An Explicit, Characteristic-Free, Equivariant Homology Equivalence Between Koszul Complexes

Abstract

Let E and G be free modules of rank e and g, respectively, over a commutative noetherian ring R. The identity map on E*⊗ G induces the Koszul complex and its dual Let H𝒩(m,n,p) be the homology of the top complex at Sym m E*⊗ Sym n G⊗ ∧ p (E*⊗ G) and Hℳ(m,n,p) the homology of the bottom complex at D m E⊗ D n G*⊗ ∧ p (E⊗ G*). It is known that H𝒩(m,n,p)≅ Hℳ(m′, n′, p′), provided m + m′ = g − 1, n + n′ = e − 1, p + p′ = (e − 1)(g − 1), and 1 − e ≤ m − n ≤ g − 1. In this article, we exhibit a complex 𝕐 and explicit quasi-isomorphisms from 𝕐 to two complexes, as described above, for the appropriate choice of parameters, which give rise to this isomorphism. Our quasi-isomorphisms may be formed over the ring of integers; they can be passed to an arbitrary ring or field by base change. All of our work is equivariant under the action of the group GL(E) × GL(G); that is, everything we do is independent of the choice of basis. Knowledge of the homology of the top complex is equivalent to knowledge of the modules in the resolution of the Segre module Segre(e,g, ℓ), for ℓ = m − n. The modules {Segre (e,g, ℓ)|ℓ ∈ ℤ} are a set of representatives of the divisor class group of the determinantal ring defined by the 2 × 2 minors of an e × g matrix of indeterminates. If R is the ring of integers, then the homology H𝒩(m,n,p) is not always a free abelian group. In other words, if R is a field, then the dimension of H𝒩(m,n,p) depends on the characteristic of R. The module H𝒩(m,n,p) is known when R is a field of characteristic zero; however, this module is not yet known over arbitrary fields. The modules in the minimal resolution of the universal ring for finite length modules of projective dimension two are equal to modules of the form H𝒩(m,n,p).