In Auslander's 1962 ICM address, he announced some theorems regarding homological algebra over (unramified) regular local rings that relate to Tor-rigidity. The proofs of many of these results appeared in papers of Auslander shortly thereafter. However, a proof of one of the results never appeared; though a proof of part of the theorem in question appears in a paper of Bose (1985). In this talk, I will present this theorem, give a sketch of a proof, and provide some applications. This is joint work with Olgur Celikbas.