Some dual homological results for modules over commutative rings (Q919090)

Let R be a commutative ring, M, N, E R-modules, \(\phi: M\to N\) an R- linear map. It is shown that if Hom(\(\phi\),E) is injective, then Hom(\(\bigwedge^{n}\phi, E)\) is also injective, for any natural number n. This is used to prove a criterion for \(Ext^ i_ R(M,E)\) to be zero, for all i less than a fixed natural number, generalizing a result of \textit{D. Rees} [Proc. Camb. Philos. Soc. 53, 28-42 (1957; Zbl 0079.266)]. It is pointed out that each of the previous results has dualizations, considering \(-\otimes E\) instead of \(Hom(-,E).\) This leads to an interesting dualization of the Buchsbaum-Eisenbud criterion for the exactness of a complex.