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 R = C [ x ] / ( x 2 ) {\displaystyle R=\mathbb {C} [x]/(x^{2})} there is an infinite resolution of the R {\displaystyle R} -module C {\displaystyle \mathbb {C} } where ⋯ → ⋅ x R → ⋅ x R → ⋅ x R → C → 0 {\displaystyle \cdots {\xrightarrow {\cdot x}}R{\xrightarrow {\cdot x}}R{\xrightarrow {\cdot x}}R\to \mathbb {C} \to 0} Instead 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 2 {\displaystyle 2} after finitely many objects in the resolution.
Definition For a commutative ring S {\displaystyle S} and an element f ∈ S {\displaystyle f\in S} , a matrix factorization of f {\displaystyle f} is a pair of n-by-n matrices A , B {\displaystyle A,B} such that A B = f ⋅ Id n {\displaystyle AB=f\cdot {\text{Id}}_{n}} . This can be encoded more generally as a Z / 2 {\displaystyle \mathbb {Z} /2} -graded S {\displaystyle S} -module M = M 0 ⊕ M 1 {\displaystyle M=M_{0}\oplus M_{1}} with an endomorphism d = [ 0 d 1 d 0 0 ] {\displaystyle d={\begin{bmatrix}0&d_{1}\\d_{0}&0\end{bmatrix}}} such that d 2 = f ⋅ Id M {\displaystyle d^{2}=f\cdot {\text{Id}}_{M}} .
Examples (1) For S = C [ [ x ] ] {\displaystyle S=\mathbb {C} [[x]]} and f = x n {\displaystyle f=x^{n}} there is a matrix factorization d 0 : S ⇄ S : d 1 {\displaystyle d_{0}:S\rightleftarrows S:d_{1}} where d 0 = x i , d 1 = x n − i {\displaystyle d_{0}=x^{i},d_{1}=x^{n-i}} for 0 ≤ i ≤ n {\displaystyle 0\leq i\leq n} .
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