We observe that there is an equivalence between the singularity category of an affine complete intersection and the homotopy category of matrix factorizations over a related scheme. This relies in part on a theorem of Orlov. Using this equivalence, we give a geometric construction of the ring of cohomology operators, and a generalization of the theory of support varieties, which we call stable support sets. We settle a question of Avramov about which stable support sets can arise for a given complete intersection ring. We also use the equivalence to construct a projective resolution of a module over a complete intersection ring from a matrix factorization, generalizing the well-known result in the hypersurface case.

We study matrix factorizations of regular global sections of line bundles on schemes. If the line bundle is very ample relative to a Noetherian affine scheme we show that morphisms in the homotopy category of matrix factorizations may be computed as the hypercohomology of a certain mapping complex. Using this explicit description, we prove an analogue of Orlov's theorem that there is a fully faithful embedding of the homotopy category of matrix factorizations into the singularity category of the corresponding zero subscheme. Moreover, we give a complete description of the image of this functor.

No abstract

We give a complete description of the cone of Betti diagrams over a standard graded hypersurface ring of the form k[x, y]/ q , where q is a homogeneous quadric. We also provide a finite algorithm for decomposing Betti diagrams, including diagrams of infinite projective dimension, into pure diagrams. Boij-Söderberg theory completely describes the cone of Betti diagrams over a standard graded polynomial ring; our result provides the first example of another graded ring for which the cone of Betti diagrams is entirely understood. * β R 0,0 (M ) β R