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TitleGraded algebras and their differential graded extensions.
Author(s) D. Piontkovski
TypeArticle in Journal
AbstractIn the survey, we deal with the following situation. Let A be a graded algebra or a differential graded algebra. Adjoining a set x of free (in any sense) indeterminates, we make a new differential graded algebra A〈x〉 by setting the differential values d: x → A on x. In the general case, such a construction is called the Shafarevich complex. Beginning with classical examples like the bar-complex, Koszul complex, and Tate resolution, we discuss noncommutative (and sometimes even nonassociative) versions of these notions. The comparison with the Koszul complex leads to noncommutative regular sequences and complete intersections; Tate’s process of killing cycles gives noncommutative DG resolutions and minimal models. The applications include the Golod-Shafarevich theorem, growth measures for graded algebras, characterizations of algebras of low homological dimension, and a homological description of Gröbner bases. The same constructions for categories of algebras with identities (like Lie or Jordan algebras) allow one to give a homological description of extensions and deformations of PI-algebras.
ISSN1072-3374; 1573-8795/e
URL http://link.springer.com/article/10.1007%2Fs10958-007-0137-y
JournalJ. Math. Sci., New York
PublisherSpringer US, New York, NY
Translation No
Refereed No