McKay correspondence for linearly reductive finite group schemes in positive characteristic

Jeremy Usatine; Matthew Satriano

REVIEW 2 major objections 3 minor 14 references

McKay correspondence for linearly reductive finite group schemes in positive characteristic

T0 review · 2 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper proves that for any finite linearly reductive subgroup scheme G of SL(V) over any field, every crepant resolution of V/G has Euler number equal to the number of irreducible algebraic representations of G.

desk verdict Real extension of the McKay correspondence to positive characteristic, worth refereeing; but Lemma 7.2 has a false displayed identity that must be fixed before the main chain is internally sound. read the letter →

arxiv 2608.05020 v1 pith:TXEGCSTI submitted 2026-08-05 math.AG

Matthew Satriano , Jeremy Usatine This is my paper · ORCID

classification math.AG MSC 14E1814A2014L15

keywords McKaycorrespondencelinearlyreductivegroupschemespositivecharacteristicmotivicintegrationArtinstackscrepantresolutionstringyEulernumbercyclotomicinertia

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper establishes a McKay correspondence for quotient singularities in arbitrary characteristic, for the class of finite linearly reductive group schemes. The headline result is that if $V$ is a finite-dimensional vector space and $G$ is a finite linearly reductive subgroup scheme of $\mathrm{SL}(V)$, then every crepant resolution of $V/G$ has Euler number equal to the number of irreducible algebraic representations of $G$ (after scalar extension to an algebraic closure). This covers non-reduced group schemes such as $\mu_p$, whose quotient stacks are Artin but not Deligne-Mumford, so the proof extends the authors' motivic integration for Artin stacks to positive characteristic. A motivic refinement and a cohomological refinement are proved along the way. A reader should care because positive-characteristic quotient singularities, including many rational double points, are often only expressible through such group schemes, and the result gives a uniform representation-theoretic count on resolutions.

What carries the argument

The carrying object is the cyclotomic inertia stack $I_\mu(X)$---the stack of representable maps from $B\mu_r$ to $X$---together with its locally constant weight function $\operatorname{wt}_X$. Around this, the paper uses warping stacks $W(X)$: pairs consisting of a flat good-moduli-space cover of a disc and a representable map to $X$, which convert twisted arcs into ordinary arcs and let the motivic change-of-variables formula run. The positive-characteristic work is concentrated in proving that fibers of the twisted-to-warped map $J_n^r(X)\to L_n(W(X))$ are affine lines when nonempty, under the divisibility condition $p^s\mid n$ for $r=p^s r'$, and in computing the resulting measure comparison. The group-theoretic stage is played by $\operatorname{Conj}_\mu(G)$, which replaces conjugacy classes of elements by conjugacy classes of injections $\mu_r\hookrightarrow G$, and by the age of such a class, defined from the $\mathbb Z/r$-grading of $V$.

What would settle it

In characteristic $2$, let $G=\mu_2$ act on $V=\mathbb A^2$ with weights $(1,1)$, so that $V/G$ is the $\mathrm A_1$ hypersurface $w^2=uv$. Compute the motivic class of the crepant resolution (the total space of $O(-2)$ over $\mathbb P^1$) and compare it with the Theorem A sum $L^2\,\mathfrak e(B\mu_2)+L^1\,\mathfrak e(B\mu_2)$; if the two classes differ, the motivic McKay correspondence is false. Equivalently, the compactly supported Euler number must equal $\#\operatorname{Irrep}_k(\mu_2)$, which the equal-count assertion requires to be $2$.

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Extended reading notes

Core claim

The paper's central claim is a McKay correspondence for finite linearly reductive group schemes over fields of any characteristic. If $V$ is a finite-dimensional vector space over $k$ and $G$ is a finite linearly reductive subgroup scheme of $\mathrm{SL}(V)$, then the stringy motivic class of the quotient is $$\mathfrak $e^{{\mathrm{str}}$}(V/G)=\sum_{\phi\in\operatorname{Conj}_\mu(G)} $L^{{\operatorname{age}}$(\phi)}\,\mathfrak e(BZ_G(\phi)),$$ where $\operatorname{Conj}_\mu(G)$ indexes conjugacy classes of group-scheme homomorphisms $\mu_r\to G$ and $\operatorname{age}(\phi)$ is computed from the weights of the induced $\mu_r$-action. Specializing to Euler numbers gives $\chi^{\mathrm{str}}(V/G)=\#\operatorname{Irrep}_{\bar k}(G\otimes_k\bar k)=\#\operatorname{Conj}_\mu(G\otimes_k\bar k)$, and if $X\to V/G$ is a crepant resolution by a scheme, this Euler number is the compactly supported Euler number of $X$. The cohomological refinement asserts $\dim_{\mathbb Q_\ell}H^i_{\acute{e}t,c}(X\otimes_k k^s,\mathbb Q_\ell)=\#\{\phi:\operatorname{age}(\phi)=i/2\}$, so only even-degree cohomology occurs. These statements are derived from a global McKay correspondence for tame Artin stacks applied to the quotient stack $[V/G]$.

Load-bearing premise

The whole chain of theorems depends on the assertion that, at each finite arc approximation level satisfying the divisibility condition $p^s\mid n$ (for $r=p^s r'$), the space of ways to extend a twisted arc to a warped one is an affine line when nonempty; if this positive-characteristic fiber computation is wrong, the change-of-variables formula and the McKay theorems collapse.

Editorial extensions

If this is right

Every crepant resolution of $V/G$ has compactly supported Euler number equal to $\#\operatorname{Irrep}_{\bar k}(G\otimes_k\bar k)$, so the count is independent of the embedding $G\subset\mathrm{SL}(V)$.
The stringy motivic class of $V/G$ is directly computable from group-scheme data: it is the sum over $\phi\in\operatorname{Conj}_\mu(G)$ of $L^{\operatorname{age}(\phi)}\,\mathfrak e(BZ_G(\phi))$.
A crepant resolution $X\to V/G$ has etale cohomology only in even degrees, with the dimension in degree $2a$ equal to the number of cyclotomic conjugacy classes of age $a$.
The theorems cover quotients by non-reduced group schemes such as $\mu_p$, bringing singularities like the characteristic-2 $\mathrm A_1$ hypersurface into the McKay correspondence.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because $\operatorname{Conj}_\mu(G)$ counts maps $\mu_r\to G$ rather than group elements, it may serve as the right notion of conjugacy class for arbitrary finite group schemes, opening a route to wild McKay questions by the same counting mechanism.
The embedding independence of the Euler count suggests that stringy invariants can be attached to the abstract group scheme $G$ alone, and one can ask which positive-characteristic quotient singularities are classified by such data.
The divisibility condition in the fiber theorem indicates that a fully general motivic integration in positive characteristic may need a wild or $p$-adic variant when the jet order is not $p$-power divisible; the boundary cases are the natural place to look for corrections.
The purity statement for crepant resolutions is likely to combine with equivariant resolution techniques to yield further cohomological vanishing results for these quotient singularities.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a serious paper, and it deserves a serious referee. The main results — motivic, Euler-number, and cohomological McKay correspondence for finite linearly reductive group schemes in arbitrary characteristic — are new and significant. But the stress-test note is right: Lemma 7.2 has a false displayed identity, and because Theorem 7.1 is the positive-characteristic hinge, the main theorems are not internally justified as typeset.

The paper's real content is a positive-characteristic extension of the authors' motivic integration for Artin stacks, applied to the tame stack [V/G] where G is a finite linearly reductive group scheme. The reduction to Theorem D, the description of cyclotomic inertia in Section 12, the count of irreducible representations in Proposition 13.1, and the worked example in characteristic 3 all looked coherent to me. The derivation is parameter-free and the example is independently checkable. The authors are also honest about where characteristic zero was used in earlier proofs and what they had to replace.

The soft spot is exactly where the stress-test says it is. Lemma 7.2 displays ((1 + x t^{n/p^s})^{r'})^r = 1 + a t^n. Expanding the left gives (1 + x t^{n/q})^{q r'} = (1 + a t^n)^{r'} = 1 + r'a t^n mod t^{n+1}. Since r' need not be 1 mod p, the equality fails; the p=5, r=10, n=5 example is correct. The fix is small: adjoin x^{p^s} = a/r', or equivalently a times the inverse of r', and take h' = 1 + x t^{n/p^s}. Then h'^r = 1 + a t^n. So the intended lemma is true, but as printed the proof of Theorem 7.1 does not establish the A^1-fiber statement, and Theorems A–D depend on that.

Two secondary concerns. The paper leans heavily on the authors' own earlier framework, with several 'the proof works verbatim' imports; that is not a flaw by itself, but it makes independent verification genuinely hard, and the referee should ask which cited lemmas really are characteristic-free. Also Theorem 14.1 uses results from a book draft and a functorial resolution paper; those should be checked, not trusted.

Bottom line: send it to peer review. The right report is 'revise': ask for the corrected Lemma 7.2, a scan of every place it is used, and a note about the imported results. I would not cite the current arXiv version in this state, but I expect the post-fix version to be citable and important. This is exactly the kind of paper a reading group should dissect.

Referee Report

2 major / 3 minor

Summary. The paper develops motivic integration for Artin stacks over fields of arbitrary characteristic and applies it to prove McKay-type correspondences for quotients V/G by finite linearly reductive group schemes. The main results are: a motivic McKay correspondence (Theorem A) expressing e_str(V/G) as a sum over cyclotomic conjugacy classes Conj_mu(G); an Euler-number correspondence (Theorem B) identifying chi_str(V/G) with the number of irreducible representations of G; a cohomological refinement (Theorem C); and a change-of-variables formula (Theorem D) for crepant maps from tame Artin stacks. The architecture is to compare twisted jets to warping stacks, with Theorem 7.1 as the positive-characteristic pivot, then to identify the cyclotomic inertia of [V/G] with fixed loci indexed by maps mu_r -> G, and finally to count cyclotomic conjugacy classes via a decomposition G = Delta semidirect H. The paper also proves a purity statement for crepant resolutions and gives a worked example in characteristic 3.

Significance. If the main chain is valid, this is a substantial contribution: it extends the McKay correspondence to non-reduced linearly reductive group schemes such as mu_p and, along the way, generalizes the authors' motivic integration theory for Artin stacks to positive characteristic. The paper is careful to identify where earlier proofs used characteristic zero, and the explicit example in Section 15 provides nontrivial evidence for the formulas. However, two proof errors occur in load-bearing lemmas (Lemma 7.2 and Lemma 13.2), so the current version does not yet establish the main results as typeset.

major comments (2)

[Section 7, Lemma 7.2(b)] The displayed identity in the proof of Lemma 7.2(b) is false as written. With r = p^s r' and gcd(p,r')=1, the left-hand side ((1 + x t^{n/p^s})^{r'})^r equals (1 + x t^{n/p^s})^{p^s r'^2}, which in characteristic p is (1 + a t^n)^{r'^2}, not 1 + a t^n unless r'=1. For example, p=5, r=10 (so r'=2), n=5, a=1 in F_5[x]/(x^5-1)[t]/(t^6) gives (1+x t)^{20} = (1+t^5)^4 = 1+4t^5, not 1+t^5. Lemma 7.2 is used in Proposition 7.3(1), Proposition 7.4, and Theorem 7.1; Theorem 8.2, Theorem 9.1, Corollary 9.8, Theorem D, and Theorems A-C all depend on Theorem 7.1. The positive-characteristic pivot of the paper is therefore not established as typeset. The error appears repairable (for instance, choose an integer m with m r' congruent to 1 modulo p^s and use h = (1 + x t^{n/p^s})^m), but the corrected argument must be supplied.
[Section 13, Lemma 13.2] The proof of Lemma 13.2 contains a sign error. Conjugation by (a,1) sends (q,h) to (q+(1-h)a,h). With the displayed choice a = -(1/m) sum_{i=1}^{m-1} i h^i q, one obtains q+(1-h)a = 2q - pi_h(q), not pi_h(q); the positive choice a = +(1/m) sum_{i=1}^{m-1} i h^i q gives pi_h(q). For instance, when Q = Z/3, H = Z/2, h(q)=-q, and q=1, the displayed choice gives 2 mod 3, which is not fixed by h, whereas the positive choice gives 0. Lemma 13.2 is used in the proof of Proposition 13.1, which proves the equality #Conj_mu(G) = #Irrep_k(G) in Theorem B. The statement of the lemma is plausibly correct with the sign corrected, but as written the proof is invalid.

minor comments (3)

[Section 7, Lemma 7.2] The statement of Lemma 7.2 should explicitly assume n >= 1; the proof of the characteristic-zero case uses t^{2n}=0 in A[t]/(t^{n+1}).
[Throughout] Many results are imported from the authors' earlier papers with the phrase 'the proof works verbatim' (e.g., Theorem 3.5, Proposition 4.2, Theorem 5.6, Theorem 9.7). Because the current paper is the first place several of these statements are stated in arbitrary characteristic, a table listing the exact hypotheses used in each quoted proof would substantially aid verification.
[Section 9, Corollary 9.8] The proof invokes [BV24] for a bijection of arc sets without a theorem number; please cite the precise statement used.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivation is self-contained, with the positive-characteristic machinery generalized rather than imported as the target result.

full rationale

The paper's central claims (Theorems A, B, C, D) are obtained by a formal derivation chain whose inputs are the authors' previously developed motivic integration for Artin stacks and new positive-characteristic extensions proved here. The key new steps—Theorem 3.11, Proposition 3.8, Section 6, Theorem 7.1, Theorem 8.2, Theorem 9.1, and Corollary 9.8—are proved in the text rather than assumed, and they do not presuppose the McKay equalities. The quantity Conj_mu(G) is defined independently of the main theorems via maps from mu_r up to conjugacy, age is defined from the eigen-decomposition of the representation, and the equality #Conj_mu(G) = #Irrep_k(G) is established by an independent group-theoretic argument in Proposition 13.1 using the structure G = Delta ⋊ H and standard representation theory, not by fitting. The self-citations to [SU22, SU26, SU24a, SU23a, SU23b, SU24b] are used as previously established foundations, with explicit statements about which characteristic-0 hypotheses are unused; these are real evidence and not circular. The final Euler-number equality follows by specializing a motivic identity, with EP(e(BH)) = 1 and EP(L) = 1, so no fitted parameter is renamed as a prediction. One caveat, unrelated to circularity, is that the proof of Lemma 7.2 displays the r-th root identity ((1 + x t^{n/p^s})^{r'})^r = 1 + a t^n, which an explicit counterexample (e.g., p=5, r=10, n=5, a=1) suggests is false as written; this is a mathematical correctness risk in the positive-characteristic pivot Theorem 7.1, not a circularity, since even if corrected it would not make the target theorems inputs of the derivation.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests on no fitted parameters or postulated entities. It relies on the philosophers' prior motivic integration theory for Artin stacks, on structure theorems for finite linearly reductive group schemes, on the cyclotomic-inertia description, and on fixed-locus stratification results from a book draft. None of these inputs is circular with the McKay theorem itself, but several are substantial unverified-upstream imports.

assumptions (6)

domain assumption The motivic integration framework for Artin stacks of [SU22, SU26, SU24a, SU24b, SU23a, SU23b] is valid, and the cited proofs work without characteristic 0 or algebraic closedness of k.
Used throughout sections 3 to 10 as the base theory. The paper supplies replacements for the places where characteristic 0 was used, but the unmodified portions are imported.
standard math Every finite linearly reductive group scheme over an algebraically closed field has the form G = Delta ⋊ H, with Delta diagonalizable connected and H a finite etale tame constant group scheme.
Used in Propositions 11.3, 11.4, Lemma 12.10, and Proposition 13.1, citing [AOV08, Lemma 2.11 and Proposition 2.13].
standard math The Hom scheme Homgrp,inj_k(mu_r, G) exists as a finite scheme and describes cyclotomic inertia of quotient stacks.
Used in Notation 1.1 and Proposition 12.3, citing [Sal25, Proposition 1.1 and Proposition 1.2].
domain assumption [Alp26, Theorem 7.8.14(2)-(4)] provides smooth G_m-fixed loci, a filterable stratification, and local affine fibrations for actions on proper schemes.
Used in the proof of Theorem 14.1 to obtain purity of etale cohomology. The source is a book draft, and the proof is not reproduced in this preprint.
domain assumption Functorial resolution of singularities for finite linearly reductive quotient singularities exists in positive characteristic.
Used in Proposition 14.5 to construct an equivariant auxiliary resolution Y' -> X via [BR19, Theorem E].
domain assumption The arc bijection [BV24] for tame stacks holds in positive characteristic.
Used in the proof of Corollary 9.8 to identify arcs of a tame stack with arcs of its coarse space away from the exceptional locus.

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Cite this review

Pith. "Pith review of McKay correspondence for linearly reductive finite group schemes in positive characteristic." pith.science (2026). https://pith.science/paper/TXEGCSTI

@misc{pith2026260805020,
  author       = {Pith},
  title        = {Pith review of: McKay correspondence for linearly reductive finite group schemes in positive characteristic},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TXEGCSTI}},
  note         = {Machine review of arXiv:2608.05020}
}

abstract

We obtain a motivic and a cohomological McKay correspondence for finite linearly reductive group schemes in arbitrary characteristic. In particular, we prove that if $V$ is a finite dimensional vector space and $G$ is a finite linearly reductive subgroup scheme of $\mathrm{SL}(V)$, then the Euler number of any crepant resolution of $V/G$ is equal to the number of irreducible algebraic representations of $G$. We obtain these McKay correspondences as a consequence of a motivic change of variables formula applied to $[V/G] \to V/G$. If $G$ is non-reduced, as can happen in positive characteristic, the stack quotient $[V/G]$ is not Deligne-Mumford. Therefore in order to prove this change of variables formula and the resulting McKay correspondences, we generalize the authors' theory of motivic integration for Artin stacks to arbitrary characteristic, which may be of independent interest.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

14 extracted references · 12 canonical work pages

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [13]

Stringy Hodge numbers via crepant resolutions by Artin stacks

[SU24b] Matthew Satriano and Jeremy Usatine. Stringy Hodge numbers via crepant resolutions by Artin stacks.arXiv e-prints, page arXiv:2410.23951, October

work page Pith review arXiv

[2] [1974]

A cohomological interpretation for stringy Hodge numbers.arXiv e-prints, page arXiv:2602.19344, February

[HSU26] Jiahui Huang, Matthew Satriano, and Jeremy Usatine. A cohomological interpretation for stringy Hodge numbers.arXiv e-prints, page arXiv:2602.19344, February

work page arXiv

[3] [1998]

With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japan- ese original. [Lie24] Christian Liedtke. A McKay correspondence in positive characteristic.Forum Math. Sigma, 12:Paper No. e116, 36,

work page 1998

[4] [2002]

A geometric invariant of a finite group

[Eke09a] Torsten Ekedahl. A geometric invariant of a finite group.arXiv e-prints, page arXiv:0903.3148, March

work page Pith review arXiv

[5] [2006]

Open problems in the wild McKay correspondence and related fields

[Yas23] Takehiko Yasuda. Open problems in the wild McKay correspondence and related fields. InMcKay correspondence, mutation and related topics, volume 88 ofAdv. Stud. Pure Math., pages 279–303. Math. Soc. Japan, Tokyo, [2023]©2023. [Yas24] Takehiko Yasuda. Motivic integration over wild Deligne-Mumford stacks.Algebr. Geom., 11(2):178–255,

work page 2023

[6] [2007]

Functorial destackification and weak factorization of orbifolds.arXiv e-prints, page arXiv:1905.00872, May

[BR19] Daniel Bergh and David Rydh. Functorial destackification and weak factorization of orbifolds.arXiv e-prints, page arXiv:1905.00872, May

arXiv 1905

[7] [2008]

[Bat98] Victor V. Batyrev. Stringy Hodge numbers of varieties with Gorenstein canonical singu- larities. InIntegrable systems and algebraic geometry (Kobe/Kyoto, 1997), pages 1–32. World Sci. Publ., River Edge, NJ,

work page 1997

[8] [2009]

The Grothendieck group of algebraic stacks

[Eke09b] Torsten Ekedahl. The Grothendieck group of algebraic stacks.arXiv e-prints, page arXiv:0903.3143, March

work page Pith review arXiv