The crisp topology, a refinement of the fpqc topology

TL;DR

The paper introduces the crisp topology for schemes, refining the fpqc topology by using crisp morphisms that generalize universal injectivity, and explores its properties and behavior.

Contribution

It presents a new Grothendieck topology called the crisp topology, extending the concept of universal injectivity to arbitrary scheme morphisms.

Findings

Crisp topology is a well-behaved refinement of fpqc topology.

Crisp morphisms generalize universal injectivity.

Basic properties of the crisp topology are established.

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.