On the Vanishing of Ext

Mark Ramras
1971 Proceedings of the American Mathematical Society  
In this paper we exhibit certain modules^! over a commutative noetherian local ring (R, SÏÏÎ) which test projective dimension of finitely generated modules in the following sense: if Ext'CM", A) =0 for all j^i, then pd M<i. We also show that the module 9JÎ tests in a stronger way: if Ext'CM", m) =0, then pd M<i. In conclusion we show that if (R, 5K) is artin, then R is self-injective if and only if Ext'Crc/SJÎ". R)=0, where the index of nilpotence of 2JÎ is « + 1. Let (R, 9JÎ) be a commutative,
more » ... noetherian, local ring with maximal ideal S0Î. It is well known that for any finitely generated i?-moduIe
doi:10.2307/2036475 fatcat:coipmr525zfgvffi5vwrwx24ei