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Moduli of formal torsors II

Fabio Tonini, Takehiko Yasuda

Abstract

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 the wild McKay correspondence are well-defined.

Moduli of formal torsors II

Abstract

Applying the authors' preceding work, we construct a version of the moduli space of -torsors over the formal punctured disk for a finite group . 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 the wild McKay correspondence are well-defined.

Paper Structure

This paper contains 13 sections, 54 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Notation, terminology and convention
  3. Luin and sur topologies
  4. P-schemes and moduli spaces
  5. P-morphisms and associated functor
  6. P-moduli spaces
  7. P-schemes locally of finite type over a locally Noetherian scheme.
  8. Quotients by P-actions of finite groups
  9. Motivic integration on schemes locally of finite type
  10. Some results on power series rings
  11. Uniformization
  12. The P-moduli space of formal torsors
  13. Local constructibility of weighting functions

Key Result

Theorem 1.1

Let $G$ be a finite group and let $k$ be a field. Consider the functor from the category of affine $k$-schemes to the category of sets which sends $\mathrm{Spec}\, R$ to the set of isomorphism classes of $G$-torsors over $\mathrm{Spec}\, R((t))$. This functor has a strong P-moduli space, which is th

Theorems & Definitions (146)

  • Theorem 1.1: Theorem \ref{['thm:strong P moduli of DeltaG']}
  • Definition 3.1
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Remark 3.4
  • Definition 3.5
  • Definition 3.6: stacks-project
  • Theorem 3.7
  • Lemma 3.8
  • ...and 136 more