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[1] 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 $\mathbb {Z} /2$ 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

Categorical structure

References

Eisenbud, David (1980). "Homological Algebra on a Complete Intersection, with an Application to Group Respresentations" (PDF). Transactions of the American Mathematical Society. 260: 35–64. doi:10.1090/S0002-9947-1980-0570778-7. S2CID 27495286. Archived from the original (PDF) on 25 Feb 2020.