

6.1 Conjugacy classes

Given a finite group $G_{\mathrm {abs}}$ , a representation $\rho :G_{\mathrm {abs}}\to {\mathbf {GL}}_n({\mathbb {C}})$ and an element $g\in G_{\mathrm {abs}}$ , we have the trace $\mathrm {tr}(\rho (g))\in {\mathbb {C}}$ . This pairing $(\rho ,g)\mapsto \mathrm {tr}(\rho (g))$ induces a non-degenerate pairing between isomorphism classes of simple representations of $G_{\mathrm {abs}}$ and conjugacy classes. Thus, conjugacy classes can be thought of as being “dual” to semi-simple representations, which can be used to give an unusual definition of conjugacy classes. This definition generalises to finite and linearly reductive group schemes. More precisely, we make the following definition, which is inspired by a result of Serre [Se77, Section 11.4] and which we discuss in detail in Appendix B.2.

Definition 6.1. Let G be a finite and linearly reductive group scheme over an algebraically closed field k of characteristic $p\geq 0$ . Then, the set of conjugacy classes is defined to be $\mathrm {Spec}\: {\mathbb {C}}\otimes K_k(G)$ .

We discuss several approaches to conjugacy classes for finite group schemes in Appendix B - there, we hope to convince the reader that Definition 6.1 is the best for the purposes of this article. For example, by Proposition B.2 it is compatible with canonical lifts and there is a natural bijection with the set of conjugacy classes of the abstract group $G_{\mathrm {abs}}$ associated to G. Concerning the choice of field ${\mathbb {C}}$ in Definition 6.1, we refer to Remarks B.3.

6.2 An explicit bijection

For a finite and linearly reductive subgroup scheme G of ${\mathbf {SL}}_{2,k}$ , we have an associated embedding of the finite group $G_{\mathrm {abs}}$ into ${\mathbf {SL}}_2({\mathbb {C}})$ . Let $\{\rho _i\}_i$ be the set of isomorphism classes of semi-simple representations of G and let $\{\rho _{\mathrm {abs},i}\}_i$ be the corresponding set for $G_{\mathrm {abs}}$ obtained by Proposition 2.6. The associated McKay graphs $\Gamma (G,\rho ,\{\rho _i\}_i)$ and $\Gamma (G_{\mathrm {abs}},\rho _{\mathrm {abs}}, \{\rho _{\mathrm {abs},i}\}_i)$ coincide by Proposition 3.5 and we denote both by $\widehat {\Gamma }$ . Next, the group $G_{\mathrm {abs}}$ admits a presentation of the form

$$ \begin{align*}G_{\mathrm{abs}} \,=\, \left\langle A,B,C \,|\, A^r=B^s=C^t=ABC \right\rangle \end{align*} $$

for suitable non-negative integers $r,s,t$ . The non-trivial conjugacy classes of $G_{\mathrm {abs}}$ can be uniquely represented by the following elements

$$ \begin{align*}ABC,\quad A^i \mbox{ with }1\leq i\leq r-1, \quad B^i \mbox{ with }1\leq i\leq s-1, \quad C^i \mbox{ with }1\leq i\leq t-1. \end{align*} $$

This allows to give an explicit bijection between the conjugacy classes of $G_{\mathrm {abs}}$ and the vertices of $\widehat {\Gamma }$ . We refer to [Ki16, Section 2] for details. Using Proposition B.2, we obtain an explicit bijection between the conjugacy classes of G and the vertices of $\widehat {\Gamma }$ .

6.3 The Ito-Reid correspondence

If G is a finite subgroup of ${\mathbf {SL}}_2({\mathbb {C}})$ , then Ito and Reid [IR96] (see also [Re02]) constructed a canonical bijection. Let us run through [Re02, Section 2] and show that this can be carried over to our linearly reductive setting: let k be an algebraically closed field of characteristic $p>0$ and let G be a finite and linearly reductive subgroup scheme of ${\mathbf {SL}}_{2,k}$ .

1. 1. Associated to G, we have the abstract group $G_{\mathrm {abs}}$ . By Proposition 2.6, an embedding $G\to {\mathbf {SL}}_{n,k}$ yields an embedding $G_{\mathrm {abs}}\to {\mathbf {SL}}_n({\mathbb {C}})$ . By Proposition B.2, we can identify conjugacy classes of G (in the sense of Definition 6.1) and conjugacy classes of $G_{\mathrm {abs}}$ , which allows us to define the age of a conjugacy class of G via the corresponding notion for $G_{\mathrm {abs}}$ as, for example, in [Re02, Section 2]. A conjugacy class of age 1 is called junior and if $n=2$ , then all conjugacy classes are junior.

Remark 6.2. Using the “toric mechanism” mentioned in [Re02, Section 2], one can define the age directly and without referring to lifts, but we will not pursue this here.

1. 2. Let $x\in X:=U/G$ be the associated Klein singularity. We have the canonical lift $\mathcal {X}_{\mathrm {can}}\to \mathrm {Spec}\: W(k)$ and the simultaneous resolution of singularities $\widetilde {\pi }:\mathcal {Y}\to \mathcal {X}_{\mathrm {can}}\to \mathrm {Spec}\: W(k)$ by Theorem 4.10. Passing to geometric generic fibres and using the Lefschetz principle, we obtain the minimal resolution of singularities of ${\mathbb {C}}^2/G_{\mathrm {abs}}$ . The special fibre of $\widetilde {\pi }$ , the geometric generic fibre of $\widetilde {\pi }$ and the minimal resolution of ${\mathbb {C}}^2/G_{\mathrm {abs}}$ are crepant in the sense of [Re87] and we can identify the exceptional divisors of these three resolutions with each other, see Theorem 4.10. This way, we obtain an Ito-Reid correspondence between junior conjugacy classes of G and crepant divisors of the resolution [Re02, Theorem 2.1].

Remark 6.3. It seems reasonable that one can extend this correspondence to finite and linearly reductive subgroup schemes of ${\mathbf {SL}}_{n,k}$ with $n\geq 3$ , but we will not pursue this here.

7 Derived categories

Let G be a very small, finite and linearly reductive subgroup scheme of ${\mathbf {GL}}_{2,k}$ , let $x\in X:=U/G$ be the associated linearly reductive quotient singularity and let $\pi :Y\to X$ be its minimal resolution of singularities. Gonzalez-Sprinberg and Verdier [GV83] gave an interpretation of the McKay correspondence as an isomorphism between the K-groups $K^G(U)$ and $K(Y)$ . Kapranov and Vasserot [KV00] and Bridgeland, King and Reid [BKR01] generalised this to an equivalence of derived categories ${\mathcal {D}}^G(U)$ and ${\mathcal {D}}(Y)$ . In this section, we extend this to our setting, following Ishii, Ito, Nakamura and Ueda [Is02, IN19, IU15].

We have a commutative diagram

By Theorem 4.5, the minimal resolution $\pi $ can be constructed by the Ishii-Ito-Nakamura resolution

$$ \begin{align*}G\text{-}\mathrm{Hilb}(U) \,\to\, U/G \end{align*} $$

We let ${\mathcal {Z}}$ be the universal cluster over $G\text{-}\mathrm {Hilb}(U)$ , we identify $G\text{-}\mathrm {Hilb}(U)$ with Y and then, we have a commutative diagram

Let ${\mathcal {D}}(Y)$ be the derived category of coherent sheaves on Y. Let ${\mathcal {D}}^G(U)$ be the derived category of G-equivariant coherent sheaves on U. Following [Is02] and [IN19], we define two functors

$$ \begin{align*}\begin{array}{ccccc} \Psi &:& {\mathcal{D}}^G(U) &\to& {\mathcal{D}}(Y)\\ \Phi &:& {\mathcal{D}}(Y) &\to& {\mathcal{D}}^G(U) \end{array} \end{align*} $$

by

$$ \begin{align*}\Psi(-) \,:=\, [p_*\, \mathbf{L}q^*\, (-)]^G \end{align*} $$

and

$$ \begin{align*}\Phi(-) \,:=\, \mathbf{R}\pi_{U,*} \left( {\mathcal O}_{{\mathcal{Z}}}^\vee \,\otimes^{\mathbf{L}}\,\pi_Y^*(-\otimes\rho_0)\,\otimes^{\mathbf{L}}\, \pi_U^*K_{U}\right) [2], \end{align*} $$

where ${\mathcal O}_{{\mathcal {Z}}}^\vee :=\mathbf {R}Hom({\mathcal O}_{{\mathcal {Z}}},{\mathcal O}_{Y\times U})$ denotes the dual of ${\mathcal O}_{{\mathcal {Z}}}$ , where $-\otimes \rho _0:{\mathcal {D}}(Y)\to {\mathcal {D}}^G(Y)$ denotes the functor that attaches the trivial G-action and where $K_U$ denotes the canonical sheaf of U. We refer to [IN19, Section 3.1] for details, conventions and notations.

Theorem 7.1. Keeping assumptions and notations, $\Phi $ is fully faithful and $\Psi $ is a left adjoint of $\Phi $ .

Proof. For G is a very small subgroup of ${\mathbf {GL}}_2({\mathbb {C}})$ , this is [Is02, Section 6] and [IU15, Proposition 1.1 and Lemma 2.9]. For k algebraically closed of arbitrary characteristic and G a very small and finite subgroup of ${\mathbf {GL}}_{2}(k)$ of order prime to p, this is [IN19, Theorem 3.2]. However, these proofs also work if G is a very small, finite and linearly reductive subgroup scheme of ${\mathbf {GL}}_{2,k}$ .