Computing the homology of Koszul complexes

Details Computing the homology of Koszul complexes Computing the homology of Koszul complexes

1998-09-29

arxiv.org/abs/math/9809175v2

Mathematics

Commutative Algebra

35 pages

Scientific paper

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.

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