Two Themes in Commutative Algebra: Algebraic Dependence and Kähler Differentials

Abstract

. I reproduce an elementary proof for the jacobian criterion for algebraic dependence and give an introduction to the basics of Kahler differentials. All results presented are known, but some of them are not easily accessible, either because the references are hard to get hold of or because they are difficult to read for non-algebraists. 1 Introduction The aim of this report is twofold: ffl to provide an elementary proof for the jacobian criterion for algebraic dependence ffl to make the concept of Kahler differentials more accessible to non-experts My motivation is mainly applications to control theory; see e.g. [2, 3, 4, 5]. These applications will not be explicitly discussed here, however. Some acquaintance with basic concepts and notation of commutative algebra is required; concepts that will be used without further notice are: algebraic dependence, transcendence degree, ideal, module, k-algebra. However it is not vital that the reader understands all these concepts in detail. C..