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Computing the homology of Koszul complexes

Computing the homology of Koszul complexes
Computing the homology of Koszul complexes
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 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 --> P_0 for all n, 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 Sym^2 K(P.) where K and N denote the functors occurring in the Dold-Kan correspondence.
koszul complex, dold-kan correspondence, cross effect functor, symmetric power operation, adams-riemann-roch theorem, plethysm problem
0002-9947
3115-3147
Koeck, Bernhard
84d11519-7828-43a6-852b-0c1b80edeef9
Koeck, Bernhard
84d11519-7828-43a6-852b-0c1b80edeef9

Koeck, Bernhard (2001) Computing the homology of Koszul complexes. Transactions of the American Mathematical Society, 353 (8), 3115-3147. (doi:10.1090/S0002-9947-01-02723-4).

Record type: Article

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 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 --> P_0 for all n, 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 Sym^2 K(P.) where K and N denote the functors occurring in the Dold-Kan correspondence.

Other
KoszulAms.dvi - Accepted Manuscript
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S0002-9947-01-02723-4.pdf - Version of Record
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More information

Published date: 10 April 2001
Keywords: koszul complex, dold-kan correspondence, cross effect functor, symmetric power operation, adams-riemann-roch theorem, plethysm problem

Identifiers

Local EPrints ID: 29783
URI: http://eprints.soton.ac.uk/id/eprint/29783
ISSN: 0002-9947
PURE UUID: a2ab95f8-2cdc-442f-9ea9-5e74a471917f
ORCID for Bernhard Koeck: ORCID iD orcid.org/0000-0001-6943-7874

Catalogue record

Date deposited: 11 May 2006
Last modified: 16 Mar 2024 03:22

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