Homology over a Complete Intersection Ring via the Generic Hypersurface

We study homological properties and constructions for modules over a complete intersection ring $Q/(f_1,\ldots,f_c)$ by way of the related generic hypersurface ring $Q[T_1,\ldots,T_c]/(f_1T_1+\cdots+f_cT_c)$. The advantage of this approach is that over a hypersurface ring, free resolutions are eventually 2-periodic, given by matrix factorizations, and are thus relatively easy to understand. We approach this relationship in two ways. First, we give a correspondence between the two rings in the graded setting, where existing results are insufficient for preserving graded structures. As an application, we use this correspondence to move a functor appearing in a theorem of Orlov to the generic hypersurface setting. Second, we shift out of the graded setting to discuss the relationship between Tor groups over these rings, inspired by recent work of Bergh and Jorgensen, and building on cohomological results of Burke and Walker. This second part takes place in a scheme-theoretic context, so we develop some machinery that provides a sort of ``global Tor for complexes of sheaves that can be compared to the usual Tor for modules