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Noncommutative resolution, F-blowups and D-modules

Yukinobu Toda, Takehiko Yasuda

Abstract

We explain the isomorphism between the $G$-Hilbert scheme and the F-blowup from the noncommutative viewpoint after Van den Bergh. In doing this, we immediately and naturally arrive at the notion of $D$-modules. We also find, as a byproduct, a canonical way to construct a noncommutative resolution at least for a few classes of singularities in positive characteristic.

Noncommutative resolution, F-blowups and D-modules

Abstract

We explain the isomorphism between the -Hilbert scheme and the F-blowup from the noncommutative viewpoint after Van den Bergh. In doing this, we immediately and naturally arrive at the notion of -modules. We also find, as a byproduct, a canonical way to construct a noncommutative resolution at least for a few classes of singularities in positive characteristic.

Paper Structure

This paper contains 12 sections, 17 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Tame quotient singularities
  3. Functors between module categories
  4. Modules of covariants and Frobenius maps
  5. Moduli spaces of stable objects
  6. Stability
  7. Moduli space
  8. The maps from the $G$-Hilbert scheme to F-blowups
  9. Fourier-Mukai transform
  10. F-blowups of F-pure singularities
  11. 1-dimensional analytically irreducible singularities
  12. The simple singularity of type $A_{1}$

Key Result

Theorem 1.3

Let $M=\mathbb A_{k}^{d}=\mathop{\mathrm{Spec}}\nolimits S$, $G \subset GL _{d}(k)$ a small finite subgroup and $X:= M/G = \mathop{\mathrm{Spec}}\nolimits R$. Then for sufficiently large $e$, we have an equivalence of abelian categories Hence $D_{R,e}$ has global dimension $d$. Moreover $\mathrm{Hilb}^{G}(M)$ and $\mathop{\mathrm{FB}}\nolimits_{e} (X)$ are the moduli spaces corresponds to each ot

Theorems & Definitions (36)

  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.6
  • Proposition 2.1
  • proof
  • Corollary 2.2
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 26 more