Breadcrumbs Section. Click here to navigate to respective pages.

Chapter

Chapter

# The Koszul complex in projective dimension one

DOI link for The Koszul complex in projective dimension one

The Koszul complex in projective dimension one book

# The Koszul complex in projective dimension one

DOI link for The Koszul complex in projective dimension one

The Koszul complex in projective dimension one book

Click here to navigate to parent product.

## 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.