  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.
