ABSTRACT
Let M be a finitely generated module of rank r and finite projective dimension over a commutative Noetherian local ring S. We show that M has a free direct summand of rank r if and only if its Buchsbaum–Eisenbud invariant J(M) is a principal ideal. As an application, we consider the problem of finding rigid triples. Recall that the module M is said to be rigid if for each i ≥ 0 and each finitely generated S-module N the condition implies
for every j ≥ i. A triple of positive integers b = (b
0, b
1, b
2) is said to be rigid if for every commutative Noetherian local ring S one has that every S-module with minimal free resolution of the form 0 → S
b
2
→ S
b
1
→ S
b
0
→ 0 is rigid. We use our criterion for free direct summands of maximal rank to derive a necessary and sufficient condition for a triple b to be rigid in terms of the rigidity of the universal modules of type b over complete regular local rings. In particular, we exhibit the rigidity of any triple of the form (k, m + 1,m), thus providing a new class of rigid modules.
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Notes
Communicated by I. Swanson.