Matrix factorization (algebra) explained

In homological algebra, a branch of mathematics, a matrix factorization is a tool used to study infinitely long resolutions, generally over commutative rings.

Motivation

R=C[x]/(x2)

there is an infinite resolution of the

R

-module

C

where

\xrightarrow{x}R\xrightarrow{x}R\xrightarrow{x}R\toC\to0

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

2

after finitely many objects in the resolution.

Definition

For a commutative ring

S

and an element

f\inS

, a matrix factorization of

f

is a pair of n-by-n matrices

A,B

such that

AB=fIdn

. This can be encoded more generally as a

Z/2

-graded

S

-module

M=M0 ⊕ M1

with an endomorphism

d=\begin{bmatrix}0&d1\d0&0\end{bmatrix}

such that

d2=fIdM

.

Examples

(1) For

S=C[[x]]

and

f=xn

there is a matrix factorization

d0:S\rightleftarrowsS:d1

where
i,
d
0=x

d1=xn-i

for

0\leqi\leqn

.

(2) If

S=C[[x,y,z]]

and

f=xy+xz+yz

, then there is a matrix factorization
2
d
0:S

\rightleftarrows

2:d
S
1
where

d0=\begin{bmatrix}z&y\x&-x-y\end{bmatrix}d1=\begin{bmatrix}x+y&y\x&-z\end{bmatrix}

Periodicity

definition

Main theorem

R

and an ideal

I\subsetR

generated by an

A

-sequence, set

B=A/I

and let

\toF2\toF1\toF0\to0

be a minimal

B

-free resolution of the ground field. Then

F\bullet

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

See also

Further reading

Notes and References

  1. Eisenbud. David. Homological Algebra on a Complete Intersection, with an Application to Group Respresentations. Transactions of the American Mathematical Society. 1980 . 260. 35–64. 10.1090/S0002-9947-1980-0570778-7 . 27495286 . https://web.archive.org/web/20200225190215/https://www.ams.org/journals/tran/1980-260-01/S0002-9947-1980-0570778-7/S0002-9947-1980-0570778-7.pdf. 25 Feb 2020.