Local Analytic Geometry

Basic Theory and Applications

• Theo de Jong
• Gerhard Pfister
Book

Part of the Advanced Lectures in Mathematics book series (ALM)

1. Front Matter
Pages i-xi
2. Theo de Jong, Gerhard Pfister
Pages 1-46
3. Theo de Jong, Gerhard Pfister
Pages 47-73
4. Theo de Jong, Gerhard Pfister
Pages 74-125
5. Theo de Jong, Gerhard Pfister
Pages 126-170
6. Theo de Jong, Gerhard Pfister
Pages 171-224
7. Theo de Jong, Gerhard Pfister
Pages 225-274
8. Theo de Jong, Gerhard Pfister
Pages 275-294
9. Theo de Jong, Gerhard Pfister
Pages 295-310
10. Theo de Jong, Gerhard Pfister
Pages 311-338
11. Theo de Jong, Gerhard Pfister
Pages 339-373
12. Back Matter
Pages 374-384

Introduction

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.

Keywords

Algebraische Geometrie Algebraische Kurve Kommutative Algebra Komplexe Analysis Singularität (Math.) deformation theory singularity theory

Authors and affiliations

• Theo de Jong
• 1
• Gerhard Pfister
• 2
1. 1.Fachbereich MathematikUniversität des SaarlandesSaarbrückenGermany
2. 2.Fachbereich MathematikUniversität KaiserslauternKaiserslauternGermany

Bibliographic information

• DOI https://doi.org/10.1007/978-3-322-90159-0