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Commutative non-Koszul algebras having a linear resolution of arbitrarily high order.: Applications to torsion in loop space homology.
Stockholm University, Faculty of Science, Department of Mathematics. matematik.
1993 (English)In: Comptes Rendus de l'Académie des Sciences. Série I. Mathématique, Vol. 316, no 11, 1123-1128 p.Article in journal (Refereed) Published
Abstract [en]

Let k be a field and let S be a commutative algebra of the form k[X_1, · · · ,X_n]/(f_1, · · · , f_m),

where k[X_1, · · · ,X_n] is a polynomial ring in variables X_i of degree 1, and the f_j’s are homogeneous

polynomials of degrees >= 2 in the X_i’s. We say that S is a Koszul algebra if the graded

Tor^S_{i,j}(k.k) are 0 for i \neq j. Let us fix n (n \geq 6) and suppose that the characteristic of k is 0. We

show that, for any integer \alpha \geq 2, there exists S such that Tor^S_{i,j}(k, k) = 0 for i \neq j, i \leq \alpha (this

gives in particular that the f_j’s are quadratic forms in the X_i’s), and such that Tor^S_{\alpha+1,\alpha+2(k, k) = k. This solves a problem of Peeva and Eisenbud and also gives rise to finite simply connected CW-complexes

X such that the algebra H_*(\Omega X,Q)

is finitely generated whereas H_*(\Omega X,Z/pZ) is

not so for any prime p.

Place, publisher, year, edition, pages
1993. Vol. 316, no 11, 1123-1128 p.
Keyword [en]
Koszul algebra, Hopf algebra, loop space homology
National Category
URN: urn:nbn:se:su:diva-13301OAI: diva2:179821
Available from: 2008-03-15 Created: 2008-03-15 Last updated: 2011-01-16Bibliographically approved

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