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Auf der Grundlage einer Einführung in die kommutative Algebra, algebraische
Geometrie und komplexe Analysis werden zunächst Kurvensingularitäten
untersucht. Daran schließen Ergebnisse an, die zum ersten Mal in einem
Lehrbuch aufgenommen wurden, das Verhalten von Invarianten in Familien,
Standardbasen für konvergente Potenzreihenringe, Approximationssätze,
Grauerts Satz über die Existenz der versellen Deformation.
Das Buch richtet sich an Studenten höherer Semester, Doktoranden und
Dozenten. Es ist auf der Grundlage mehrerer Vorlesungen und Seminaren an
den Universitäten in Kaiserslautern und Saarbrücken 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 organize the first four chapters of this book as follows. In Chapter 1 we discuss the algebra we need. Here, we assume the reader attended courses on linear algebra and abstract algebra, including some Galois theory.
Geometrie und komplexe Analysis werden zunächst Kurvensingularitäten
untersucht. Daran schließen Ergebnisse an, die zum ersten Mal in einem
Lehrbuch aufgenommen wurden, das Verhalten von Invarianten in Familien,
Standardbasen für konvergente Potenzreihenringe, Approximationssätze,
Grauerts Satz über die Existenz der versellen Deformation.
Das Buch richtet sich an Studenten höherer Semester, Doktoranden und
Dozenten. Es ist auf der Grundlage mehrerer Vorlesungen und Seminaren an
den Universitäten in Kaiserslautern und Saarbrücken 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 organize the first four chapters of this book as follows. In Chapter 1 we discuss the algebra we need. Here, we assume the reader attended courses on linear algebra and abstract algebra, including some Galois theory.
"This book is an introduction to local analytic geometry, with emphasis on the study of singularities of germs of complex analytic spaces. It is very well written and the authors, assuming very little background on the side of the reader, manage to cover in less than 400 pages a large amount of beautiful material, presented in a didactical way. ( ) This seems to be an excellent introduction to the subject and a very appropriate textbook for a graduate course on these matters" Zentralblatt für Mathematik, 01/09