In homological algebra, a branch of mathematics, a matrix factorization is a tool used to study infinitely long resolutions, generally over commutative rings.
Motivation
[edit]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
[edit]For a commutative ring and an element , a matrix factorization of is a pair of n-by-n matrices such that . This can be encoded more generally as a -graded -module with an endomorphism
such that .
Examples
[edit](1) For and there is a matrix factorization where for .
(2) If and , then there is a matrix factorization where
Periodicity
[edit]definition
Main theorem
[edit]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
[edit]page 18 of eisenbud article
Categorical structure
[edit]Support of matrix factorizations
[edit]See also
[edit]- Derived noncommutative algebraic geometry
- Derived category
- Homological algebra
- Triangulated category
References
[edit]- ^ 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.