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Further Properties and Applications of Koszul Pairs
Symmetry, integrability and geometry: methods and applications, Tome 12 (2016) Cet article a éte moissonné depuis la source Math-Net.Ru

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Koszul pairs were introduced in [arXiv:1011.4243] as an instrument for the study of Koszul rings. In this paper, we continue the enquiry of such pairs, focusing on the description of the second component, as a follow-up of the study in [arXiv:1605.05458]. As such, we introduce Koszul corings and prove several equivalent characterizations for them. As applications, in the case of locally finite $R$-rings, we show that a graded $R$-ring is Koszul if and only if its left (or right) graded dual coring is Koszul. Finally, for finite graded posets, we obtain that the respective incidence ring is Koszul if and only if the incidence coring is so.
Keywords: Koszul rings; Koszul corings; Koszul pairs; incidence (co)ring of a poset. 
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Adrian Manea; Dragoş Ştefan. Further Properties and Applications of Koszul Pairs. Symmetry, integrability and geometry: methods and applications, Tome 12 (2016). http://geodesic.mathdoc.fr/item/SIGMA_2016_12_a91/

[1] Beilinson A., Ginzburg V., Soergel W., “Koszul duality patterns in representation theory”, J. Amer. Math. Soc., 9 (1996), 473–527 | DOI | MR | Zbl

[2] Berger R., “Koszulity for nonquadratic algebras”, J. Algebra, 239 (2001), 705–734 | DOI | MR | Zbl

[3] Brzezinski T., Wisbauer R., Corings and comodules, London Mathematical Society Lecture Note Series, 309, Cambridge University Press, Cambridge, 2003 | DOI | MR | Zbl

[4] Green E. L., Reiten I., Solberg Ø., Dualities on generalized Koszul algebras, Mem. Amer. Math. Soc., 159, 2002, xvi+67 pp. | DOI | MR

[5] Jara Martínez P., López Peña J., Ştefan D., “Koszul pairs and applications”, J. Noncommut. Geom. (to appear) , arXiv: 1011.4243

[6] Keller B., Koszul duality and coderived categories (after K. {L}efèvre), https://webusers.imj-prg.fr/b̃ernhard.keller/publ/kdcabs.html

[7] Madsen D. O., “On a common generalization of Koszul duality and tilting equivalence”, Adv. Math., 227 (2011), 2327–2348, arXiv: 1007.3282 | DOI | MR | Zbl

[8] Madsen D. O., “Quasi-hereditary algebras and generalized Koszul duality”, J. Algebra, 395 (2013), 96–110, arXiv: 1201.0441 | DOI | MR | Zbl

[9] Manea A., Ştefan D., “On Koszulity of finite graded posets”, J. Algebra Appl. (to appear) , arXiv: 1605.05458

[10] May J. P., Bialgebras and Hopf algebras, http://www.math.uchicago.edu/\allowbreakm̃ay/TQFT/HopfAll.pdf

[11] Mazorchuk V., Ovsienko S., Stroppel C., “Quadratic duals, Koszul dual functors, and applications”, Trans. Amer. Math. Soc., 361 (2009), 1129–1172, arXiv: math.RT/0603475 | DOI | MR | Zbl

[12] Piontkovski D., “Graded algebras and their differentially graded extensions”, J. Math. Sci., 142 (2007), 2267–2301 | DOI | MR | Zbl

[13] Polishchuk A., Positselski L., Quadratic algebras, University Lecture Series, 37, Amer. Math. Soc., Providence, RI, 2005 | DOI | MR | Zbl

[14] Polo P., “On Cohen–Macaulay posets, Koszul algebras and certain modules associated to Schubert varieties”, Bull. London Math. Soc., 27 (1995), 425–434 | DOI | MR | Zbl

[15] Priddy S. B., “Koszul resolutions”, Trans. Amer. Math. Soc., 152 (1970), 39–60 | DOI | MR | Zbl

[16] Reiner V., Stamate D. I., “Koszul incidence algebras, affine semigroups, and Stanley–Reisner ideals”, Adv. Math., 224 (2010), 2312–2345, arXiv: 0904.1683 | DOI | MR | Zbl

[17] Woodcock D., “Cohen–Macaulay complexes and Koszul rings”, J. London Math. Soc., 57 (1998), 398–410 | DOI | MR | Zbl