  Let p be a prime ideal in a commutative noetherian ring R and denote by k(p)
the residue field of the local ring R_p. We prove that if an R-module M
satisfies Ext_R^n(k(p),M) = 0 for some n >= dim R, then Ext_R^i(k(p),M) = 0
holds for all i >= n. This improves a result of Christensen, Iyengar, and
Marley by lowering the bound on n. We also improve existing results on
Tor-rigidity. This progress is driven by the existence of minimal
semi-flat-cotorsion replacements in the derived category as recently proved by
Nakamura and Thompson.
