Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Sheafyness Property of Spectra of Banach Rings

Federico Bambozzi, Kobi Kremnizer

Abstract

Let R be a non-Archimedean Banach ring, satisfying some mild technical hypothesis that we will specify later on. We prove that to R one can associate a homotopical Huber spectrum Spa^h(R) via the introduction of the notion of derived rational localizations. The spectrum so obtained is endowed with a derived structural sheaf O_{Spa^h(R)} of simplicial Banach algebras for which the derived Tate-Cech complex is strictly exact. Under some hypothesis we can prove that there is a canonical morphism of sites Spa(R) -> |Spa^h(R)| that is an equivalence in some well-known examples of non-sheafy Banach rings. This permits to use the tools of derived geometry to understand the geometry of Spa(R) also when H^0(O_{Spa(R)}) is not a sheaf.

On the Sheafyness Property of Spectra of Banach Rings

Abstract

Let R be a non-Archimedean Banach ring, satisfying some mild technical hypothesis that we will specify later on. We prove that to R one can associate a homotopical Huber spectrum Spa^h(R) via the introduction of the notion of derived rational localizations. The spectrum so obtained is endowed with a derived structural sheaf O_{Spa^h(R)} of simplicial Banach algebras for which the derived Tate-Cech complex is strictly exact. Under some hypothesis we can prove that there is a canonical morphism of sites Spa(R) -> |Spa^h(R)| that is an equivalence in some well-known examples of non-sheafy Banach rings. This permits to use the tools of derived geometry to understand the geometry of Spa(R) also when H^0(O_{Spa(R)}) is not a sheaf.

Paper Structure

This paper contains 15 sections, 56 theorems, 201 equations.

Table of Contents

  1. Introduction
  2. Relative algebraic geometry over quasi-abelian categories
  3. The left-heart of a quasi-abelian category
  4. The homotopy Zariski topology
  5. Quasi-abelian categories for analytic geometry
  6. The category of Banach modules
  7. The contracting category of Banach modules
  8. The category of bornological modules
  9. The homotopical Huber spectrum of Banach and bornological rings
  10. Localizations of affinoid algebras over strongly Noetherian Tate rings
  11. Localizations of general Banach rings
  12. The homotopical Huber spectrum of a Banach ring
  13. Localizations of bornological rings
  14. Examples and applications
  15. Conclusions

Key Result

Proposition 2.4

Let $F: {\mathbf C} \to {\mathbf D}$ be a functor between quasi-abelian categories, then Dually for left exact and strictly left exact functors.

Theorems & Definitions (137)

  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • proof
  • Corollary 2.7
  • ...and 127 more