Generalized local cohomology and the canonical element conjecture

We study a generalization of the Canonical Element Conjecture. In particular we show that given a nonregular local ring (A,m) and an i>0, there exist finitely generated A-modules M such that the canonical map from ExtAi(M/mM,Syzi(M/mM)) to Hmi(M,Syzi(M/mM)) is nonzero. Moreover, we show that even when M has an infinite projective dimension and i>dim(A), studying these maps sheds light on the Canonical Element Conjecture.