Rigidity of Ext and Tor with Coefficients in Residue Fields of a Commutative Noetherian Ring

Abstract

AbstractLet 𝔭 be a prime ideal in a commutative noetherian ringR. It is proved that if anR-moduleMsatisfies${\rm Tor}_n^R $(k(𝔭),M) = 0 for somen⩾R𝔭, wherek(𝔭) is the residue field at 𝔭, then${\rm Tor}_i^R $(k(𝔭),M) = 0 holds for alli⩾n. Similar rigidity results concerning${\rm Tor}_R^{\ast} $(k(𝔭),M) are proved, and applications to the theory of homological dimensions are explored.