In homological algebra, a branch of mathematics, a matrix factorization is a tool used to study infinitely long resolutions, generally over commutative rings.
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
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.
For a commutative ring
and an element
, a matrix factorization of
is a pair of n-by-n matrices
Id
This can be encoded more generally as a
with an endomorphism
{\displaystyle d={\begin{bmatrix}0&d_{1}\\d_{0}&0\end{bmatrix}}}
there is a matrix factorization
{\displaystyle d_{0}=x^{i},d_{1}=x^{n-i}}
, then there is a matrix factorization
{\displaystyle d_{0}={\begin{bmatrix}z&y\\x&-x-y\end{bmatrix}}{\text{ }}d_{1}={\begin{bmatrix}x+y&y\\x&-z\end{bmatrix}}}
definition Given a regular local ring
and an ideal
generated by an
-sequence, set
and let be a minimal
-free resolution of the ground field.
becomes periodic after at most
dim
{\displaystyle 1+{\text{dim}}(B)}
steps.
https://www.youtube.com/watch?v=2Jo5eCv9ZVY page 18 of eisenbud article