Complexes Which Arise from a Matrix and a Vector: Resolutions of Divisors on Certain Varieties of Complexes

Abstract

Consider the polynomial ring R = R0[X, Y] where R0 is a normal domain, and X1 × g and Yg × f are matrices of indeterminates. The R-ideal J = I1(XY) + Imin {f, g}(Y) defines a variety of complexes over R0. The divisor class group of R/J is isomorphic to Cl(R0)⊕Z[I′], where I′ is an ideal of R/J generated by appropriately chosen lower order minors of Y. We produce the minimal R-free resolution of i[I′] for all integers i ≥ −1. If f is greater than or equal to g, then J is a generic residual intersection of the generic grade g complete intersection I1(X). The resolutions that we produce in this case are, in many ways, analogous to resolutions of divisors on generic residual intersections of grade two perfect ideals or grade three Gorenstein ideals.