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.

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 ${\displaystyle R=\mathbb {C} [x]/(x^{2})}$ there is an infinite resolution of the ${\displaystyle R}$-module ${\displaystyle \mathbb {C} }$ where

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

Instead of looking at only the derived category of the module category, David Eisenbud¹ studied such resolutions by looking at their periodicity. In general, such resolutions are periodic with period ${\displaystyle 2}$ after finitely many objects in the resolution.

For a commutative ring ${\displaystyle S}$ and an element ${\displaystyle f\in S}$, a matrix factorization of ${\displaystyle f}$ is a pair of n-by-n matrices ${\displaystyle A,B}$ such that ${\displaystyle AB=f\cdot {\text{Id}}_{n}}$. This can be encoded more generally as a ${\displaystyle \mathbb {Z} /2}$-graded ${\displaystyle S}$-module ${\displaystyle M=M_{0}\oplus M_{1}}$ with an endomorphism

${\displaystyle d={\begin{bmatrix}0&d_{1}\\d_{0}&0\end{bmatrix}}}$

such that ${\displaystyle d^{2}=f\cdot {\text{Id}}_{M}}$.

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

(2) If ${\displaystyle S=\mathbb {C} [[x,y,z]]}$ and ${\displaystyle f=xy+xz+yz}$, then there is a matrix factorization ${\displaystyle d_{0}:S^{2}\rightleftarrows S^{2}:d_{1}}$ where

${\displaystyle d_{0}={\begin{bmatrix}z&y\\x&-x-y\end{bmatrix}}{\text{ }}d_{1}={\begin{bmatrix}x+y&y\\x&-z\end{bmatrix}}}$

definition

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

${\displaystyle \cdots \to F_{2}\to F_{1}\to F_{0}\to 0}$

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

page 18 of eisenbud article