Orlov’s Equivalence and Tensor Products: from Sheaves to Matrix Factorizations and Back

A special case of a theorem due to Orlov states that for a hypersurface X ⊂ Pn−1 of degree n given by the equation W = 0, there exists an equivalence between the bounded derived category D(cohX) of coherent sheaves on X and the homotopy category HMF(W ) of graded matrix factorizations. We first give a description of this result, and present some methods for doing calculations with it. In the last two sections, we consider some possible product-like structures on these two triangulated categories, and provide possible directions for further inquiry as to what these product-like structures correspond to in either category, highlighting possible connections to tensor triangular geometry.