×

Linear forms on modules of projective dimension one. (English) Zbl 1058.13007

From the paper: Let \(R\) be a noetherian ring and \(M\) an \(R\)-module which has a presentation \(0\to F@>\psi>> G\to M\to 0\) with finite free \(R\)-modules \(F\) and \(G\) of rank \(m\) and \(n\). W. Bruns and M. Vetter [in: Geometric and combinatorial aspects of commutative algebra, Pap. Int. Conf. Commutative Algebra Algebraic Geometry, Messina 1999, Lect. Notes Pure Appl. Math. 217, 89–97 (2001; Zbl 0987.13006)] proved:
Assume that \(r=n-m>1\) and that the first nonvanishing Fitting ideal of \(M\) has grade \(r+1\). Then the following conditions are equivalent:
(1) There is a \(\chi\in M^*=\text{Hom}_R(M,R)\) such that the ideal \(\text{Im}\,\chi\) has grade \(r+1\).
(2) There exists a submodule \(U\) of \(M\) with the following properties:
(i) \(\text{rank}\, U=r-1\);
(ii) \(U\) is reflexive, orientable, and \(U_{\mathfrak p}\) is a free direct summand of \(M_{\mathfrak p}\) for all primes \({\mathfrak p}\) of \(R\) such that \(\text{grade}\,{\mathfrak p}\leq r\).
(3) \(m=1\) and \(r\) is odd.
The aim of this note is to give an explicit construction of \(U\) as a submodule of the free module \(R^{{n\choose 2}-1}\).

MSC:

13D02 Syzygies, resolutions, complexes and commutative rings
13D05 Homological dimension and commutative rings
13E15 Commutative rings and modules of finite generation or presentation; number of generators

Citations:

Zbl 0987.13006
PDFBibTeX XMLCite
Full Text: EuDML