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 whereInstead of looking at only the derived category of the module category, David Eisenbud 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 n-by-n 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
Derived noncommutative algebraic geometry Derived category Homological algebra Triangulated category
References
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