Analytic space

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, ..., f_k be a collection of analytic functions on U. Denote by Z the common vanishing locus of f₁, ..., f_k, that is, let Z = { x | f₁(x) = ... = f_k(x) = 0 }. Z is an analytic variety.

Suppose that the structure sheaf of U is . Then Z has a structure sheaf , where is the ideal generated by f₁, ..., f_k. In other words, the structure sheaf of Z consists of all functions on U modulo the possible ways they can differ outside of Z.