Kahler Differentials (Advanced lectures in mathematics): Ernst Kunz

Preface:
This book is based on a lecture course that I gave at the University of Regensburg. The purpose of these lectures was to explain the role of Kahler differential forms in ring theory, to prepare the road for their application in algebraic geometry, and to lead up to some research problems The text discusses almost exclusively local questions and is therefore written in the language of commutative algebra. The translation into the language of algebraic geometry is easy for the reader who is familiar with sheaf theory and the theory of schemes.

The principal goals of the monograph are: To display the information contained in the algebra of Kahler differential forms (de Rham algebra) of a commutative algebra, to introduce and discuss "differential invariants" of algebras, and to prove theorems about algebras with "differential methods" The most important object we study is the module of Kahler differentials Omega^1_{S/R} of an algebra S/R. Like the differentials of analysis, differential modules "linearize" problems, i.e. reduce questions about algebras (non-linear problems) to questions of linear algebra.

We are mainly interested in algebras of finite type. Results about arbitrary algebras are only given, if no extra effort is necessary. However, by working with the technic of "admissible derivations of the ground ring", I have tried to present the results free from "separability assumptions". Later (in Section 16) traces of differential forms are constructed in the case of locally complete intersection algebras. These traces are strongly related to "regular differential forms", "residues" and "duality theory", subjects that are not touched upon in this book, except for the one-dimensional case (Section 17). But I hope to come back to the higher dimensional applications of the theory elsewhere.

For standard results of commutative algebra the general references are Matsumura [M1] and Bourbaki [B2]. In the ear sections of the book, in which the functorial properties of derivation modules and differential algebras are described, only very little is needed. Then gradually we have to rely more and more on facts from other sources. Appendices A-G collect, for easier reference, some material of ring theory mainly about complete intersections and traces. These appen dices of together 96 pages do no require knowledge of the rest of the book and can be read at the beginning. Appendix describes the language we use and introduces some notation. The reader should begin by having a brief look at this appendix.

I wish to thank the participants of my lectures and semi nars, and the readers of preliminary versions of these note for their critical interest and valuable comments. Thanks are also due to my present and former students S.Bruderle, T.Grimier, R.Hubl, Dr.H.Knebl, B.Kock, Dr.J.Koch, M.Kreuzer G.Seibert, and Dr.R.Waldi, who have read part of the manuscript and have corrected many errors. Last but not least I have to thank Frau Eva Riitz for the patience and skill she showed while preparing various versions of this text.