Universality of the category of schemes

TL;DR

This paper extends the concept of schemes to broader algebraic categories, characterizing coherent schemes via a universal property and introducing $ ext{ extbackslash scr ext{ extbackslash C}}$-schemes as a further generalization.

Contribution

It generalizes the construction of schemes to other algebraic categories and introduces $ ext{ extbackslash scr ext{ extbackslash C}}$-schemes, broadening the scope of scheme theory.

Findings

Coherent schemes characterized by a universal property.

Introduction of $ ext{ extbackslash scr ext{ extbackslash C}}$-schemes as a generalization.

Shared properties between $ ext{ extbackslash scr ext{ extbackslash C}}$-schemes and ordinary schemes.

Abstract

In this paper, we generalize the construction method of schemes to other algebraic categories, and show that the category of coherent schemes can be characterized by a universal property, if we fix the class of Grothendieck topology. Also, we introduce the notion of $\scr{C}$-schemes, which is a further generalization of coherent schemes and still shares common properties with ordinary schemes.