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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

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 60 , Issue 2 , May 2018 , pp. 361 - 399
Copyright
Copyright © Glasgow Mathematical Journal Trust 2017 

References

REFERENCES

1. Berger, R., Weakly confluent quadratic algebras, Algeb. Represent. Theory 1 (1998), 189–213.CrossRefGoogle Scholar
2. Berger, R., Gerasimov's theorem and N-Koszul algebras, J. Lond. Math. Soc. 79 (2009), 631–648.Google Scholar
3. Chouhy, S. and Solotar, A., Projective resolutions of associative algebras and ambiguities, J. Algebra 432 (2015), 22–61.Google Scholar
4. Gerasimov, V. N., Free associative algebras and inverting homomorphisms of rings, Am. Math. Soc. Transl. 156 (1993), 1–76.Google Scholar
5. Gerstenhaber, M., The cohomology structure of an associative ring, Ann. Math. 78 (1963), 267–288.CrossRefGoogle Scholar
6. Gerstenhaber, M. and Schack, S. D., Simplicial cohomology in Hochschild cohomology, J. Pure Appl. Algebra 30 (1983), 143–156.Google Scholar
7. Ginzburg, V., Calabi-Yau algebras, arXiv:math.AG/0612139v3.Google Scholar
8. Goodwillie, T. G., Cyclic homology, derivations and the free loop space, Topology 24 (1985), 187–215.Google Scholar
9. Guiraud, Y., Hoffbeck, E. and Malbos, P., Linear polygraphs and Koszulity of algebras, arXiv:1406.0815.Google Scholar
10. Keller, B., Derived invariance of higher structures on the Hochschild complex, https://webusers.imj-prg.fr/~bernhard.keller/publ/dih.pdfGoogle Scholar
11. Lambre, T., Dualité de Van den Bergh et structure de Batalin-Vilkoviskiǐ sur les algèbres de Calabi-Yau, J. Noncommut. Geom. 4 (2010), 441–457.Google Scholar
12. Loday, J.-L., Cyclic homology, Grundlehren der mathematischen Wissenschaften vol. 301, 2nd edition (Springer-Verlag, Berlin, 1998).Google Scholar
13. Loday, J.-L. and Vallette, B., Algebraic operads, Grundlehren der mathematischen Wissenschaften, vol. 346 (Springer, Heidelberg, 2012).Google Scholar
14. Manin, Yu. I., Quantum groups and non-commutative geometry (Publications du Centre de Recherches Mathématiques, Montréal, 1988).Google Scholar
15. Polishchuk, A. and Positselski, L., Quadratic algebras, University lecture series, vol. 37 (AMS, Providence, RI, 2005).Google Scholar
16. Priddy, S., Koszul resolutions, Trans. Am. Math. Soc. 152 (1970), 39–60.CrossRefGoogle Scholar
17. Rinehart, G. S., Differential forms on general commutative algebras, Trans. Am. Math. Soc. 108 (1963), 195–222.Google Scholar
18. Tamarkin, D. and Tsygan, B., The ring of differential operators on forms in noncommutative calculus, in Proceedings of Symposia in Pure Mathematics, vol. 73 (American Mathematical Society, Providence, RI, 2005), 105–131.Google Scholar
19. Van den Bergh, M., Non-commutative homology of some three-dimensional quantum spaces, K-Theory 8 (1994), 213–220.Google Scholar
20. Van den Bergh, M., A relation between Hochschild homology and cohomology for Gorenstein rings, Proc. Am. Math. Soc. 126 (1998), 1345–1348; erratum Van den Bergh, M., Proc. Amer. Math. Soc. 130 (2002), 2809–2810.Google Scholar
21. Weibel, C. A., An introduction to homological algebra (Cambridge University Press, Cambridge, 1994).Google Scholar