Algebra/Topology seminar

Speaker: Luchezar Avramov

Title:  Vanishing of (co)homology over commutative noetherian rings

Abstract: Finiteness of the projective dimension of a finitely generated module $M$ over a commutative ring $R$ is characterized by the vanishing of $Ext_R^i(M,N)$ or $Tor^R_i(M,N)$ for $i\gg0$ and for all finitely generated $R$-modules $N$.  For a number of reasons, it is important to be able to recognize that property from vanishing against a single $N$. Forty years ago, Auslander and Reiten conjectured that if$Ext_R^i(M,R\oplus M)=0$ for $i\gg0$, then $M$ has finite projective dimension.  The question, whether this also follows from $Tor^R_i(M,M)=0$ for $i\gg0$, was raised only recently. Results on that question will be discussed.