Matrix factorizations over elementary divisor domains

TL;DR

This paper investigates the structure of the homotopy category of matrix factorizations over elementary divisor domains, revealing a decomposition into categories of singularities and providing explicit descriptions, even for non-Noetherian rings.

Contribution

It establishes a decomposition of the homotopy category into singularity categories for certain elementary divisor domains, extending results to non-Noetherian cases and explicitly describing these categories.

Findings

Decomposition of homotopy categories into singularity categories

Explicit descriptions of triangulated categories of singularities

Additive generation of cocycle categories by elementary factorizations

Abstract

We study the homotopy category $\mathrm{hmf}(R,W)$ of matrix factorizations of non-zero elements $W\in R^\times$, where $R$ is an elementary divisor domain. When $R$ has prime elements and $W$ factors into a square-free element $W_0$ and a finite product of primes of multiplicity greater than one and which do not divide $W_0$, we show that $\mathrm{hmf}(R,W)$ is triangle-equivalent with an orthogonal sum of the triangulated categories of singularities $\mathrm{D}_{\mathrm sing}(A_n(p))$ of the local Artinian rings $A_n(p)=R/\langle p^n\rangle$, where $p$ runs over the prime divisors of $W$ of order $n\geq 2$. This result holds even when $R$ is not Noetherian. The triangulated categories $\mathrm{D}_{\mathrm sing}(A_n(p))$ are Krull-Schmidt and we describe them explicitly. We also study the cocycle category $\mathrm{zmf}(R,W)$, showing that it is additively generated by elementary matrix factorizations. Finally, we discuss a few classes of examples.