Complete intersections and equivalences with categories of matrix factorizations

TL;DR

This paper establishes a connection between triangulated subcategories of singularity categories of complete intersections and homotopy categories of matrix factorizations, also exploring embeddings into acyclic complexes.

Contribution

It demonstrates that certain subcategories of singularity categories can be realized as homotopy categories of matrix factorizations and shows embeddings into acyclic complexes for any commutative ring with a non-zerodivisor.

Findings

Triangulated subcategories of singularity categories are realizable as matrix factorizations.

Homotopy category of matrix factorizations embeds into totally acyclic complexes.

Results apply to any commutative ring with a non-zerodivisor.

Abstract

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.