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The crisp topology, a refinement of the fpqc topology

Saskia Kern

Abstract

We introduce the crisp topology for schemes as a refinement of the fpqc topology. This Grothendieck topology uses the new notion of crisp morphisms, which generalise universal injectivity from ring homomorphisms to arbitrary morphisms of schemes. We study basic properties and demonstrate that this topology is well-behaved.

The crisp topology, a refinement of the fpqc topology

Abstract

We introduce the crisp topology for schemes as a refinement of the fpqc topology. This Grothendieck topology uses the new notion of crisp morphisms, which generalise universal injectivity from ring homomorphisms to arbitrary morphisms of schemes. We study basic properties and demonstrate that this topology is well-behaved.

Paper Structure

This paper contains 12 sections, 27 theorems, 68 equations.

Table of Contents

  1. On crisp morphisms of rings and modules
  2. Permanence properties
  3. Equivalent characterisations of crisp ring homomorphisms
  4. Some examples
  5. Crisp descent for modules and algebras
  6. On crisp morphisms of schemes
  7. Permanence properties
  8. Crisp morphisms in context
  9. Excursion: subtrusive and submersive morphisms
  10. The crisp topology
  11. Crisp-local properties
  12. The crisp topology is subcanonical

Key Result

Theorem V

Crisp morphisms of schemes satisfy the following properties:

Theorems & Definitions (73)

  • Definition I: Definition \ref{['crisp-module-map']}
  • Definition II: Definition \ref{['crisp-ring-hom']}
  • Definition IV: Definition \ref{['crisp-morphism']}
  • Theorem V: Theorem \ref{['permanence-crisp']}
  • Theorem VI: Propositions \ref{['faithfully-flat-crisp']}, \ref{['pureqcqs-crisp']} and \ref{['crisp-subtrusive']}
  • Definition VII: Definition \ref{['crisp-top']}
  • Theorem VII: Theorem \ref{['crispt-descent']} and Corollary \ref{['crisp-loc-props']}
  • Definition 1.1
  • Definition 1.2
  • Lemma 1.3
  • ...and 63 more