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

$R=\mathbb{ℂ}[x]/(x^2)$

there is an infinite resolution of the

$R$

-module

$\mathbb{ℂ}$

where

$\cdots \stackrel{\cdot x}{\to }R\stackrel{\cdot x}{\to }R\stackrel{\cdot x}{\to }R\to \mathbb{ℂ}\to 0$

Instead of looking at only the derived category of the module category, David Eisenbud studied such resolutions by looking at their periodicity. In general, such resolutions are periodic with period

$2$

after finitely many objects in the resolution.

Definition

For a commutative ring

$S$

and an element

$f\in S$

, a matrix factorization of

$f$

is a pair of

$n\times n$

square matrices

$A,B$

such that

$AB=f\cdot {\text{Id}}_n$

. This can be encoded more generally as a

$\mathbb{\mathbb{Z}}/2$

graded

$S$

-module

$M=M_0\oplus M_1$

with an endomorphism

$d=\left[\begin{array}{cc}0& d_1\\ d_0& 0\end{array}\right]$

such that

$d^2=f\cdot {\text{Id}}_M$

.

Examples

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

(2) If

$S=\mathbb{ℂ}x,y,z$

and

$f=xy+xz+yz$

, then there is a matrix factorization

$d_0:S^2\rightleftarrows S^2:d_1$

where

$d_0=\left[\begin{array}{cc}z& y\\ x& -x-y\end{array}\right]d_1=\left[\begin{array}{cc}x+y& y\\ x& -z\end{array}\right]$

Periodicity

definition

Main theorem

Given a regular local ring $R$ and an ideal $I\subset R$ generated by an $A$-sequence, set $B=A/I$ and let

$\cdots \to F_2\to F_1\to F_0\to 0$

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

Maximal Cohen-Macaulay modules

page 18 of eisenbud article

See also

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

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