Abstract
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 Ext A i ( M / m M , Syz i ( M / m M ) ) to H m i ( M , Syz i ( M / m M ) ) 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.
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