Let R and S be arbitrary associative rings. Given a bimodule RWS , we denote by ∆? and Γ? the functors Hom?(−, W ) and Ext?(−, W ), where ? = R or S. We say that RWS is a finitistic weakly cotilting bimodule (briefly FWC) if for each module M cogenerated by W , finitely generated or homomorphic image of a finite direct sum of copies of W , ΓM = 0 = Ext(M, W ). We are able to describe, on a larg...

It is becoming increasingly difficult for geometers and even physicists to avoid papers containing phrases like “triangulated category”, not to mention derived functors. I will give some motivation for such things from algebraic geometry, and show how the concepts are already familiar from topology. This gives a natural and simple way to look at cohomology and other scary concepts in homologica...

It is becoming increasingly difficult for geometers and even physicists to avoid papers containing phrases like “triangulated category”, not to mention derived functors. I will give some motivation for such things from algebraic geometry, and show how the concepts are already familiar from topology. This gives a natural and simple way to look at cohomology and other scary concepts in homologica...

It was the best of times, it was the worst of times, it was the age of covariance, it was the age of contravariance, it was the epoch of homology, it was the epoch of cohomology, it was the season of Ext, it was the season of Tor, it was the spring of short exact sequences, it was the winter of long exact sequences, we had right exactness, we had left exactness, our arrows were all going in one...

We prove that if M , N are finite modules over a Gorenstein local ring R of codimension at most 4, then the vanishing of Ext R (M,N) for n ≫ 0 is equivalent to the vanishing of Ext R (N,M) for n ≫ 0. Furthermore, if b R has no embedded deformation, then such vanishing occurs if and only if M or N has finite projective dimension.