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Cohen's theorem in tensor triangular geometry

Tobias Barthel

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.

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 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.

Paper Structure

This paper contains 6 sections, 8 theorems, 7 equations.

Table of Contents

  1. Introduction
  2. Topological preliminaries
  3. The main result
  4. Application to tt-stratification
  5. Examples
  6. Acknowledgements

Key Result

Theorem A

Suppose $\mathscr{K}$ is an essentially small tt-category with weakly Noetherian spectrum. Then the following are equivalent:

Theorems & Definitions (21)

  • Theorem A
  • Theorem B
  • Definition 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Definition 4
  • Lemma 5
  • proof
  • ...and 11 more