Nonnoetherian geometry

TL;DR

This paper develops a new geometric framework for nonnoetherian algebras, introducing concepts like depiction and geometric codimension, and applies it to quiver algebras to characterize noetherianity and analyze their structure.

Contribution

It introduces a novel geometric theory for nonnoetherian algebras, including new notions of normalization and height, and applies these to characterize properties of quiver algebras.

Findings

Vertex corner rings are isomorphic iff the algebra is noetherian.

Center of the algebra is depicted by a cycle-generated algebra.

Example where projective dimension equals geometric codimension.

Abstract

We introduce a theory of geometry for nonnoetherian commutative algebras with finite Krull dimension. In particular, we establish new notions of normalization and height: depiction (a special noetherian overring) and geometric codimension. The resulting geometries are algebraic varieties with positive dimensional points, and are thus inherently nonlocal. These notions also give rise to new equivalent characterizations of noetherianity that are primarily geometric. We then consider an application to quiver algebras whose simple modules of maximal dimension are one dimensional at each vertex. We show that the vertex corner rings of $A$ are all isomorphic if and only if $A$ is noetherian, if and only if the center $Z$ of $A$ is noetherian, if and only if $A$ is a finitely generated $Z$-module. Furthermore, we show that $Z$ is depicted by a commutative algebra generated by the cycles in its quiver. We conclude with an example of a quiver algebra where projective dimension and geometric codimension, rather than height, coincide.