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KOSZUL CALCULUS

Part of: Homological methods Rings and algebras arising under various constructions

Published online by Cambridge University Press:  18 October 2017

ROLAND BERGER ,
THIERRY LAMBRE  and
ANDREA SOLOTAR
ROLAND BERGER
Affiliation:
Univ Lyon, UJM-Saint-Étienne, CNRS UMR 5208, Institut Camille Jordan, F-42023, Saint-Étienne, France e-mail: roland.berger@univ-st-etienne.fr
THIERRY LAMBRE
Affiliation:
Laboratoire de Mathématiques Blaise Pascal, UMR 6620 CNRS & UCA, Campus universitaire des Cézeaux, 3 place Vasarely, TSA 60026, CS 60026, 63178 Aubière Cedex, France e-mail: thierry.lambre@uca.fr
ANDREA SOLOTAR
Affiliation:
IMAS and Dto de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, Pabellòn 1, 1428 Buenos Aires, Argentina e-mail: asolotar@dm.uba.ar
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Abstract

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We present a calculus that is well-adapted to homogeneous quadratic algebras. We define this calculus on Koszul cohomology – resp. homology – by cup products – resp. cap products. The Koszul homology and cohomology are interpreted in terms of derived categories. If the algebra is not Koszul, then Koszul (co)homology provides different information than Hochschild (co)homology. As an application of our calculus, the Koszul duality for Koszul cohomology algebras is proved for any quadratic algebra, and this duality is extended in some sense to Koszul homology. So, the true nature of the Koszul duality theorem is independent of any assumption on the quadratic algebra. We compute explicitly this calculus on a non-Koszul example.

MSC classification

Primary: 16S37: Quadratic and Koszul algebras
Secondary: 16E35: Derived categories 16E40: (Co)homology of rings and algebras (e.g. Hochschild, cyclic, dihedral, etc.) 16E45: Differential graded algebras and applications
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 60 , Issue 2 , May 2018 , pp. 361 - 399
Copyright
Copyright © Glasgow Mathematical Journal Trust 2017 

References

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