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A note on quasi-perfect morphisms

Timothy De Deyn, Pat Lank, Kabeer Manali-Rahul

Abstract

This note records two results concerning quasi-perfect morphisms between Noetherian algebraic spaces. First, we give a new characterization of regular Noetherian algebraic spaces as those for which blowups at closed points are quasi-perfect. Secondly, we study the local behavior of quasi-perfect proper morphisms. In particular, we show that quasi-perfectness of a proper morphism can be detected at their étale local rings, completions, and (strict) Henselizations. As a corollary, the locus of points where a proper morphism is quasi-perfect is Zariski open.

A note on quasi-perfect morphisms

Abstract

This note records two results concerning quasi-perfect morphisms between Noetherian algebraic spaces. First, we give a new characterization of regular Noetherian algebraic spaces as those for which blowups at closed points are quasi-perfect. Secondly, we study the local behavior of quasi-perfect proper morphisms. In particular, we show that quasi-perfectness of a proper morphism can be detected at their étale local rings, completions, and (strict) Henselizations. As a corollary, the locus of points where a proper morphism is quasi-perfect is Zariski open.

Paper Structure

This paper contains 12 sections, 17 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Generation
  4. Algebraic Spaces
  5. Modules and derived categories
  6. Perfect complexes
  7. Local rings
  8. Residue fields
  9. Regularity
  10. Morphisms
  11. Quasi-perfect blowups detect regularity
  12. On proper quasi-perfect morphisms

Key Result

Proposition 1

A Noetherian algebraic space $X$ is regular if and only if the blowup of $X$ along any closed point is a quasi-perfect morphism.

Theorems & Definitions (35)

  • Proposition : \ref{['prop:regularity_via_blowups']}
  • Theorem : \ref{['thm:quasi-perfectness_locality']}
  • Corollary : \ref{['cor:q-perfect_locus']}
  • Lemma 2.1
  • Definition 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • ...and 25 more