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On the characterisation of Koszul algebras. Four counter-examples.
Stockholm University, Faculty of Science, Department of Mathematics. matematik.
1995 (English)In: Comptes Rendus de l'Académie des Sciences. Série I. Mathématique, ISSN 0764-4442, Vol. 321, no 1, 15-20 p.Article in journal (Refereed) Published
Abstract [en]

We consider rings of the form R=k[X_1,...,X_n]/(f_1,..,f_s), where k is a field, k[X_1,...,X_n] is the polynomial ring in variables X_i of degree 1, and the f_i:s are quadratic forms. The dual algebra R^! is the subalgebra of the Yoneda algebra Ext*_R(k,k), that is generated by Ext^1_R(k,k). We say that R is a Koszul algebra if Tor_{i,j}^R(k,k) = 0 for i \neq j. If R is a Koszul algebra, then the Hilbert series R(t) of R and R^!(t) of R^! are related by R(t)R^!(-t)=1. The converse is true for n \leq 3. We give four counter-examples in the case n = 5. This answers negatively a question in the PhD thesis (1983) by Jörgen Backelin.

Place, publisher, year, edition, pages
1995. Vol. 321, no 1, 15-20 p.
Keyword [en]
Koszul algebra, Golod maps, Hopf algebras, Yoneda Ext-algebra, Hilbert series, Koszul dual
National Category
URN: urn:nbn:se:su:diva-13299OAI: diva2:179819
Available from: 2008-03-15 Created: 2008-03-15 Last updated: 2011-01-16Bibliographically approved

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Roos, Jan-Erik
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