Irreducible components of rigid spaces. (English) Zbl 0928.32011

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 work of Coleman and Mazur on Fredholm series, i.e., global sections (entire functions) on affine \(n\)-space \(\mathbf A_X^n=:\mathbf A_k^n\times_k X\) over a rigid analytic variety \(X\).


32P05 Non-Archimedean analysis
32C18 Topology of analytic spaces
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