The local homological conjectures

Summary

AcknowledgmeAnts

The author wishes to thank once more the organizers of the Durham Symposium, Professor D.G.Northcott and Dr. R. Y. Sharp, for putting together an excellent conference.

Notes taken by Dr.Sharp were a great aid to the author in preparing this article.

The author was supported in part by a grant from the National Science Foundation, U.S.A.

1. Rigidity, multiplicities, and other conjectures

What follows is a modified version of a series of three lectures presented during the Durham Symposium on Commutative Algebra, which was held during the period July 15–25, 1981. Some proofs given during those lectures but which are to appear elsewhere have been omitted, while others which were sketched or not given in the original lectures have been provided in greater detail here.

I would like first to recall two conjectures which I feel have played a pivotal role in the development of local homological commutative algebra, and which have inspired much work and many results in several directions, not all of which are “homological”.

In the sequel all rings are commutative, associative, with identity, all modules are unital, and “local ring” means Noetherian ring R with a unique maximal ideal m.

The first is as follows.

RIGIDITY CONJECTURE. Let R be a local ring, let M, N be finitely generated R–modules, and suppose that pd_RM is finite. Suppose that Tor^Ri(M, N) = O. Then Tor^R_j(M, N) = O for all j ≥ i.