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Categorical matrix factorizations and monomorphism categories

Jonas Frank, Mathias Schulze

TL;DR

This work develops a purely categorical generalization of Eisenbud–Yoshino’s matrix factorization correspondence by replacing hypersurface rings with a general exact category $\mathcal{A}$ equipped with a twisted homothety $(\tau,\omega)$. It defines hypersurface categories $\mathcal{A}/\omega$ and studies factorizations over Frobenius subcategories, showing these form a Frobenius category and giving a triangle equivalence between their stable category and the stable category of chains of monomorphisms in $\mathcal{E}_{\omega}$. The core construction uses a generalized cokernel functor and a suite of adjunctions between factorization diagrams and monomorphism chains, and it interprets factorizations as comma/diagram categories to connect to derivations in $\mathcal{E}_{\omega}$. Under suitable Ext-vanishing conditions, the main result yields a triangle equivalence between $\ul{\mathrm{Fac}}^{\mathcal{E}}_{l+1}(\omega)$ and $\ul{\mathrm{Mor}}^{\mathrm{m}}_{l-1}(\mathcal{E}_{\omega})$, unifying previous module-based results and extending to multi-factor factorizations. This framework provides a powerful categorical lens for understanding matrix factorizations and their relations to representation theory via Frobenius and Morita-theoretic structures.

Abstract

This article generalizes the correspondence between matrix factorizations and maximal Cohen-Macaulay modules over hypersurface rings due to Eisenbud and Yoshino. We consider factorizations with several factors in a purely categorical context, extending results of Sun and Zhang for Gorenstein projective module factorizations. Our formulation relies on a notion of hypersurface category and replaces Gorenstein projectives by objects of general Frobenius exact subcategories. We show that factorizations over such categories form again a Frobenius category. Our main result is then a triangle equivalence between the stable category of factorizations and that of chains of monomorphisms.

Categorical matrix factorizations and monomorphism categories

TL;DR

This work develops a purely categorical generalization of Eisenbud–Yoshino’s matrix factorization correspondence by replacing hypersurface rings with a general exact category equipped with a twisted homothety . It defines hypersurface categories and studies factorizations over Frobenius subcategories, showing these form a Frobenius category and giving a triangle equivalence between their stable category and the stable category of chains of monomorphisms in . The core construction uses a generalized cokernel functor and a suite of adjunctions between factorization diagrams and monomorphism chains, and it interprets factorizations as comma/diagram categories to connect to derivations in . Under suitable Ext-vanishing conditions, the main result yields a triangle equivalence between and , unifying previous module-based results and extending to multi-factor factorizations. This framework provides a powerful categorical lens for understanding matrix factorizations and their relations to representation theory via Frobenius and Morita-theoretic structures.

Abstract

This article generalizes the correspondence between matrix factorizations and maximal Cohen-Macaulay modules over hypersurface rings due to Eisenbud and Yoshino. We consider factorizations with several factors in a purely categorical context, extending results of Sun and Zhang for Gorenstein projective module factorizations. Our formulation relies on a notion of hypersurface category and replaces Gorenstein projectives by objects of general Frobenius exact subcategories. We show that factorizations over such categories form again a Frobenius category. Our main result is then a triangle equivalence between the stable category of factorizations and that of chains of monomorphisms.

Paper Structure

This paper contains 6 sections, 32 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Categories of monomorphisms
  3. Hypersurface categories
  4. Categories of factorizations
  5. Equivalence with comma categories
  6. Categorical Eisenbud-Yoshino Theorem

Key Result

Theorem 1.2

Let $S$ be a commutative ring and $R = S/\langle f_1, \dots, f_k \rangle$, where $f_1, \dots, f_k$ is an $S$-regular sequence. Then $\mathop{\mathrm{gdim}}\nolimits_S(M) = \mathop{\mathrm{gdim}}\nolimits_{R}(M)+k$ for any finitely generated $R$-module $M$.

Theorems & Definitions (78)

  • Theorem 1.2: Change of Rings, Chr00
  • Theorem A
  • Theorem B
  • Theorem C
  • Proposition 2.1: Buh10
  • Proposition 2.2: Buh10
  • Lemma 2.3: Noether lemma, Buh10
  • Definition 2.4
  • Lemma 2.5: FS24
  • Proposition 2.6: Buh10
  • ...and 68 more