Denseness of the restricted Knörrer functor on matrix factorizations

Determine whether the restricted functor Kn: underline{MF(A; ω)} → underline{MF(A2; y^2 − x^2 + ω)} is dense in general for arbitrary rings A with automorphisms σ and τ satisfying τ(ω)=ω and τ^2=σ, assuming 2 is invertible in A. Equivalently, ascertain whether every object in the stable category of matrix factorizations underline{MF(A2; y^2 − x^2 + ω)} is a retract of an object in the essential image of Kn under these hypotheses.

Background

The paper establishes a noncommutative Knörrer periodicity equivalence Kn between the stable categories of projective-module factorizations PF(A; ω) and PF(A2; y2 − x2 + ω), under the assumption that 2 is invertible in A. It provides an explicit quasi-inverse A1⊗_{A2}− for projective-module factorizations.

The equivalence Kn restricts to the stable categories of matrix factorizations MF up to retracts, but the authors point out that the quasi-inverse does not restrict to matrix factorizations. They explicitly state that the denseness of the restricted functor on matrix factorizations is unknown in general, although it holds when underline{MF(A; ω)} is idempotent-split, yielding a triangle equivalence in that case.

References

The quasi-inverse $A_1\otimes_{A_2}-$ above is brand new, since it does not restrict to matrix factorizations. We do not know the denseness of the restricted functor of ${\rm Kn}$ between matrix factorizations, in general. If the stable category $\underline{\mathbf{MF}(A; \omega)$ is idempotent-split, the denseness holds true, and thus we obtain a triangle equivalence between matrix factorizations.

— Noncommutative Knörrer periodicity via equivariantization  (2509.05725 - Chen et al., 6 Sep 2025) in Introduction, after Theorem I