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Geometric purity and the frame of smashing ideals

Juan Omar Gómez, Maurcio Medina-Barcenas, Greg Stevenson, Bernardo Villarreal, Ángel Zaldivar-Corichi

TL;DR

The paper introduces geometric purity in rigidly-compactly generated tt-categories by enforcing purity after tt-stalk localization at all points of the Balmer spectrum, yielding a strictly stronger notion than ordinary purity and enabling the construction of geometrically pure-injective objects via pushforwards from tt-stalks. It proves a reduction principle: every indecomposable g-pure-injective is the pushforward of a pure-injective from some stalk ${\mathcal T}_{\mathcal P}$, and establishes a localization-based framework to test and analyze purity locally, including a reduction to closed points and to qc open covers. The authors compute and illustrate these ideas in the derived category of the projective line $\mathbb P^1$, showing g-purity can differ from classical purity, and describe the g-pure-injective objects in this setting. They develop a local-to-global strategy for the spatiality of the frame of smashing ideals via the geometric Ziegler spectrum, proving that spatiality can be inferred from stalk data and holds in several key examples (e.g., qcqs schemes, hereditary/von Neumann regular rings, Dedekind domains). As an application, this approach rules out Balchin–Stevenson counterexamples to existing methods and provides a partial global criterion for spatiality through the geometry of the geometric Ziegler spectrum.

Abstract

We introduce the notion of geometric purity in rigidly-compactly generated tt-categories by considering exact triangles that are pure at each tt-stalk. We develop a systematic study of this concept, including examples and applications. In particular, we show that geometric purity is, in general, strictly stronger than ordinary purity, and that it naturally leads to the notion of geometrically pure-injective objects. We prove that such objects arise as pushforwards of pure-injective objects from suitable tt-stalks. Moreover, we give a detailed analysis of indecomposable geometrically pure-injective objects in the derived category of the projective line. Under mild additional assumptions, we identify the geometric part of the Ziegler spectrum as a closed subset. As an application, we demonstrate that this new notion of purity can be used to tackle the problem of spatiality of the frame of smashing ideals via the geometric Ziegler spectrum. In particular, we show that our approach rules out the counterexamples of Balchin and Stevenson to existing methods.

Geometric purity and the frame of smashing ideals

TL;DR

The paper introduces geometric purity in rigidly-compactly generated tt-categories by enforcing purity after tt-stalk localization at all points of the Balmer spectrum, yielding a strictly stronger notion than ordinary purity and enabling the construction of geometrically pure-injective objects via pushforwards from tt-stalks. It proves a reduction principle: every indecomposable g-pure-injective is the pushforward of a pure-injective from some stalk , and establishes a localization-based framework to test and analyze purity locally, including a reduction to closed points and to qc open covers. The authors compute and illustrate these ideas in the derived category of the projective line , showing g-purity can differ from classical purity, and describe the g-pure-injective objects in this setting. They develop a local-to-global strategy for the spatiality of the frame of smashing ideals via the geometric Ziegler spectrum, proving that spatiality can be inferred from stalk data and holds in several key examples (e.g., qcqs schemes, hereditary/von Neumann regular rings, Dedekind domains). As an application, this approach rules out Balchin–Stevenson counterexamples to existing methods and provides a partial global criterion for spatiality through the geometry of the geometric Ziegler spectrum.

Abstract

We introduce the notion of geometric purity in rigidly-compactly generated tt-categories by considering exact triangles that are pure at each tt-stalk. We develop a systematic study of this concept, including examples and applications. In particular, we show that geometric purity is, in general, strictly stronger than ordinary purity, and that it naturally leads to the notion of geometrically pure-injective objects. We prove that such objects arise as pushforwards of pure-injective objects from suitable tt-stalks. Moreover, we give a detailed analysis of indecomposable geometrically pure-injective objects in the derived category of the projective line. Under mild additional assumptions, we identify the geometric part of the Ziegler spectrum as a closed subset. As an application, we demonstrate that this new notion of purity can be used to tackle the problem of spatiality of the frame of smashing ideals via the geometric Ziegler spectrum. In particular, we show that our approach rules out the counterexamples of Balchin and Stevenson to existing methods.
Paper Structure (17 sections, 40 theorems, 69 equations)

This paper contains 17 sections, 40 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Balmer spectrum
  4. Big categories and geometric functors
  5. Smashing localizations
  6. Purity in triangulated categories
  7. Geometric purity in tt-categories
  8. An alternative definition of g-purity
  9. Examples of g-pure-injective objects
  10. Purity at closed points and open covers
  11. Closed points
  12. Covers by quasi-compact opens
  13. Reduction to stalks
  14. Geometric purity in the derived category of the projective line
  15. Local to global for spatiality of the frame of smashing ideals
  16. ...and 2 more sections

Key Result

Theorem 1.3

Let ${\mathcal{T}}$ be a rigidly-compactly generated tt-category, and let $x$ be an indecomposable g-pure injective object in ${\mathcal{T}}$. Then there exists a prime ${\mathcal{P}} \in \operatorname{Spc}\nolimits({\mathcal{T}}^\omega)$ and a pure-injective object $y \in {\mathcal{T}}_{\mathcal{P} where $\iota_{{\mathcal{P}},\ast}$ denotes the right adjoint of $\iota_{\mathcal{P}}^\ast$.

Theorems & Definitions (112)

  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 1.6
  • Theorem 1.7
  • Remark 2.4
  • Remark 2.5
  • Example 2.7
  • Remark 2.9
  • Definition 2.11
  • ...and 102 more