# Cosupport computations for finitely generated modules over commutative noetherian rings

## TL;DR

This paper characterizes the cosupport of finitely generated modules over noetherian rings, showing its relation to minimal pure-injective resolutions, and explores properties and limitations of cosupport in this context.

## Contribution

It establishes the precise relationship between cosupport and minimal pure-injective resolutions for noetherian rings, and addresses open questions about cosupport's topological properties.

## Key findings

- Cosupport equals primes in minimal pure-injective resolutions.
- Every countable noetherian ring has full cosupport.
- Cosupport of finitely generated modules may not be closed.

## Abstract

We show that the cosupport of a commutative noetherian ring is precisely the set of primes appearing in a minimal pure-injective resolution of the ring. As an application of this, we prove that every countable commutative noetherian ring has full cosupport. We also settle the comparison of cosupport and support of finitely generated modules over any commutative noetherian ring of finite Krull dimension. Finally, we give an example showing that the cosupport of a finitely generated module need not be a closed subset of $\operatorname{Spec}(R)$, providing a negative answer to a question of Sather-Wagstaff and Wicklein.