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

… \xrightarrow{ ⋅ x} R \xrightarrow{ ⋅ x} R \xrightarrow{ ⋅ x} R \to 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 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 ⋅ Id _{n} . This can be encoded more generally as a Z / 2 -graded S -module M = M _{0 ⊕} M _{1} with an endomorphism

d = \begin{bmatrix} 0 & d _{1} \ d _{0} & 0 \end{bmatrix}

such that d ^{2} = f ⋅ Id _{M} .

Examples

(1) For

S = C [[x]] and f = x ^{n} there is a matrix factorization d _{0:S} \rightleftarrows S:d _{1} where d ^{i,} _{0= x} d _{1} = x ^{n-i} for 0 \leq i \leq n .

(2) If

S = C [[x,y,z]] and f = xy + xz + yz , then there is a matrix factorization d ^{2} _{0 :S} \rightleftarrows S ^{2 :d} _{1} where

d _{0} = \begin{bmatrix} z & y \ x & - x - y \end{bmatrix} d _{1} = \begin{bmatrix} x + y & y \ x & - z \end{bmatrix}

Periodicity

definition

Main theorem

R and an ideal I \subset R generated by an A -sequence, set B = A/I and let… \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 + dim (B) steps. https://www.youtube.com/watch?v=2Jo5eCv9ZVY

Maximal Cohen-Macaulay modules

page 18 of eisenbud article

See also

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

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

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.