Misplaced Pages

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 ${\displaystyle R=\mathbb {C} /(x^{2})}$ 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 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 n-by-n 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 ${\displaystyle S=\mathbb {C} ]}$ 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 ${\displaystyle S=\mathbb {C} ]}$ 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

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

Support of matrix factorizations

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

See also

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

References

1. 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.

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