Cosupports and minimal pure-injective resolutions of affine rings

TL;DR

This paper demonstrates that affine rings over a field have full cosupport and provides a detailed description of their minimal pure-injective resolutions under certain conditions, partially addressing a conjecture by Gruson.

Contribution

It establishes that affine rings over a field have full cosupport and characterizes their minimal pure-injective resolutions in specific cases.

Findings

Affine rings over a field have full cosupport.

Complete description of minimal pure-injective resolutions for certain affine rings.

Partial resolution of Gruson's conjecture.

Abstract

We prove that any affine ring $R$ over a field $k$ has full cosupport, i.e., the cosupport of $R$ is equal to ${\rm Spec}\, R$. Using this fact, we give a complete description of all terms in a minimal pure-injective resolution of $R$, provided that $|k|=\aleph_1$ and ${\rm dim}\, R\geq 2$, or $|k|\geq \aleph_1$ and ${\rm dim}\, R=2$. As a corollary, we obtain a partial answer to a conjecture by Gruson.