Details:
Title  Graded algebras and their differential graded extensions.  Author(s)  D. Piontkovski  Type  Article in Journal  Abstract  In 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 barcomplex, 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 GolodShafarevich 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 PIalgebras.  ISSN  10723374; 15738795/e 
URL 
http://link.springer.com/article/10.1007%2Fs109580070137y 
Language  English  Journal  J. Math. Sci., New York  Volume  142  Number  4  Pages  22672301  Publisher  Springer US, New York, NY  Year  2007  Edition  0  Translation 
No  Refereed 
No 
