Matrix factorizations over elementary divisor domains

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 ${\mathrm{D}}_{\mathrm{s}ing}(A_n(p))$ of the local Artinian rings $A_n(p)=R/⟨p^n⟩$, where $p$ runs over the prime divisors of $W$ of order $n\ge 2$. This result holds even when $R$ is not Noetherian. The triangulated categories ${\mathrm{D}}_{\mathrm{s}ing}(A_n(p))$ 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.

37 pages