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The frame of smashing tensor-ideals

Paul Balmer, Henning Krause, Greg Stevenson

Abstract

Given a tensor-triangulated category $T$, we prove that every flat tensor-idempotent in the module category over $T^c$ (the compacts) comes from a unique smashing ideal in $T$. We deduce that the lattice of smashing ideals forms a frame.

The frame of smashing tensor-ideals

Abstract

Given a tensor-triangulated category , we prove that every flat tensor-idempotent in the module category over (the compacts) comes from a unique smashing ideal in . We deduce that the lattice of smashing ideals forms a frame.

Paper Structure

This paper contains 6 sections, 19 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Quick review of right-idempotents
  3. From right-idempotents to Serre subcategories
  4. From Serre subcategories to smashing subcategories
  5. Distributivity of the lattice of smashing subcategories
  6. The module category $\mathop{\mathrm{Mod}}\nolimits\text{-}\mathscr{T}^{c}$ and its tensor structure

Key Result

Theorem 1.1

Let $\mathscr{T}$ be a tensor-triangulated category, assumed to be rigidly-compactly generated. Then the lattice of smashing $\otimes$-ideals of $\mathscr{T}$ is a frame, i.e. it is complete (all meets and joins exist) and the meet distributes over arbitrary joins.

Theorems & Definitions (62)

  • Theorem 1.1
  • Theorem 1.3
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Corollary 2.5
  • Definition 2.6
  • Remark 2.7
  • ...and 52 more