A generalization of formal schemes and rigid analytic varieties

1994 ◽  
Vol 217 (1) ◽  
pp. 513-551 ◽  
Author(s):  
R. Huber
Author(s):  
Peter Scholze ◽  
Jared Weinstein

This chapter reviews the theory of adic spaces as developed by Huber. There are two familiar categories of geometric objects which arise in nonarchimedean geometry: formal schemes and rigid-analytic varieties. The goal is to construct a category of adic spaces which contains both formal schemes and rigid-analytic spaces as full subcategories. Just as formal schemes are built out of affine formal schemes associated to adic rings, and rigid-analytic spaces are built out of affinoid spaces associated to affinoid algebras, adic spaces are built out of affinoid adic spaces, which are associated to pairs of topological rings. The affinoid adic space associated to such a pair is the adic spectrum. The chapter then looks at Huber rings and defines the set of continuous valuations on a Huber ring, which constitute the points of an adic space.


2018 ◽  
Vol 155 (1) ◽  
pp. 38-88 ◽  
Author(s):  
Alberto Vezzani

We establish a tilting equivalence for rational, homotopy-invariant cohomology theories defined over non-archimedean analytic varieties. More precisely, we prove an equivalence between the categories of motives of rigid analytic varieties over a perfectoid field $K$ of mixed characteristic and over the associated (tilted) perfectoid field $K^{\flat }$ of equal characteristic. This can be considered as a motivic generalization of a theorem of Fontaine and Wintenberger, claiming that the Galois groups of $K$ and $K^{\flat }$ are isomorphic.


1991 ◽  
Vol 112 (1) ◽  
pp. 125-125
Author(s):  
Donna Kumagai ◽  
Zbigniew Slodkowski

1972 ◽  
Vol 45 ◽  
pp. 1-38 ◽  
Author(s):  
David Meredith

Throughout this paper, (R, m) denotes a (noetherian) local ring R with maximal ideal m.In [5], Monsky and Washnitzer define weakly complete R-algebras with respect to m. In brief, an R-algebra A† is weakly complete if


Phronimon ◽  
2018 ◽  
Vol 18 ◽  
pp. 95-114
Author(s):  
Daniel Francois Strauss

Sometimes systematic theoretical thinking is identified with abstract (formal) schemes. This opposition is also found in Malan and Goosen’s dismissal of Dooyeweerdian reformational thinking. This article aims at making a contribution to this issue by analysing the indispensable role of systematic philosophical reflection within the world of scholarship. One way in which systematic thinking could be justified is to highlight the need for consistency and the role of logical principles in achieving it. It is argued that, since we are living in the same world, all philosophical orientations have to account for shared states of affairs. At this point attention is given to the question whether or not these “states of affairs” are “static or dynamic.” An alternative for the distinction between static and dynamic is proposed by alternatively considering the relationship between constancy and change. Von Weizsäcker articulates the problem aptly by pointing out that although our experience exhibits constant change, something exists that remains unchanged through all these changes. In conclusion it is pointed out that scholars have only two options: either they give an account of the philosophical presuppositions and systematic distinctions with which they work—in which case they have a philosophical view of reality, or implicitly (and uncritically) they proceed from one or another philosophical view of reality—in which case they are the victims of a philosophical view. The primary aim of this article is, therefore, to highlight the indispensability of systematic thought by referring to some of the main distinctions included in such a system of thought.


2016 ◽  
Vol 4 ◽  
Author(s):  
PETER SCHOLZE

The author would like to make some changes to the previously published article [1] by correcting two definitions.


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