Moduli of formal torsors

Tonini, Fabio; Yasuda, Takehiko

doi:10.1090/jag/771

Cited by 4 publications

(13 citation statements)

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“…We say that a representable morphism f : Y → X of stacks has property P if for every morphism U → X from an algebraic space, the projection Y × X U → U has property P. Definition 2.2. An ind-DM stack (respectively, ind-algebraic space, ind-scheme, ind-affine scheme) is a fibered category over Aff isomorphic to the limit lim − → X n of an inductive system X 0 → X 1 → • • • indexed by N of DM stacks (respectively, algebraic spaces, schemes, affine schemes) in the sense of [TY20,Appendix A]. An ind-DM stack (respectively, ind-algebraic space, ind-scheme, ind-affine scheme) is called a formal DM stack (respectively, formal algebraic space, formal scheme, formal affine scheme) if we can further assume that the transition maps X n → X n+1 are (necessarily representable) thickenings.…”

Section: Ind-dm Stacks and Formal Dm Stacksmentioning

confidence: 99%

Motivic integration over wild Deligne–Mumford stacks

Yasuda

2024

Alg. Geom.

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We develop the motivic integration theory over formal Deligne-Mumford stacks over a power series ring of arbitrary characteristic. This is a generalization of the corresponding theory for tame and smooth Deligne-Mumford stacks constructed in earlier papers of the author. As an application, we obtain the wild motivic McKay correspondence for linear actions of arbitrary finite groups, which has been known only for cyclic groups of prime order. In particular, this implies the motivic version of Bhargava's mass formula as a special case. In fact, we prove a more general result, the invariance of stringy motives of (stacky) log pairs under crepant morphisms.

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Section: Ind-dm Stacks and Formal Dm Stacksmentioning

confidence: 99%

Motivic integration over wild Deligne–Mumford stacks

Yasuda

2024

Alg. Geom.

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“…As a consequence of [10, Lemma 3.2] we have the following. The colimit Z ∞ −→ colim n ( Zn ) = Z∞ of the coarse moduli maps Z n −→ Zn is a coarse indalgebraic space map in the sense of [10,Definition 3.1]. Moreover if the transition maps Z n −→ Z n+1 are finite and universally injective then so are the maps Zn −→ Zn+1 : they are universally injective, thus quasi-finite, by [10, Lemma 3.2], they are proper because so are the coarse moduli maps Z n −→ Zn .…”

Section: Just Before Proposition A2])mentioning

confidence: 99%

“…Remark 8.3. -Notation above slightly differs to the notation used in [10], where ∆ G denotes the analogous fiber category. Corollary 8.10.…”

Section: The P-moduli Space Of Formal Torsorsmentioning

confidence: 99%

“…In the preceding paper [10], the authors constructed the moduli stack of G-torsors over Spec k((t)), where k is a field of characteristic p > 0 and G is a group of the form H C for a p-group H and a tame cyclic group C, which generalizes and refines Harbater's work for p-groups [7]. The motivation of the authors came from the wild McKay correspondence.…”

Section: Introductionmentioning

confidence: 99%

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Moduli of formal torsors II

Tonini¹,

Yasuda²

2023

Annales De l'Institut Fourier

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Applying the authors' preceding work, we construct a version of the moduli space of G-torsors over the formal punctured disk for a finite group G. To do so, we introduce two Grothendieck topologies, the sur (surjective) and luin (locally universally injective) topologies, and define P-schemes using them as variants of schemes. Our moduli space is defined as a P-scheme approximating the relevant moduli functor. We then prove that Fröhlich's module resolvent gives a locally constructible function on this moduli space, which implies that motivic integrals appearing in the wild McKay correspondence are well-defined.Résumé. -En appliquant le travail précédent des auteurs, nous construisons une version d'un espace de modules de G-torseurs sur le disque perforé formel pour un groupe fini G. Dans ce but, nous introduisons deux topologies de Grothendieck, les topologies sur (surjective) et luin (localement universellement injective), et définissons des P-schémas en les utilisant comme variantes de schémas. Notre espace de modules est défini comme un P-schéma approximant le foncteur de modules pertinent. On montre alors que la résolvante de modules de Fröhlich donne une fonction localement constructible sur cet espace de modules, ce qui implique que les intégraux motiviques apparaissant dans la correspondance sauvage de McKay sont bien définis.

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“…Since G-torsors have nontrivial automorphisms, as is usual in the moduli problem, the fine moduli space should be a stack if it exists. When group G is a semidirect product H ⋊ C of a p-group H and a tame cyclic group C, it was proved in [TY20b] that the moduli stack ∆ G is an inductive limit of Deligne-Mumford stacks of finite type. For a general finite group, we can construct the "moduli space" of ∆ G as a rough geometric structure called P-scheme [TY19a].…”

Section: The V-function/fröhlich's Module Resolventmentioning

confidence: 99%

Open problems in the wild McKay correspondence and related fields

Yasuda¹

2021

Preprint

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The wild McKay correspondence is a form of McKay correspondence in terms of stringy invariants that is generalized to arbitrary characteristics. It gives rise to an interesting connection between the geometry of wild quotient varieties and arithmetic on extensions of local fields. The principal purpose of this article is to collect open problems on the wild McKay correspondence, as well as those in related fields that the author believes are interesting or important. It also serves as a survey on the present state of these fields. Contents 1. Introduction Convention 2. Brief guide to the wild McKay correspondence and stringy motives 2.1. Wild actions and wild quotients 2.2. Stringy motives 2.3. The wild McKay correspondence 3. Log-terminal singularities 4. Crepant resolutions 5. Rationality 6. Duality 7. The v-function/Fröhlich's module resolvent 8. The moduli space ∆ G 9. Non-linear actions 10. Generalization in various directions 11. Relation to other theories on wild ramification 12. The derived wild McKay correspondence 13. The McKay correspondence over global fields 14. Stringy motives and quasi-étale Galois coverings References

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Moduli of formal torsors

Cited by 4 publications

References 17 publications

Motivic integration over wild Deligne–Mumford stacks

Motivic integration over wild Deligne–Mumford stacks

Moduli of formal torsors II

Open problems in the wild McKay correspondence and related fields

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