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Conservative geometric functors via purity

Natàlia Castellana, Juan Omar Gómez

TL;DR

This work develops a purity-based criterion for joint conservativity of families of geometric tensor-triangulated functors. By introducing pure descendability, the authors show that joint conservativity follows once the pure-closure $\mathbf{pure}^\Delta_\Pi\big(\bigcup_i (f_i)_*(\mathcal S_i)\big)$ contains the monoidal unit and is a tensor-ideal. Two main applications are developed: (i) pure monomorphisms for sequential limits of ring spectra, providing criteria for the unit to lie in the pure-closure; and (ii) pure descendability for cochains on classifying spaces of locally finite Artinian groups, yielding Chouinard-type theorems and conservative induction from elementary abelian subgroups. The framework advances descent theory in tensor-triangulated geometry and yields practical criteria for joint conservativity in complex settings, including $p$-local groups and discrete $p$-toral groups.

Abstract

We establish a criterion for determining when a family of geometric functors is jointly conservative through the lens of purity in compactly generated triangulated categories. We introduce the notion of pure descendability and we apply it to two particular situations involving sequential limits of ring spectra.

Conservative geometric functors via purity

TL;DR

This work develops a purity-based criterion for joint conservativity of families of geometric tensor-triangulated functors. By introducing pure descendability, the authors show that joint conservativity follows once the pure-closure contains the monoidal unit and is a tensor-ideal. Two main applications are developed: (i) pure monomorphisms for sequential limits of ring spectra, providing criteria for the unit to lie in the pure-closure; and (ii) pure descendability for cochains on classifying spaces of locally finite Artinian groups, yielding Chouinard-type theorems and conservative induction from elementary abelian subgroups. The framework advances descent theory in tensor-triangulated geometry and yields practical criteria for joint conservativity in complex settings, including -local groups and discrete -toral groups.

Abstract

We establish a criterion for determining when a family of geometric functors is jointly conservative through the lens of purity in compactly generated triangulated categories. We introduce the notion of pure descendability and we apply it to two particular situations involving sequential limits of ring spectra.
Paper Structure (6 sections, 18 theorems, 52 equations)

This paper contains 6 sections, 18 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Purity in triangulated categories
  4. Conservative geometric functors
  5. Pure monomorphisms for sequential limits of ring spectra
  6. Pure descendability for cochains on classifying spaces

Key Result

Theorem 1.3

If $(f_i^\ast\colon {\mathcal{T}}\to {\mathcal{S}}_i)_{i\in \mathcal{I}}$ is a pure descendable family of geometric functors, then it is jointly conservative.

Theorems & Definitions (55)

  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.6
  • ...and 45 more