Traces of Differential Forms and Hochschild Homology
This monograph provides an introduction to, as well as a unification and extension of the published work and some unpublished ideas of J. Lipman and E. Kunz about traces of differential forms and their relations to duality theory for projective morphisms.
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1368
Reinhold Hubl
Traces of Differential Forms and Hochschild Homology
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Lecture Notes in Mathematics Edited by A. Oold and B. Eckmann
1368
Reinhold Hubl
Traces of Differential Forms and Hochschild Homology
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Author
Reinhold Hubl Purdue University, Department of Mathematics Mathematical Sciences Building West Lafayette, IN 47907, USA
Mathematics Subject Classification (1980): 13099, 14F 10, 16A61, 32A27 ISBN 3-540-50985-2 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-50985-2 Springer-Verlag New York Berlin Heidelberg
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© Springer-Verlag Berlin Heidelberg 1989 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210
Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . §1.
1
The Hochschild homology and the Hochschild cohomology of a topological algebra
8
§2. Differential forms and Hochschild homology
28
§3. Traces in Hochschild homology
47
§4. Traces of differential forms .
56
§5. Traces in complete intersections
70
§6. The topological residue homomorphism
84
§7. Trace formulas for residues of differential forms .
94
References .
106
Symbol Index
108
Subject Index
109
Introduction. Differential forms and their traces have a long tradition in mathematics. In analysis they were used to study finite coverings of Riemann surfaces. Algebraic analogues appeared first in the theory of algebraic function fields in one variable, for example in M. Deuring's "Theorie der Korrespondenzen algebraischer Funktionenkorper II" (Journal fur die reine und angewandte Mathematik, Bd. 183, 1941) and in C. Chevalley's "Introduction to algebraic functions in one variable." These papers are strongly oriented towards the corresponding analytic theory. In particular they only consider differentials on function fields with respect to the field of coefficients, so that strong results could be achieved only in case of separable field extensions. Later E. Kunz and H. J. Nastold studied one-dimensional function fields Kl k; emphasizing the inseparable case. They showed that it is necessary to replace the field of coefficients k of K by a suitable subfield ko of k, a so-called admissible field for K / k; and to look at
differential forms of K/k o in order to get for instance a general duality theorem for function fields of dimension 1 ([KaJ, [Na]). However they still only
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