Matrix Factorization (Algebra)

In homological algebra, a branch of mathematics, a matrix factorization is a tool used to study infinitely long resolutions, generally over commutative rings.

Motivation

One of the problems with non-smooth algebras, such as Artin algebras, are their derived categories are poorly behaved due to infinite projective resolutions. For example, in the ring [math]\displaystyle{ R = \mathbb{C}[x]/(x^2) }[/math] there is an infinite resolution of the [math]\displaystyle{ R }[/math]-module [math]\displaystyle{ \mathbb{C} }[/math] where

[math]\displaystyle{ \cdots \xrightarrow{\cdot x} R \xrightarrow{\cdot x} R \xrightarrow{\cdot x} R \to \mathbb{C} \to 0 }[/math]

Instead of looking at only the derived category of the module category, David Eisenbud^[1] studied such resolutions by looking at their periodicity. In general, such resolutions are periodic with period [math]\displaystyle{ 2 }[/math] after finitely many objects in the resolution.

Definition

For a commutative ring [math]\displaystyle{ S }[/math] and an element [math]\displaystyle{ f \in S }[/math], a matrix factorization of [math]\displaystyle{ f }[/math] is a pair of [math]\displaystyle{ n\times n }[/math] square matrices [math]\displaystyle{ A,B }[/math] such that [math]\displaystyle{ AB = f \cdot \text{Id}_n }[/math]. This can be encoded more generally as a [math]\displaystyle{ \mathbb{Z}/2 }[/math] graded [math]\displaystyle{ S }[/math]-module [math]\displaystyle{ M = M_0\oplus M_1 }[/math] with an endomorphism

[math]\displaystyle{ d = \begin{bmatrix}0 & d_1 \\ d_0 & 0 \end{bmatrix} }[/math]

such that [math]\displaystyle{ d^2 = f \cdot \text{Id}_M }[/math].

Examples

(1) For [math]\displaystyle{ S = \mathbb{C}x }[/math] and [math]\displaystyle{ f = x^n }[/math] there is a matrix factorization [math]\displaystyle{ d_0:S \rightleftarrows S:d_1 }[/math] where [math]\displaystyle{ d_0=x^i, d_1 = x^{n-i} }[/math] for [math]\displaystyle{ 0 \leq i \leq n }[/math].

(2) If [math]\displaystyle{ S = \mathbb{C}x,y,z }[/math] and [math]\displaystyle{ f = xy + xz + yz }[/math], then there is a matrix factorization [math]\displaystyle{ d_0:S^2 \rightleftarrows S^2:d_1 }[/math] where

[math]\displaystyle{ d_0 = \begin{bmatrix} z & y \\ x & -x-y \end{bmatrix} \text{ } d_1 = \begin{bmatrix} x+y & y \\ x & -z \end{bmatrix} }[/math]

Periodicity

definition

Main theorem

Given a regular local ring [math]\displaystyle{ R }[/math] and an ideal [math]\displaystyle{ I \subset R }[/math] generated by an [math]\displaystyle{ A }[/math]-sequence, set [math]\displaystyle{ B = A/I }[/math] and let

[math]\displaystyle{ \cdots \to F_2 \to F_1 \to F_0 \to 0 }[/math]

be a minimal [math]\displaystyle{ B }[/math]-free resolution of the ground field. Then [math]\displaystyle{ F_\bullet }[/math] becomes periodic after at most [math]\displaystyle{ 1 + \text{dim}(B) }[/math] steps. https://www.youtube.com/watch?v=2Jo5eCv9ZVY

Maximal Cohen-Macaulay modules

page 18 of eisenbud article

Categorical structure

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Support of matrix factorizations

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See also

• Derived noncommutative algebraic geometry
• Derived category
• Homological algebra
• Triangulated category

References

Further reading

• Homological Algebra on a Complete Intersection with an Application to Group Representations
• Geometric Study of the Category of Matrix Factorizations
• https://web.math.princeton.edu/~takumim/takumim_Spr13_JP.pdf
• https://arxiv.org/abs/1110.2918

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