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.

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 $R = \mathbb{C}[x]/(x^2)$ there is an infinite resolution of the $R$ -module $\mathbb{C}$ where

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

Instead of looking at only the derived category of the module category, David Eisenbud{{Cite journal|last=Eisenbud|first=David|title=Homological Algebra on a Complete Intersection, with an Application to Group Respresentations|url=https://www.ams.org/journals/tran/1980-260-01/S0002-9947-1980-0570778-7/S0002-9947-1980-0570778-7.pdf|journal=Transactions of the American Mathematical Society|year=1980 |volume=260|pages=35–64|doi=10.1090/S0002-9947-1980-0570778-7 |s2cid=27495286 |archive-url=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|archive-date=25 Feb 2020|via=}} 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 $f \in S$ , a matrix factorization of $f$ is a pair of n-by-n matrices $A,B$ such that $AB = f \cdot \text{Id}_n$ . This can be encoded more generally as a $\mathbb{Z}/2$ -graded $S$ -module $M = M_0\oplus M_1$ with an endomorphism

$d = \begin{bmatrix}0 & d_1 \\ d_0 & 0 \end{bmatrix}$

such that

d^2 = f \cdot \text{Id}_M

.

= Examples =

(1) For $S = \mathbb{C} x$ and $f = x^n$ there is a matrix factorization $d_0:S \rightleftarrows S:d_1$ where $d_0=x^i, d_1 = x^{n-i}$ for $0 \leq i \leq n$ .

(2) If $S = \mathbb{C} x,y,z$ and $f = xy + xz + yz$ , then there is a matrix factorization $d_0:S^2 \rightleftarrows S^2:d_1$ where

$d_0 = \begin{bmatrix} z & y \\ x & -x-y \end{bmatrix} \text{ } d_1 = \begin{bmatrix} x+y & y \\ x & -z \end{bmatrix}$

Periodicity

definition

= Main theorem =

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

: $\cdots \to F_2 \to F_1 \to F_0 \to 0$

be a minimal $B$ -free resolution of the ground field. Then $F_\bullet$ becomes periodic after at most $1 + \text{dim}(B)$ steps. https://www.youtube.com/watch?v=2Jo5eCv9ZVY

Matrix factorization (algebra)

Motivation

Definition

= Examples =

Periodicity

= Main theorem =

= Maximal Cohen-Macaulay modules =

Categorical structure

Support of matrix factorizations

See also

References

Further reading