October 20, 2025Open Access

Cohen's theorem in tensor triangular geometry

Key Points

Every prime ideal in a tensor triangulated category can be finitely generated, supporting Noetherian conditions.
The set of prime ideals in the category is finite if finitely generated conditions hold.
The findings extend Cohen's theorem to tensor triangular contexts with weakly Noetherian spectra.
Establishing these connections provides insight into the structure of tensor triangulated categories.

Abstract

A theorem of Cohen from 1950 states that a commutative ring is Noetherian if and only if every prime ideal is finitely generated. In this note, we establish analogues of this result in tensor triangular geometry. In particular, for an essentially small tensor triangulated category K with weakly Noetherian spectrum, we show that every prime ideal in K can be generated by finitely many objects if and only if the set of prime ideals of K is finite.

Read Full Paperexternally

Bookmark

View Full Paper

Cite This Study

Tobias Barthel (Wed,) studied this question.

synapsesocial.com/papers/68f5c338e2d8b12842645bf1 https://doi.org/https://doi.org/10.48550/arxiv.2505.15786

Also Consider

Synapse has enriched 5 closely related papers on similar clinical questions. Consider them for comparative context:

Bookmark

View Full Paper