Irreducible components of rigid spaces
type: Item
instance of: scholarly article
title: Irreducible components of rigid spaces
zbMATH Open document ID: 0928.32011
publication date: 1999-05-16T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/75345
review text: The paper introduces a theory of irreducible decomposition for rigid analytic varieties (over a field \(k\) complete with respect to a non-trivial non-archimedean norm). For affinoid varieties \(X=\text{Sp}A\), there is no problem in defining the irreducible components; they are just the closed analytic subsets defined by the minimal primes of \(A\). For global rigid analytic varieties, e.g., the zero-locus of a Fredholm series (see below), the definition is no longer obvious. A similar difficulty is found in complex analysis; however, two extra difficulties present themselves in the rigid case: the base field need not be algebraically closed and instead of a locally compact Hausdorff topology, we only have a Grothendieck topology. The main observation in the paper is that for a normal rigid analytic variety \(X\) (= all local rings are normal), we have that \(X\) is connected (= not the union of two disjoint proper admissible open sets, or, equivalently, the ring of global sections \(\Gamma(X,\mathcal O_X)\) has no non-trivial idempotents), if and only if, \(X\) is irreducible (= not the union of two proper closed analytic subsets). Therefore, the author defines the irreducible components of an arbitrary rigid analytic variety \(X\) to be the images under \(\pi\) of the connected components of \(\widetilde X\), where \(\pi:\widetilde X\to X\) is the normalization of \(X\). Such a normalization is shown to exist and to be unique up to isomorphism. The theory of irreducible components is then developed (behavior with respect to base change of the ground field, irreducible components of the analytification of a scheme of finite type over \(k\), etc.). In order to obtain these results, the author needs to resort to some commutative algebra. In particular, he shows that the local ring \(\mathcal O_{X,x}\) of a point \(x\) on a rigid analytic variety \(X\) is excellent. As an application, the paper presents an alternative treatment (and some partial extensions) of the wo…
reviewed by: Hans Schoutens
Mathematics Subject Classification ID: 32P05
Mathematics Subject Classification ID: 32C18
zbMATH DE Number: 1287528
zbMATH Keywords: irreducible component
zbMATH Keywords: rigid analysis
zbMATH Keywords: excellence
zbMATH Keywords: Fredholm series
MaRDI profile type: Publication
OpenAlex ID: W1993756907
cites work: Differentialrechnung in der analytischen Geometrie
cites work: Q3661717
cites work: Orthonormalbasen in der nichtarchimedischen Funktionentheorie
cites work: Q3325851
cites work: Formal and rigid geometry. I: Rigid spaces
cites work: Formal and rigid geometry. II: Flattening techniques
cites work: Formal and rigid geometry. III: The relative maximum principle
cites work: Q4249385
cites work: Crystalline Dieudonné module theory via formal and rigid geometry
cites work: Q5602063
cites work: Q3322387
cites work: Q3871181
cites work: Q2551490
cites work: Theorem A und B in der nichtarchimedischen Funktionentheorie
cites work: Die analytische Normalität affinoider Ringe
cites work: Ausgezeichnete Ringe in der nichtarchimedischen analytischen Geometrie.
cites work: Local constancy in \(p\)-adic families of Galois representations
cites work: Q5183625
cites work: Der Satz von Remmert-Stein in der nichtarchimedischen Funktionentheorie
cites work: Formal-algebraic and rigid-analytic geometry
cites work: Q3882554
cites work: Q3739243
cites work: On the excellent property for power series rings over polynomial rings
cites work: A few theorems on completion of excellent rings
DOI: 10.5802/AIF.1681
author: Brian Conrad
published in: Annales de l’institut Fourier
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description: scientific article; zbMATH DE number 1287528