Complete intersections and equivalences with categories of matrix factorizations

We prove that one can realize certain triangulated subcategories of the singularity category of a complete intersection as homotopy categories of matrix factorizations. Moreover, we prove that for any commutative ring and non-zerodivisor, the homotopy category of matrix factorizations embeds into the homotopy category of totally acyclic complexes of finitely generated projective modules over the factor ring.