MINIMAL FREE RESOLUTIONS AND (G, N)-COMPLEXES FOR FINITE ABELIAN-GROUPS

Let G be a finite abelian group and let n be a non-negative integer. A (G, n)-resolution is a partial free resolution of the ZG-module Z with n + 1 intermediate finitely generated free ZG-modules F0,...,F(n). The kernel K of the map from F(n) to F(n-1) is called a (G, n)-module, and K is called a minimal (G, n)-module if it has the smallest possible Z-rank. We shall calculate the number of isomorphism types of minimal (G, n)-modules and various related numbers. In particular, we shall calculate the number of minimal (G, n)-complexes for all n greater-than-or-equal-to 2. This extends work of Browning, Dyer and Sieradski who had already dealt with the case n = 2.