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Homological support of big objects in tensor-triangulated categories

Paul Balmer

Abstract

Using homological residue fields, we define supports for big objects in tensor-triangulated categories and prove a tensor-product formula.

Homological support of big objects in tensor-triangulated categories

Abstract

Using homological residue fields, we define supports for big objects in tensor-triangulated categories and prove a tensor-product formula.

Paper Structure

This paper contains 6 sections, 23 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Yoneda and modules
  3. Maximal localizing tensor-ideals
  4. Support for big objects
  5. The image of $\textrm{Spc}(F)$
  6. Comparing homological and triangular spectra

Key Result

Theorem 1.2

One can assign to every object $X$ of $\mathscr{T}$ a subset $\mathop{\mathrm{Supp}}\nolimits(X)$ of the homological spectrum $\mathop{\mathrm{Spc}}\nolimits^{\mathop{\mathrm{h}}\nolimits}(\mathscr{T}^c)$ of Balmer20a, with the following properties:

Theorems & Definitions (59)

  • Theorem 1.2: \ref{['sec:support']}
  • Definition 1.7
  • Theorem 1.8
  • Theorem 1.9: \ref{['Thm:Spc-image']}
  • Proposition 2.9
  • proof
  • Remark 2.10
  • Proposition 2.14
  • proof
  • Lemma 2.15
  • ...and 49 more