Matrix factorization (algebra)

In mathematics, matrix factorizations are tools used to study infinitely long resolutions in homological algebra, 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 ${\displaystyle R=\mathbb {C} [x]/(x^{2})}$  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 Eisenbud^[1] 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.

Definition

For a commutative ring ${\displaystyle S}$  and an element ${\displaystyle f\in S}$ , a matrix factorization of ${\displaystyle f}$  is a pair of ${\displaystyle n\times n}$  square 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 ${\displaystyle d^{2}=f\cdot {\text{Id}}_{M}}$ .

Examples

(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

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

Periodicity

definition

Main theorem

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

Maximal Cohen-Macaulay modules

page 18 of eisenbud article

Categorical structure

Support of matrix factorizations

References

1. Eisenbud, David. "Homological Algebra on a Complete Intersection, with an Application to Group Respresentations" (PDF). Transactions of the American Mathematical Society. 260: 35–64. Archived from the original (PDF) on 25 Feb 2020.

See also

• Derived category
• Homological algebra
• Triangulated category

External links

• Homological Algebra on a Complete Intersection with an Application to Group Representations
• Geometric Study of the Category of Matrix Factorizations