Pushouts of affine algebraic sets

TL;DR

This paper investigates the existence of pushouts in algebraic sets over infinite fields, linking it to the preservation of finitely generated algebras and Noetherian rings under intersections, and explores specific cases.

Contribution

It provides a complete characterization of pushouts in algebraic sets via the preservation of algebraic properties under intersections.

Findings

Pushouts exist if and only if finitely generated algebras are preserved under intersections.

The problem reduces to understanding the behavior of Noetherian rings under intersections.

Specific cases of pushouts and intersections of Noetherian rings are analyzed.

Abstract

We study the problem of existence of pushouts in the category of algebraic sets over an infinite field. This problem can be reduced to asking whether the property of being a finitely generated algebra over a field, or a Noetherian ring in general, is preserved under taking pullbacks. We show that this problem can be fully solved in terms of preservation of this property under taking intersections. In addition, we also discuss specific cases of pushouts of algebraic sets and intersections of Noetherian rings.