Local theory of complex spaces. (English) Zbl 0808.32008

Remmert, R.

Grauert, H. (ed.) et al., Several complex variables VII. Sheaf- theoretical methods in complex analysis. Berlin: Springer-Verlag. Encycl. Math. Sci. 74, 7-96 (1994).

From the introduction: “This introductory chapter is a rambling through basic notions and results of local complex analysis based on local function theory, local algebra and sheaves. In focus are coherent analytic sheaves. We discuss four fundamental results: Coherence of structure sheaves in §7, Finite mapping theorems in §8, Coherence of ideal sheaves in §9, Coherence of normalization sheaves in §14. All local function theory originates from the Weierstrass Preparation and Division Theorems. These theorems form the cornerstones of §1. In sections 2 to 6 we introduce and discuss basic notions. Dimension theory is developed in §10, while §11 is devoted to homological codimension, Cohen-Macaulay spaces, Noether property and analytic spectra. Pure dimensional reduced complex spaces look locally like analytically branched coverings of domains in \(\mathbb{C}^ n\). In §12 we study such coverings. Sections 13 to 15 are dealing with normal spaces and (semi- )normalizations”.
For the entire collection see [Zbl 0793.00010].

Reviewer: K.Dabrowski (Piastow)

MSC:

32Cxx	Analytic spaces
32Bxx	Local analytic geometry
32-02	Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces
14F10	Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials

Keywords:

Cohen-Macaulay space; analytic spectrum; homological codimension; meromorphic function; normalization sheaf; Weierstrass algebra; local complex analysis; coherent analytic sheaves