Matrix Factorizations (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 (x^2) there is an infinite resolution of the $R$-module $\mathbb$ where

$\cdots \xrightarrow R \xrightarrow R \xrightarrow R \to \mathbb \to 0$

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 $AB = f \cdot \text_n$. This can be encoded more generally as a $\mathbb/2$ graded $S$-module $M = M_0\oplus M_1$ with an endomorphism

$d = \begin0 & d_1 \\ d_0 & 0 \end$

such that $d^2 = f \cdot \text_M$.

Examples

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

(2) If S = \mathbb x,y,z and $f = xy + xz + yz$, then there is a matrix factorization $d_0:S^2 \rightleftarrows S^2:d_1$ where

$d_0 = \begin z & y \\ x & -x-y \end \text d_1 = \begin x+y & y \\ x & -z \end$

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 $1 + \text(B)$ steps. https://www.youtube.com/watch?v=2Jo5eCv9ZVY

Maximal Cohen-Macaulay modules

page 18 of eisenbud article

Categorical structure

Support of matrix factorizations

See also

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

References

{{Reflist

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

Homological algebra