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.

## Contents

* 1 Motivation
* 2 Definition
* 2.1 Examples
* 3 Periodicity
* 3.1 Main theorem
* 3.2 Maximal Cohen-Macaulay modules
* 4 Categorical structure
* 5 Support of matrix factorizations
* 6 See also
* 7 References
* 8 Further reading

## 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=ℂ[x]/(x2)

there is an infinite resolution of the

-module

where

> ⋯→⋅xR→⋅xR→⋅xR->ℂ->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.

## Definition

For a commutative ring

and an element

f∈S

, a matrix factorization of

is a pair of

n×n

square matrices

A,B

such that

AB=f⋅Idn

. This can be encoded more generally as a

ℤ/2

graded

-module

M=M0⊕M1

with an endomorphism

> d=[0d1d00]

such that

d2=f⋅IdM

### Examples

(1) For S=ℂx and f=xn there is a matrix factorization d0:S⇄S:d1 where d0=xi,d1=xn−i for 0≤i≤n.

(2) If

S=ℂx,y,z

and

f=xy+xz+yz

, then there is a matrix factorization

d0:S2⇄S2:d1

where

> d0=[zyx−x−y] d1=[x+yyx−z]

## Periodicity

definition

### Main theorem

Given a regular local ring R and an ideal I⊂R generated by an A-sequence, set B=A/I and let

⋯->F2->F1->F0->0

be a minimal B-free resolution of the ground field. Then F∙ becomes periodic after at most 1+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)

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## Support of matrix factorizations

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

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## 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". Transactions of the American Mathematical Society 260: 35–64. doi:10.1090/S0002-9947-1980-0570778-7. https://www.ams.org/journals/tran/1980-260-01/S0002-9947-1980-0570778-7/S0002-9947-1980-0570778-7.pdf.

## 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

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