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Powers of the maximal ideal and vanishing of (co)homology
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We prove that each positive power of the maximal ideal of a commutative Noetherian local ring is Tor-rigid, and strongly-rigid. This gives new characterizations of regularity and, in particular, shows that such ideals satisfy the torsion condition of a long-standing conjecture of Huneke and Wiegand.
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On an example concerning the second rigidity theorem
Over a hypersurface ring, if M has finite projective dimension and N is locally free at primes of height at most one, then reflexivity of M tensor N implies both M and N are reflexive.
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