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.

We have fixed some mistakes pointed out to us by Peter Scholze. The main results remain unchanged

References in corpus (4)

• Higher Topos Theory
• A Perspective on the Foundations of Derived Analytic Geometry
• Fréchet Modules and Descent
• An Open Mapping Theorem for rings which have a zero sequence of units

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• Topics on Geometric and Representation Theoretic Aspects of Period Rings I
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• Analytic Geometry and Hodge-Frobenius Structure Continued
• Derived Analytic Geometry for Z-Valued Functions. Part I -- Topological Properties