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 there is an infinite resolution of the -module where
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 , a matrix factorization of is a pair of square matrices such that . This can be encoded more generally as a graded -module with an endomorphism
such that .
Examples
(1) For and there is a matrix factorization where for .
(2) If and , then there is a matrix factorization where
Periodicity
definition
Main theorem
Given a regular local ring and an ideal generated by an -sequence, set and let
be a minimal -free resolution of the ground field. Then becomes periodic after at most steps. https://www.youtube.com/watch?v=2Jo5eCv9ZVY
Maximal Cohen-Macaulay modules
page 18 of eisenbud article
Categorical structure
Support of matrix factorizations
See also
References
- ↑ 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.