Elementary matrix factorizations over B\'ezout domains

TL;DR

This paper investigates the structure and classification of elementary matrix factorizations over Bézout domains, providing criteria for isomorphisms, formulas for class counts, and exploring their relation to the broader homotopy category of matrix factorizations.

Contribution

It offers new criteria for isomorphism detection, formulas for counting classes, and insights into the structure of elementary factorizations within the homotopy category over Bézout domains.

Findings

Criteria for detecting isomorphisms in homotopy categories

Formulas for the number of isomorphism classes of elementary factorizations

The subcategory of elementary factorizations is Krull-Schmidt and possibly equals the entire homotopy category

Abstract

We study the homotopy category $\mathrm{hef}(R,W)$ (and its $\mathbb{Z}_2$-graded version $\mathrm{HEF}(R,W)$) of elementary factorizations, where $R$ is a B\'ezout domain which has prime elements and $W=W_0 W_c$, where $W_0\in R^\times$ is a square-free element of $R$ and $W_c\in R^\times$ is a finite product of primes with order at least two. In this situation, we give criteria for detecting isomorphisms in $\mathrm{hef}(R,W)$ and $\mathrm{HEF}(R,W)$ and formulas for the number of isomorphism classes of objects. We also study the full subcategory $\mathbf{hef}(R,W)$ of the homotopy category $\mathrm{hmf}(R,W)$ of finite rank matrix factorizations of $W$ which is additively generated by elementary factorizations. We show that $\mathbf{hef}(R,W)$ is Krull-Schmidt and we conjecture that it coincides with $\mathrm{hmf}(R,W)$. Finally, we discuss a few classes of examples.