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The Quarterly Journal of Mathematics Advance Access published online on January 18, 2008
The Quarterly Journal of Mathematics, doi:10.1093/qmath/ham056
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© 2007. Published by Oxford University Press. All rights reserved For permissions, please email: journals.permissions@oxfordjournals.org

MULTIPLICATIVE STRUCTURES FOR KOSZUL ALGEBRAS

Ragnar-Olaf Buchweitz {dagger}

Department of Mathematics, University of Toronto at Scarborough, 1265 Military Trail, Toronto, ON, Canada M1C 1A4

Edward L. Green {ddagger}

Department of Mathematics, Virginia Tech, Blacksburg, VA 24061, USA

Nicole Snashall§

Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH

Øyvind Solberg ¶

Institutt for matematiske fag, NTNU, N-7034 Trondheim, Norway

§ Corresponding author. E-mail: N.Snashall{at}mcs.le.ac.uk

Received 5 October 2007; revised 10 November 2007
  Abstract

Let {Lambda} = kQ/I be a Koszul algebra over a field k, where Q is a finite quiver. An algorithmic method for finding a minimal projective resolution F of the graded simple modules over {Lambda} is given in [E. L. Green and Ø. Solberg, An algorithmic approach to resolutions, J. Symbolic Comput., 42 (2007), 1012–1033]. This resolution is shown to have a ‘comultiplicative’ structure in [E. L. Green, G. Hartman, E. N. Marcos and Ø. Solberg, Resolutions overKoszul algebras, Arch. Math. 85 (2005), 118–127.], and this is used to find a minimal projective resolution P of {Lambda}over the enveloping algebra {Lambda}e. Using these results, we showthat the multiplication in the Hochschild cohomology ring of {Lambda} relative to the resolution P is given as a cup product and also provide a description of this product. This comultiplicative structure also yields the structure constants of the Koszul dual of {Lambda} with respect to a canonical basis over k associated to the resolution F. The natural map from the Hochschild cohomology to the Koszul dual of {Lambda} is shown to be surjective onto the graded centre of the Koszul dual.

{dagger} E-mail: ragnar{at}math.toronto.edu

{ddagger} E-mail: green{at}math.vt.edu

E-mail: oyvinso{at}math.ntnu.no


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