Geodesic
KOSZUL CALCULUS
Glasgow mathematical journal, Tome 60 (2018) no. 2, pp. 361-399

Voir la notice de l'article provenant de la source Cambridge University Press

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. 
BERGER, ROLAND; LAMBRE, THIERRY; SOLOTAR, ANDREA. KOSZUL CALCULUS. Glasgow mathematical journal, Tome 60 (2018) no. 2, pp. 361-399. doi: 10.1017/S0017089517000167
@article{10_1017_S0017089517000167,
     author = {BERGER, ROLAND and LAMBRE, THIERRY and SOLOTAR, ANDREA},
     title = {KOSZUL {CALCULUS}},
     journal = {Glasgow mathematical journal},
     pages = {361--399},
     year = {2018},
     volume = {60},
     number = {2},
     doi = {10.1017/S0017089517000167},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089517000167/}
}
TY  - JOUR
AU  - BERGER, ROLAND
AU  - LAMBRE, THIERRY
AU  - SOLOTAR, ANDREA
TI  - KOSZUL CALCULUS
JO  - Glasgow mathematical journal
PY  - 2018
SP  - 361
EP  - 399
VL  - 60
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.1017/S0017089517000167/
DO  - 10.1017/S0017089517000167
ID  - 10_1017_S0017089517000167
ER  - 
%0 Journal Article
%A BERGER, ROLAND
%A LAMBRE, THIERRY
%A SOLOTAR, ANDREA
%T KOSZUL CALCULUS
%J Glasgow mathematical journal
%D 2018
%P 361-399
%V 60
%N 2
%U http://geodesic.mathdoc.fr/articles/10.1017/S0017089517000167/
%R 10.1017/S0017089517000167
%F 10_1017_S0017089517000167

[1] 1. Berger, R., Weakly confluent quadratic algebras, Algeb. Represent. Theory 1 (1998), 189–213. Google Scholar | DOI

[2] 2. Berger, R., Gerasimov's theorem and N-Koszul algebras, J. Lond. Math. Soc. 79 (2009), 631–648. Google Scholar

[3] 3. Chouhy, S. and Solotar, A., Projective resolutions of associative algebras and ambiguities, J. Algebra 432 (2015), 22–61. Google Scholar

[4] 4. Gerasimov, V. N., Free associative algebras and inverting homomorphisms of rings, Am. Math. Soc. Transl. 156 (1993), 1–76. Google Scholar

[5] 5. Gerstenhaber, M., The cohomology structure of an associative ring, Ann. Math. 78 (1963), 267–288. Google Scholar | DOI

[6] 6. Gerstenhaber, M. and Schack, S. D., Simplicial cohomology in Hochschild cohomology, J. Pure Appl. Algebra 30 (1983), 143–156. Google Scholar

[7] 7. Ginzburg, V., Calabi-Yau algebras, arXiv:math.AG/0612139v3. Google Scholar

[8] 8. Goodwillie, T. G., Cyclic homology, derivations and the free loop space, Topology 24 (1985), 187–215. Google Scholar

[9] 9. Guiraud, Y., Hoffbeck, E. and Malbos, P., Linear polygraphs and Koszulity of algebras, arXiv:1406.0815. Google Scholar

[10] 10. Keller, B., Derived invariance of higher structures on the Hochschild complex, https://webusers.imj-prg.fr/~bernhard.keller/publ/dih.pdf Google Scholar

[11] 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] 12. Loday, J.-L., Cyclic homology, Grundlehren der mathematischen Wissenschaften vol. 301, 2nd edition (Springer-Verlag, Berlin, 1998). Google Scholar

[13] 13. Loday, J.-L. and Vallette, B., Algebraic operads, Grundlehren der mathematischen Wissenschaften, vol. 346 (Springer, Heidelberg, 2012). Google Scholar

[14] 14. Manin, Yu. I., Quantum groups and non-commutative geometry (Publications du Centre de Recherches Mathématiques, Montréal, 1988). Google Scholar

[15] 15. Polishchuk, A. and Positselski, L., Quadratic algebras, University lecture series, vol. 37 (AMS, Providence, RI, 2005). Google Scholar

[16] 16. Priddy, S., Koszul resolutions, Trans. Am. Math. Soc. 152 (1970), 39–60. Google Scholar | DOI

[17] 17. Rinehart, G. S., Differential forms on general commutative algebras, Trans. Am. Math. Soc. 108 (1963), 195–222. Google Scholar

[18] 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] 19. Van Den Bergh, M., Non-commutative homology of some three-dimensional quantum spaces, K-Theory 8 (1994), 213–220. Google Scholar

[20] 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. (2002), 2809–2810. Google Scholar

[21] 21. Weibel, C. A., An introduction to homological algebra (Cambridge University Press, Cambridge, 1994). Google Scholar

Cité par Sources :