Free Direct Summands of Maximal Rank and Rigidity in Projective Dimension Two

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.

Key Words:

• Free rank
• Rigidity
• Universal resolutions

Mathematics Subject Classification:

• Primary 13C05
• Secondary 13D02, 13D07, 13D22

Notes

Communicated by I. Swanson.