Concept

Analytic space

Summary
An analytic space is a generalization of an analytic manifold that allows singularities. An analytic space is a space that is locally the same as an analytic variety. They are prominent in the study of several complex variables, but they also appear in other contexts. Definition Fix a field k with a valuation. Assume that the field is complete and not discrete with respect to this valuation. For example, this includes R and C with respect to their usual absolute values, as well as fields of Puiseux series with respect to their natural valuations. Let U be an open subset of kn, and let f1, ..., fk be a collection of analytic functions on U. Denote by Z the common vanishing locus of f1, ..., fk, that is, let Z = { x | f1(x) = ... = fk(x) = 0 }. Z is an analytic variety. Suppose that the structure sheaf of U is \mathcal{O}_U. Then Z has a structure sheaf \mathcal{O}_Z = \mathcal{O}_U / \mathcal{I}_Z, where \mathcal{I}_Z is the
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