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Transactions of the American Mathematical Society

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Computing the homology of Koszul complexes

Author: Bernhard Köck
Journal: Trans. Amer. Math. Soc. 353 (2001), 3115-3147
MSC (2000): Primary 13D25, 19E20, 14C40, 13D15
Published electronically: April 10, 2001
MathSciNet review: 1828601
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Let $R$ be a commutative ring and $I$ an ideal in $R$ which is locally generated by a regular sequence of length $d$. Then, each f. g. projective $R/I$-module $V$ has an $R$-projective resolution $P.$ of length $d$. In this paper, we compute the homology of the $n$-th Koszul complex associated with the homomorphism $P_1 \rightarrow P_0$ for all $n \ge 1$, if $d=1$. This computation yields a new proof of the classical Adams-Riemann-Roch formula for regular closed immersions which does not use the deformation to the normal cone any longer. Furthermore, if $d=2$, we compute the homology of the complex $N\, \operatorname{Sym}^2 \, \Gamma(P.)$ where $\Gamma$ and $N$ denote the functors occurring in the Dold-Kan correspondence.

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Additional Information

Bernhard Köck
Affiliation: Mathematisches Institut II, Universität Karlsruhe, 76128 Karlsruhe, Germany
Address at time of publication: Faculty of Mathematical Studies, University of Southampton, Southampton SO17 1BJ, United Kingdom

Keywords: Koszul complex, Dold-Kan correspondence, cross effect functor, symmetric power operation, Adams-Riemann-Roch theorem, plethysm problem
Received by editor(s): May 30, 1999
Received by editor(s) in revised form: January 30, 2000
Published electronically: April 10, 2001
Article copyright: © Copyright 2001 American Mathematical Society