Matrix factorization (algebra): Difference between revisions

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In [[homological algebra]], a branch of [[mathematics]], a '''matrix factorizationsfactorization''' areis toolsa tool used to study infinitely long resolutions in homological[[resolution (algebra)|resolution]]s, generally over [[commutative ringsring]]s.

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In mathematics, matrix factorizations are tools used to study infinitely long resolutions in homological algebra, generally over commutative rings.

== Motivation ==

One of the problems with non-smooth algebras, such as [[Artin algebra|Artin algebras]]s, are their [[derived category|derived categories]] are poorly behaved due to infinite [[projective resolutionsresolution]]s. For example, in the [[ring (mathematics)|ring]] <math>R = \mathbb{C}[x]/(x^2)</math> there is an infinite resolution of the <math>R</math>-[[module (mathematics)|module]] <math>\mathbb{C}</math> where<blockquote><math>\cdots \xrightarrow{\cdot x} R \xrightarrow{\cdot x} R \xrightarrow{\cdot x} R \to \mathbb{C} \to 0</math></blockquote>Instead of looking at only the derived category of the module category, [[David Eisenbud]]<ref>{{Cite journal|last=Eisenbud|first=David|first2=|date=|title=Homological Algebra on a Complete Intersection, with an Application to Group Respresentations|url=https://www.ams.org/journals/tran/1980-260-01/S0002-9947-1980-0570778-7/S0002-9947-1980-0570778-7.pdf|journal=Transactions of the American Mathematical Society|year=1980 |volume=260|pages=3535–64|doi=10.1090/S0002-649947-1980-0570778-7 |s2cid=27495286 |archive-url=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|archive-date=25 Feb 2020|via=}}</ref> studied such resolutions by looking at their periodicity. In general, such resolutions are periodic with period <math>2</math> after finitely many objects in the resolution.

== Definition ==

For a commutative ring <math>S</math> and an element <math>f \in S</math>, a '''matrix factorization''' of <math>f</math> is a pair of <math>''n\times ''-by-''n</math> square'' matrices <math>A,B</math> such that <math>AB = f \cdot \text{Id}_n</math>. This can be encoded more generally as a <math>\mathbb{Z}/2</math>-[[graded module|graded]] <math>S</math>-module <math>M = M_0\oplus M_1</math> with an [[endomorphism]]<blockquote><math>d = \begin{bmatrix}0 & d_1 \\ d_0 & 0 \end{bmatrix}</math> </blockquote>such that <math>d^2 = f \cdot \text{Id}_M</math>.

=== Examples ===

Line 17 ⟶ 16:

=== Main theorem ===

Given a [[regular local ring]] <math>R</math> and an [[ideal (ring theory)|ideal]] <math>I \subset R</math> generated by an <math>A</math>-sequence, set <math>B = A/I</math> and let

:<math>\cdots \to F_2 \to F_1 \to F_0 \to 0</math>

be a minimal <math>B</math>-free resolution of the ground [[field (mathematics)|field]]. Then <math>F_\bullet</math> becomes periodic after at most <math>1 + \text{dim}(B)</math> steps. https://www.youtube.com/watch?v=2Jo5eCv9ZVY

=== Maximal Cohen-Macaulay modules ===

page 18 of eisenbud article

<br />

== Categorical structure ==

{{section expand|date=February 2022}}

== Support of matrix factorizations ==

{{section expand|date=February 2022}}

== References ==

{{Reflist}}

== See also ==

* [[Derived noncommutative algebraic geometry]]

* [[Derived category]]

* [[Homological algebra]]

* [[Triangulated category]]

== External linksReferences ==

{{Reflist}}

== Further reading ==

* [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 Homological Algebra on a Complete Intersection with an Application to Group Representations]

* [https://web.archive.org/web/20200225192403/https://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1046&context=mathstudent Geometric Study of the Category of Matrix Factorizations]

* https://web.math.princeton.edu/~takumim/takumim_Spr13_JP.pdf

* http://www.math.ucr.edu/~jbergner/burke

* https://arxiv.org/abs/1110.2918

[[Category:Homological algebra]]