### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 092, 24 pages      arXiv:1504.03548      https://doi.org/10.3842/SIGMA.2016.092

### Further Properties and Applications of Koszul Pairs

Faculty of Mathematics and Computer Science, University of Bucharest, 14 Academiei Str., Bucharest Ro-010014, Romania

Received May 19, 2016, in final form September 08, 2016; Published online September 14, 2016

Abstract
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.

Key words: Koszul rings; Koszul corings; Koszul pairs; incidence (co)ring of a poset.

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