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 Izv. RAN. Ser. Mat., 2001, Volume 65, Issue 3, Pages 139–152 (Mi izv339)

On graded algebras of global dimension 3

D. I. Piontkovskii

Central Economics and Mathematics Institute, RAS

Abstract: Assume that a graded associative algebra $A$ over a field $k$ is minimally presented as the quotient algebra of a free algebra $F$ by the ideal $I$ generated by a set $f$ of homogeneous elements. We study the following two extensions of $A$: the algebra $\overline F=F/I\oplus I/I^2\oplus\dotsb$ associated with $F$ with respect to the $I$-adic filtration, and the homology algebra $H$ of the Shafarevich complex $\operatorname{Sh}(f,F)$ (which is a non-commutative version of the Koszul complex). We obtain several characterizations of algebras of global dimension 3. In particular, the $A$-algebra $H$ in this case is free, and the algebra $\overline F$ is isomorphic to the quotient algebra of a free $A$-algebra by the ideal generated by a so-called strongly free (or inert) set.

DOI: https://doi.org/10.4213/im339

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English version:
Izvestiya: Mathematics, 2001, 65:3, 557–568

Bibliographic databases:

MSC: 16W50, 16E40

Citation: D. I. Piontkovskii, “On graded algebras of global dimension 3”, Izv. RAN. Ser. Mat., 65:3 (2001), 139–152; Izv. Math., 65:3 (2001), 557–568

Citation in format AMSBIB
\Bibitem{Pio01} \by D.~I.~Piontkovskii \paper On graded algebras of global dimension~3 \jour Izv. RAN. Ser. Mat. \yr 2001 \vol 65 \issue 3 \pages 139--152 \mathnet{http://mi.mathnet.ru/izv339} \crossref{https://doi.org/10.4213/im339} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1853369} \zmath{https://zbmath.org/?q=an:1003.16003} \elib{http://elibrary.ru/item.asp?id=13373556} \transl \jour Izv. Math. \yr 2001 \vol 65 \issue 3 \pages 557--568 \crossref{https://doi.org/10.1070/IM2001v065n03ABEH000339} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-28244484301} 

• http://mi.mathnet.ru/eng/izv339
• https://doi.org/10.4213/im339
• http://mi.mathnet.ru/eng/izv/v65/i3/p139

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. D. I. Piontkovskii, “On differential graded Lie algebras”, Russian Math. Surveys, 58:1 (2003), 189–190
2. Piontkovski D., “Linear equations over noncommutative graded rings”, J. Algebra, 294:2 (2005), 346–372
3. D. Piontkovski, “Graded algebras and their differential graded extensions”, J Math Sci, 142:4 (2007), 2267
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