Abstract
In this paper we give elementary lemmas on the vanishing of Tor and Ext for modules over a commutative ring, which may be regarded as generalizations of known theorems on the rigidity of Tor for finitely generated modules over a complete intersection. In fact, these lemmas supply new theorems on the rigidity of Tor and Ext for finitely generated modules having finite complete intersection dimension. We also give several examples illustrating the fact that the hypotheses of these rigidity theorems are essential.
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