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. 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 . Then Z has a structure sheaf , where is the ideal generated by f1, ..., fk. In other words, the structure sheaf of Z consists of all functions on U modulo the possible ways they can differ outside of Z. An analytic space is a locally ringed space such that around every point x of X, there exists an open neighborhood U such that is isomorphic (as locally ringed spaces) to an analytic variety with its structure sheaf. Such an isomorphism is called a local model for X at x. An analytic mapping or morphism of analytic spaces is a morphism of locally ringed spaces. This definition is similar to the definition of a scheme. The only difference is that for a scheme, the local models are spectra of rings, whereas for an analytic space, the local models are analytic varieties. Because of this, the basic theories of analytic spaces and of schemes are very similar. Furthermore, analytic varieties have much simpler behavior than arbitrary commutative rings (for example, analytic varieties are defined over fields and are always finite-dimensional), so analytic spaces behave very similarly to finite-type schemes over a field. Every point in an analytic space has a local dimension.