Local Analytic Geometry Basic Theory and Applications [Paperback]

Auf der Grundlage einer Einf?hrung in die kommutative Algebra, algebraische
Geometrie und komplexe Analysis werden zun?chst Kurvensingularit?ten
untersucht. Daran schlie?en Ergebnisse an, die zum ersten Mal in einem
Lehrbuch aufgenommen wurden, das Verhalten von Invarianten in Familien,
Standardbasen f?r konvergente Potenzreihenringe, Approximationss?tze,
Grauerts Satz ?ber die Existenz der versellen Deformation.

Das Buch richtet sich an Studenten h?herer Semester, Doktoranden und
Dozenten. Es ist auf der Grundlage mehrerer Vorlesungen und Seminaren an
den Universit?ten in Kaiserslautern und Saarbr?cken entstanden.

Algebraic geometry is, loosely speaking, concerned with the study of zero sets of polynomials (over an algebraically closed field). As one often reads in prefaces of int- ductory books on algebraic geometry, it is not so easy to develop the basics of algebraic geometry without a proper knowledge of commutative algebra. On the other hand, the commutative algebra one needs is quite difficult to understand without the geometric motivation from which it has often developed. Local analytic geometry is concerned with germs of zero sets of analytic functions, that is, the study of such sets in the neighborhood of a point. It is not too big a surprise that the basic theory of local analytic geometry is, in many respects, similar to the basic theory of algebraic geometry. It would, therefore, appear to be a sensible idea to develop the two theories simultaneously. This, in fact, is not what we will do in this book, as the commutative algebra one needs in local analytic geometry is somewhat more difficult: one has to cope with convergence questions. The most prominent and important example is the substitution of division with remainder. Its substitution in local analytic geometry is called the Weierstraft Division Theorem. The above remarks motivated us to orgl³n