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 formula_1 there is an infinite resolution of the formula_2-module formula_3 whereformula_4Instead 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 formula_5 after finitely  many objects in the resolution. Definition. For a commutative ring formula_6 and an element formula_7, a matrix factorization of formula_8 is a pair of formula_9 square matrices formula_10 such that formula_11. This can be encoded more generally as a formula_12 graded formula_6-module formula_14 with an endomorphismformula_15 such that formula_16. Examples. (1) For formula_17 and formula_18 there is a matrix factorization formula_19 where formula_20 for formula_21. (2) If formula_22 and formula_23, then there is a matrix factorization formula_24 whereformula_25 Periodicity. definition Main theorem. Given a regular local ring formula_2 and an ideal formula_27 generated by an formula_28-sequence, set formula_29  and let formula_30 be a minimal formula_31-free resolution of the ground field. Then formula_32 becomes periodic after at most formula_33 steps. https://www.youtube.com/watch?v=2Jo5eCv9ZVY Maximal Cohen-Macaulay modules. page 18 of eisenbud article