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Torsion-free modules and syzygies. (English) Zbl 0820.13011

Let \(R\) be a Noetherian ring and \((F_ \bullet, \varphi)\) a finite free resolution of length \(n\) of the \(R\)-module \(M\). Let \(I (\varphi_ i)\) denote the ideal generated by the minors of \(\varphi_ i\) of size \(\text{rank} (\varphi_ i)\). A simple inductive proof is given that \(M\) is torsion-free if and only if \(\text{depth} (I (\varphi_ i)) \geq i + 1\) for \(i = 1, \dots, n\). Several known results are shown to follow from this. These include the Buchsbaum-Eisenbud exactness criterion, and conditions for a module to be \(k\)-th syzygy. It is also shown that if \(M\) is a finitely generated module of finite projective dimension over a local ring \((R,m)\), and \(M\) is a \(k\)-th syzygy of rank \(k + s\) with \(s > 1\), then \(M\) has a free submodule \(F\) of rank \(s\) such that \(F \cap mM = mF\) and \(M/F\) is a \(k\)-th syzygy. This improves a result of W. Bruns [cf. Commun. Algebra 15, 873-925 (1987; Zbl 0623.13006)] by removing his hypothesis that \(R\) is Cohen-Macaulay.

MSC:

13D02 Syzygies, resolutions, complexes and commutative rings
13C40 Linkage, complete intersections and determinantal ideals
13D15 Grothendieck groups, \(K\)-theory and commutative rings
13E05 Commutative Noetherian rings and modules

Citations:

Zbl 0623.13006
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