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Frobenius morphisms of noncommutative blowups

Takehiko Yasuda

Abstract

We define the Frobenius morphism of certain class of noncommutative blowups in positive characteristic. Thanks to a nice property of the class, the defined morphism is flat. Therefore we say that the noncommutative blowups in this class are Kunz regular. One of such blowups is the one associated to a regular Galois alteration. As a consequence of de Jong's theorem, we see that for every variety over an algebraically closed field of positive characteristic, there exists a noncommutative blowup which is Kunz regular. We also see that a variety with F-pure and FFRT (finite F-representation type) singularities has a Kunz regular noncommutative blowup which is associated to an iteration of the Frobenius morphism of the variety.

Frobenius morphisms of noncommutative blowups

Abstract

We define the Frobenius morphism of certain class of noncommutative blowups in positive characteristic. Thanks to a nice property of the class, the defined morphism is flat. Therefore we say that the noncommutative blowups in this class are Kunz regular. One of such blowups is the one associated to a regular Galois alteration. As a consequence of de Jong's theorem, we see that for every variety over an algebraically closed field of positive characteristic, there exists a noncommutative blowup which is Kunz regular. We also see that a variety with F-pure and FFRT (finite F-representation type) singularities has a Kunz regular noncommutative blowup which is associated to an iteration of the Frobenius morphism of the variety.

Paper Structure

This paper contains 13 sections, 15 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Convention
  3. Acknowledgment
  4. Noncommutative schemes
  5. Pseudo-schemes
  6. Noncommutative schemes
  7. Alterations and noncommutative blowups
  8. Frobenius morphisms
  9. Equivalent modules
  10. F-steady modules and Frobenius morphisms
  11. Compatibilities of Frobenius morphisms
  12. D-blowups
  13. Comparing Frobenius morphisms of commutative and noncommutative blowups

Key Result

Theorem 2.2

If $X^\mathrm{ps} \cong Y^\mathrm{ps}$, then $X\cong Y$.

Theorems & Definitions (45)

  • Definition 2.1
  • Theorem 2.2: Reconstruction of schemes
  • Proposition 2.3: Reconstruction of morphisms
  • proof
  • Definition 2.4
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Definition 3.3
  • Definition 3.4
  • ...and 35 more