ABSTRACT
UDO VETTER, Universitat Oldenburg, FB Mathematik, 26111 Oldenburg, Germany, vetterQmathematik.uni-oldenburg.de
Let R be a noetherian ring and M a finite .R-module. With a linear form \ on M one associates the Koszul complex K(x)- If M is a free module, then the homology of K(x) is well-understood, and in particular it is grade sensitive with respect to Imx-
In this note we investigate the case of a module M of projective dimension 1 (more precisely, M has a free resolution of length 1) for which the first nonvanishing Fitting ideal I M has the maximally possible grade r + 1, r = rankM. Then h = grade Imx < r + 1 for all linear forms x on M, and it turns out that Hr.i(K(x)) = 0 for all even t < h and Hr-i(K(x)) = S('~1)/2((7) for all odd i < h where S denotes symmetric power and C — Ext^j(M, R), in other words, C = Cokt/i* for a presentation
Moreover, if h < r, then Hr-/l(K(x)) is neither 0 nor isomorphic to a symmetric power of (7, so that it is justified to say that K(x) is grade sensitive for the modules M under consideration.
