[Paper Review] Matrix factorizations with more than two factors

Tim Tribone|arXiv (Cornell University)|Feb 12, 2021

Algebraic structures and combinatorial models12 references4 citations

TL;DR

This paper generalizes matrix factorizations from two to $d \geq 2$ factors in a regular local ring, establishing that the stable category of such factorizations is triangulated with an explicit suspension functor. It extends Knörrer and Solberg's results by identifying the category of $d$-factor matrix factorizations with maximal Cohen-Macaulay modules over a skew group algebra, which behaves homologically like a non-commutative hypersurface ring with eventually 2-periodic resolutions.

ABSTRACT

Given an element $f$ in a regular local ring, we study matrix factorizations of $f$ with $d \ge 2$ factors, that is, we study tuples of square matrices $(φ_1,φ_2,\dots,φ_d)$ such that their product is $f$ times an identity matrix of the appropriate size. Several well known properties of matrix factorizations with $2$ factors extend to the case of arbitrarily many factors. For instance, we show that the stable category of matrix factorizations with $d\ge 2$ factors is naturally triangulated and we give explicit formula for the relevant suspension functor. We also extend results of Knörrer and Solberg which identify the category of matrix factorizations with the full subcategory of maximal Cohen-Macaulay modules over a certain non-commutative algebra $Γ$. As a consequence of our findings, we observe that the ring $Γ$ behaves, homologically, like a "non-commutative hypersurface ring" in the sense that every finitely generated module over $Γ$ has an eventually $2$-periodic projective resolution.

Motivation & Objective

• To extend the theory of matrix factorizations from two to $d \geq 2$ factors in a regular local ring.
• To show that the stable category of $d$-factor matrix factorizations is naturally triangulated with an explicit suspension functor.
• To generalize Knörrer and Solberg's equivalence between matrix factorizations and maximal Cohen-Macaulay modules over a non-commutative algebra $\Gamma$ to $d \geq 2$ factors.
• To demonstrate that the algebra $\Gamma$ is Iwanaga-Gorenstein and that all finitely generated modules over it have eventually 2-periodic projective resolutions.

Proposed method

• Define $d$-factor matrix factorizations as tuples $(\varphi_1, \dots, \varphi_d)$ of endomorphisms of free $S$-modules such that $\varphi_1 \cdots \varphi_d = f \cdot \text{id}$.
• Use the cyclic symmetry of the factorization condition to define morphisms and isomorphisms in the category $\text{MF}_S^d(f)$.
• Prove that $\text{MF}_S^d(f)$ is a Frobenius category, so its stable category inherits a natural triangulated structure.
• Construct an equivalence between $\text{MF}_S^d(f)$ and the category of maximal Cohen-Macaulay modules over the skew group algebra of the $d$-fold branched cover of $R = S/(f)$.
• Explicitly compute the suspension functor on the stable category using the cyclic permutation of the factorization components.
• Use determinant arguments and irreducibility of $f$ to bound the minimal size of indecomposable $d$-factor factorizations.

Experimental results

Research questions

• RQ1 How can matrix factorizations be generalized from two to $d \geq 2$ factors in a regular local ring?
• RQ2 Is the stable category of $d$-factor matrix factorizations naturally triangulated, and if so, what is the explicit form of the suspension functor?
• RQ3 Can the equivalence between matrix factorizations and maximal Cohen-Macaulay modules over a non-commutative algebra $\Gamma$ be extended to $d \geq 2$ factors?
• RQ4 Does the algebra $\Gamma$ associated to $d$-factor factorizations exhibit properties of a non-commutative hypersurface ring, such as Iwanaga-Gorenstein condition and eventually 2-periodic resolutions?
• RQ5 What constraints does the irreducibility of $f$ impose on the size of indecomposable $d$-factor matrix factorizations?

Key findings

• The stable category of $d$-factor matrix factorizations is naturally triangulated, with an explicit suspension functor given by cyclic permutation of the factorization components.
• The category $\text{MF}_S^d(f)$ is equivalent to the category of maximal Cohen-Macaulay modules over the skew group algebra of the $d$-fold branched cover of $R = S/(f)$, generalizing Knörrer's result.
• The algebra $\Gamma$ associated to $d$-factor factorizations is Iwanaga-Gorenstein, and every finitely generated $\Gamma$-module admits an eventually 2-periodic projective resolution.
• If $f$ is irreducible, any indecomposable $d$-factor matrix factorization of size less than $d$ must be pseudoprojective, so the minimal size of a non-pseudoprojective indecomposable is at least $d$.
• Explicit examples of indecomposable $3$-factor matrix factorizations are constructed for $\mathbb{E}_6$, $\mathbb{E}_7$, and $\mathbb{E}_8$ singularities, with entries in the maximal ideal, confirming their minimality and indecomposability.