Titlebook: Kähler Differentials; Ernst Kunz Book 1986 Springer Fachmedien Wiesbaden 1986 Algebra.Endlichkeit.Funktion.Variable.differential algebra.p

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Differential Modules of Field Extensions,The purpose of this section is to investigate, which informations are contained in the differential modules of field extensions. Here we obtain the first applications of the theory of differential modules: Some results about separability and the structure of field extensions.

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Differential Modules of Affine Algebras,The results of the last section will now be applied to affine algebras and their localizations. Let A be an affine algebra over a field K. We shall use the following facts about the Krull dimension (dim A) of A and the dimension of the localizations of A (see appendix B for generalizations)

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Smooth Algebras,We shall consider now algebras S/R that are essentially of finite type where R is an arbitrary noetherian ring. S/R is called ., if the following conditions are satisfied:

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Differential Algebras and Completion,The universally finite differential algebra exists under rather mild assumptions for certain complete algebras. We shall show this after having studied the behaviour of differential algebras under completion in general.

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Differential Modules of Semianalytic Algebras,In this section k is a complete noetherian local ring whose maximal ideal is generated by p•1, where p is the characteristic of the residue field of k. In particular, k can be a Cohen ring (§ 12).

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Residues in Algebraic Function Fields of One Variable,Let L/K be an algebraic function field with Trdeg(L/K) = 1. The set X of all discrete valuation rings R with K ⊂ R and Q(R) = L is called the “non-singular model” of L/K, or the “abstract Riemann surface” of L/K. It is well-known that X together with L is also the set of local rings of a non-singular projective algebraic curve defined over K.

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