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, w
e 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 locally free coherent sheaves on a scheme. For a scheme that is projective over an affine scheme, we show that homomorphisms in the homotopy category of matrix factorizations may be computed as the hypercohomology of
a certain mapping complex. Using this explicit description, we give another proof of Orlovs theorem that there is a fully faithful embedding of the homotopy category of matrix factorizations into the singularity category of the corresponding zero subscheme. We also give a complete description of the image of this functor.
We study rings which have Noetherian cohomology under the action of a ring of cohomology operators. The main result is a criterion for a complex of modules over such a ring to have finite injective dimension. This criterion generalizes, by removing f
initeness conditions, and unifies several previous results. In particular we show that for a module over a ring with Noetherian cohomology, if all higher self-extensions of the module vanish then it must have finite injective dimension. Examples of rings with Noetherian cohomology include commutative complete intersection rings and finite dimensional cocommutative Hopf algebras over a field.
