Divisors over determinantal rings defined by two by two minors

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^* tensor G induces the Koszul complex ... -> S_mE^* tensor S_nG tensor Wedge^p(E^* tensor G) -> S_{m+1}E^* tensor S_{n+1}G tensor Wedge^{p-1}(E^* tensor G) -> ... and its dual ... -> D_{m+1}E tensor D_{n+1}G^* tensor Wedge^{p-1}(E tensor G^*) -> D_mE tensor D_nG^* tensor Wedge^p(E tensor G^*)-> ... Let H_{m,n,p} be the homology of the top complex at S_m tensor S_n tensor Wedge^p and H^{m,n,p} the homology of the bottom complex at D_m tensor D_n tensor Wedge^p. It is known that H_{m,n,p} is isomorphic to H^{m',n',p'}, provided m+m'=g-1, n+n'=e-1, p+p'=(e-1)(g-1), and m-n is between 1-e and g-1. In this paper we exhibit an explicit quasi-isomorphism M of complexes which gives rise to this isomorphism. The mapping cone of M is a split exact complex. Our complexes may be formed over the ring of integers; they can be passed to an arbitrary ring or field by base change. 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,m-n). The Segre modules are a set of representatives of the divisor class group of the determinantal ring defined by the 2 by 2 minors of an e by g matrix of indeterminants. If R is the ring of integers, then the homology H_{m,n,p} is not always a free abelian group. In other words, if R is a field, then the dimension of H_{m,n,p} depends on the characteristic of R. The module H_{m,n,p} is known when R is a field of characteristic zero; however, this module is not yet known over arbitrary fields.