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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 Kähler 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 alge bra. 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 Kähler differential forms (de Rham algebra) of a commutative algebra, to int- duce 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 Kähler differentials n~/R of an algebra SIR. 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.
List of contents
1. Derivations.-
2. Differential Algebras.-
3. Universal Extension of a Differential Algebra.-
4. Description of the Universal Extension in Special Cases.-
5. Differential Modules of Field Extensions.-
6. Differential Modules of Local Rings.-
7. Differential Modules of Affine Algebras.-
8. Smooth Algebras.-
9. Differential Modules of Complete Intersections.-
10. The Kahler Differents (Jacobian Ideals) of an Algebra.-
11. Universally Finite Differential Algebras.-
12. Differential Algebras and Completion.-
13. Differential Modules of Semianalytic Algebras.-
14. Regularity Criteria for Semianalytic Algebras.-
15. Existence of p-Bases.-
16. Traces of Differential Forms.-
17. Residues in Algebraic Function Fields of one Variable.- Appendices.- A. Commutative Algebras.- B. Dimension Formulas in Algebras of Finite Type.- C. Complete Intersections.- D. The Fitting Ideals of a Module.- E. The Dual of a Module over a Noetherian Ring.- F. Traces.- G. Differents.- Symbol Index.
