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"$\cdots \xrightarrow{\cdot x} R \xrightarrow{\cdot x} R \xrightarrow{\cdot x} R \to \mathbb{C} \to 0$"Instead of looking at only the derived category of the module category, David 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-by-n matrices $$A,B$$ such that $$AB = f \cdot \text{Id}_n$$. This can be encoded more generally as a $$\mathbb{Z}/2$$-graded $$S$$-module $$M = M_0\oplus M_1$$ with an endomorphism"$d = \begin{bmatrix}0 & d_1 \\ d_0 & 0 \end{bmatrix}$"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 $$0 \leq i \leq n$$.

(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"$d_0 = \begin{bmatrix} z & y \\ x & -x-y \end{bmatrix} \text{ } d_1 = \begin{bmatrix} x+y & y \\ x & -z \end{bmatrix}$"

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