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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 Eisenbud¹ 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} .

(2) If S = C [ [ x , y , z ] ] {\displaystyle S=\mathbb {C} [[x,y,z]]} and f = x y + x z + y z {\displaystyle f=xy+xz+yz} , then there is a matrix factorization d 0 : S 2 ⇄ S 2 : d 1 {\displaystyle d_{0}:S^{2}\rightleftarrows S^{2}:d_{1}} where

d 0 = [ z y x − x − y ]   d 1 = [ x + y y x − z ] {\displaystyle d_{0}={\begin{bmatrix}z&y\\x&-x-y\end{bmatrix}}{\text{ }}d_{1}={\begin{bmatrix}x+y&y\\x&-z\end{bmatrix}}}

Periodicity

definition

Main theorem

Given a regular local ring R {\displaystyle R} and an ideal I ⊂ R {\displaystyle I\subset R} generated by an A {\displaystyle A} -sequence, set B = A / I {\displaystyle B=A/I} and let

⋯ → F 2 → F 1 → F 0 → 0 {\displaystyle \cdots \to F_{2}\to F_{1}\to F_{0}\to 0}

be a minimal B {\displaystyle B} -free resolution of the ground field. Then F ∙ {\displaystyle F_{\bullet }} becomes periodic after at most 1 + dim ( B ) {\displaystyle 1+{\text{dim}}(B)} 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

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

References

1. 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. https://web.archive.org/web/20200225190215/https://www.ams.org/journals/tran/1980-260-01/S0002-9947-1980-0570778-7/S0002-9947-1980-0570778-7.pdf ↩