What question did this study set out to answer?

The research aims to explore the conditions under which ideals in tensor triangular geometry can be finitely generated.

April 19, 2026Bulletin of the London Mathematical Society0 citationsOpen Access

On the finite generation of ideals in tensor triangular geometry

Key Points

The research aims to explore the conditions under which ideals in tensor triangular geometry can be finitely generated.
Examined properties of essentially small tensor triangulated categories.
Analyzed the weakly Noetherian spectrum of these categories.
Established conditions linking finite generation of prime ideals to the finiteness of the prime ideals set.
Every prime ideal can be generated by finitely many objects under specific conditions.
Proved that a finite set of prime ideals leads to finite generation in the examined categories.

Abstract

Abstract Inspired by Cohen's characterization of Noetherian commutative rings, we study the finite generation of ideals in tensor triangular geometry. In particular, for an essentially small tensor triangulated category with weakly Noetherian spectrum, we show that every prime ideal in can be generated by finitely many objects if and only if the set of prime ideals of is finite.

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