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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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.