Kahler’s Differentials

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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 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 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 Kahler 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 Kahler 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.