On the perfection of schemes

TL;DR

This paper provides an in-depth exposition on the concept of perfection of schemes, addressing a gap in literature and laying groundwork for future applications and research in algebraic geometry.

Contribution

It offers a detailed exposition on the perfection of schemes, a topic not thoroughly covered in existing literature, and prepares for future applications.

Findings

Clarifies the concept of perfect schemes

Establishes foundational results for further research

Prepares groundwork for subsequent applications

Abstract

This is a chiefly expository paper on the subject of the title which, in our view, has not received a detailed treatment in the literature which is commensurate with its importance. We expect the results presented here to be useful in a number of contexts. For example, several of them will be applied in a forthcoming paper by the authors.