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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Computing the homology of Koszul complexes
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by Bernhard Köck PDF
Trans. Amer. Math. Soc. 353 (2001), 3115-3147 Request permission

Abstract:

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
  • Email: Bernhard.Koeck@math.uni-karlsruhe.de
  • Received by editor(s): May 30, 1999
  • Received by editor(s) in revised form: January 30, 2000
  • Published electronically: April 10, 2001
  • © Copyright 2001 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 353 (2001), 3115-3147
  • MSC (2000): Primary 13D25, 19E20, 14C40, 13D15
  • DOI: https://doi.org/10.1090/S0002-9947-01-02723-4
  • MathSciNet review: 1828601