Matrix factorizations in higher codimension

Abstract

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.

Complete intersections and equivalences with categories of matrix factorizations

“…Along the way we also prove a more general embedding result analogous to [Or2,Theorem 1] and [BuW,Example B.5]. It shows 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.…”

“…However, first we show 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. This result is analogous to [Or2,Theorem 1] and [BuW,Example B.5], and will be established through a series of results.…”

Motivic and $$\ell $$-adic realizations of the category of singularities of the zero locus of a global section of a vector bundle

“…One can view the former generalization as a particular case of the latter thanks to a theorem of Orlov [42] and Burke and Walker [11], which tells us that the dg category of singularities of (X , π • p) is equivalent to the dg category of singularities of (P n−1 X , (π…”

D-brane probes, branched double covers, and noncommutative resolutions

“…The sheafy matrix factorizations that we will utilize have been discusssed in a few recent mathematics papers (see e.g. [10,29,30,31,32,33]), but not often, and we will need a number of properties of these matrix factorizations. Since they are still not entirely common, and we will need a number of results, in this section we will review their properties.…”