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Abstract Let 𝔭 be a prime ideal in a commutative noetherian ring R . It is proved that if an R -module M satisfies $${\rm Tor}_n^R $$ ( k (𝔭), M ) = 0 for some n ⩾ R 𝔭 , where k (𝔭) is the residue field at 𝔭, then $${\rm Tor}_i^R $$ ( k (𝔭), M ) = 0 holds for all i ⩾ n . Similar rigidity results concerning $${\rm Tor}_R^{\ast} $$ ( k (𝔭), M ) are proved, and applications to the theory of homological dimensions are explored.
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The modular group algebra of an elementary abelian p-group is isomorphic to the restricted enveloping algebra of a commutative restricted Lie algebra. The different ways of regarding this algebra result in different Hopf algebra structures that determine cup products on cohomology of modules. However, it is proved in this paper that the products with elements of the polynomial subring of the cohomology ring generated by the Bocksteins of the degree one elements are independent of the choice of these coalgebra structures.more » « less
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