Decent intersection and Tor-rigidity for modules over local hypersurfaces

Abstract

We study two properties for a pair of finitely generated modules over a local hypersurface R R : decency, which is close to proper intersection of the supports, and Tor \operatorname {Tor} -rigidity. We show that the vanishing of Hochster’s function θ R ( M , N ) \theta ^R(M,N) , known to imply decent intersection, also implies rigidity. We investigate the vanishing of θ R ( M , N ) \theta ^R(M,N) to obtain new results about decency and rigidity over hypersurfaces. Many applications are given.