Dualizing complexes and homomorphisms vanishing in Koszul homology

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 f. g. projective R/I-module V has an Rprojective resolution P. of length d. In this paper, we compute the homology of the n-th Koszul complex associated with the homomorphism P1 → P0 for all n ≥ 1, if d = 1. This computation yields a new proof of the classical Adams-Riemann-R...

Dualizing Complexes

Length Formulas for the Homology of General- Ized Koszul Complexes

Let M be a finite module over a noetherian ring R with a free resolution of length 1. We consider the generalized Koszul complexes Cλ̄(t) associated with a map λ̄ : M → H into a finite free R-module H (see [IV], section 3), and investigate the homology of Cλ̄(t) in the special setup when grade IM = rankM = dimR. (IM is the first non-vanishing Fitting ideal of M .) In this case the (interesting) ho...

Collapsibility and Vanishing of Top Homology in Random Simplicial Complexes

Let ∆n−1 denote the (n − 1)-dimensional simplex. Let Y be a random d-dimensional subcomplex of ∆n−1 obtained by starting with the full (d − 1)-dimensional skeleton of ∆n−1 and then adding each d-simplex independently with probability p = c n . We compute an explicit constant γd, with γ2 ' 2.45, γ3 ' 3.5 and γd = Θ(log d) as d → ∞, so that for c < γd such a random simplicial complex either colla...

RIGID DUALIZING COMPLEXES

Let $X$ be a sufficiently nice scheme. We survey some recent progress on dualizing complexes. It turns out that a complex in $kinj X$ is dualizing if and only if tensor product with it induces an equivalence of categories from Murfet&#039;s new category $kmpr X$ to the category $kinj X$. In these terms, it becomes interesting to wonder how to glue such equivalences.