Generalized Koszul Complexes

This article should be viewed as a survey of generalized Koszul complexes and Koszul bicomplexes with an application to generalized Koszul complexes in projective dimension one. We shall try to give detailed information on the basic definitions and a summary of the main results. Concerning proofs the reader is invited to have a look into [I] or [IV]. Introduction. We start with the following question: given finite free modules F , G, H over a noetherian ring R and a complex F χ → G λ → H; in which way does grade Iλ depend on grade Iχ and on the ranks of F , G, H? Here Iχ is the ideal of maximal minors of χ, and grade Iχ is the maximal length of a regular sequence contained in Iχ. If, for example, rankF = 1, rankG = n, and χ is given by a regular sequence x1, . . . , xn in R, i.e. χ(1) = (x1, . . . , xn), then one knows that grade Iλ = n is possible if and only if rankH = 1 and n is even. (The if part is trivial: Set λ(a1, . . . , an) = ∑n i=1(−1) aixn−i+1; for the other direction see [BV2] or Corollary 7 at the end of the article) Assume that m = rankF ≤ n = rankG. Then it turns out that the problem just described is closely connected with the homology of the (generalized) Koszul complex associated with the induced map λ̄ : Cokχ→ H. 1. Classical Koszul complexes. Let G be a module over an arbitrary commutative ring R, and let ψ : G→ R be a linear form. Define ∂ψ(y1 ∧ . . . ∧ yp) = p ∑ i=1 (−1)ψ(yi)y1 ∧ . . . ŷi . . . ∧ yp