Matrix factorization (algebra): Difference between revisions

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== Motivation ==

One of the problems with non-smooth algebras, such as [[Artin algebra]]s, are their derived categories are poorly behaved due to infinite projective resolutions. For example, in the ring <math>R = \mathbb{C}[x]/(x^2)</math> there is an infinite resolution of the <math>R</math>-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=35–64|doi=10.1090/S0002-9947-1980-0570778-7 |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 ==