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

there is an infinite resolution of the

-module

where

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

Instead of looking at only the derived category of the module category, David Eisenbud^[1] studied such resolutions by looking at their periodicity. In general, such resolutions are periodic with period

after finitely many objects in the resolution.

For a commutative ring

and an element

, a matrix factorization of

is a pair of

square matrices

such that

. This can be encoded more generally as a

graded

-module

with an endomorphism

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

such that

.

(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 ${\displaystyle 0\le i\le n}$.

(2) If

and

, then there is a matrix factorization

where

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

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

This section needs expansion. You can help by adding to it. (February 2022)

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

• 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

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