Cohen's theorem in tensor triangular geometry

2025/05/21 by Tobias Barthel, Barthel, Tobias · 1 citation
Computer Science · Mathematics · #Algebraic Topology (math.AT) #Category Theory (math.CT) #Computational Geometry and Mesh Generation #FOS: Mathematics #Mathematics and Applications #Matrix Theory and Algorithms #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2505.15786

openalex publication_date 2025/05/21 · openalex created_date 2025/10/19 · openalex updated_date 2026/07/28

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

Citations

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Cited by

• Re-framing the classification of ideals in noncommutative tensor-triangular geometry

Related

• The spectrum of prime ideals in tensor triangulated categories
• Commutative rings with every non-maximal ideal finitely generated
• On functoriality and the tensor product property in noncommutative tensor-triangular geometry
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• Support theories for non-Noetherian tensor triangulated categories