Cohen's theorem in tensor triangular geometry

TL;DR

This paper extends Cohen's classical theorem to tensor triangular geometry, showing that in certain categories, prime ideals are finitely generated if and only if the spectrum is finite.

Contribution

It establishes a tensor triangular analogue of Cohen's theorem, linking finite generation of prime ideals to the finiteness of the spectrum.

Findings

Prime ideals in the category can be generated by finitely many objects

Finite spectrum implies finitely generated prime ideals

Analogues of Cohen's theorem in tensor triangular geometry

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 $\mathscr{K}$ with weakly Noetherian spectrum, we show that every prime ideal in $\mathscr{K}$ can be generated by finitely many objects if and only if the set of prime ideals of $\mathscr{K}$ is finite.