Some observations on local and projective hypersurfaces

Let $R$ be a hypersurface in an equicharacteristic or unramified regular local ring. For a pair of modules $(M,N)$ over $R$ we study applications of rigidity of $\Tor^R(M,N)$, based on ideas by Huneke, Wiegand and Jorgensen. We then focus on the hypersurfaces with isolated singularity and even dimension, and show that modules over such rings behave very much like those over regular local rings. Connections and applications to projective hypersurfaces such as intersection dimension of subvarieties and cohomological criterion for splitting of vector bundles are discussed.