# Journal of Noncommutative Geometry

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**Volume 11, Issue 4, 2017, pp. 1289–1350**

**DOI: 10.4171/JNCG/11-4-3**

Published online: 2017-12-15

Koszul pairs and applications

Pascual Jara^{[1]}, Javier López Peña

^{[2]}and Dragoş Ştefan

^{[3]}(1) Universidad de Granada, Spain

(2) University College London, UK

(3) University of Bucharest, Romania

Let $R$ be a semisimple ring. A pair $(A,C)$ is called almost-Koszul if $A$ is a connected graded $R$-ring and $C$ is a compatible connected graded $R$-coring. To an almost-Koszul pair one associates three chain complexes and three cochain complexes such that one of them is exact if and only if the others are so. In this situation $(A,C)$ is said to be Koszul. One proves that a connected $R$-ring $A$ is Koszul if and only if there is a connected $R$-coring $C$ such that $(A,C)$ is Koszul. This result allows us to investigate the Hochschild (co)homology of Koszul rings. We apply our method to show that the twisted tensor product of two Koszul rings is Koszul. More examples and applications of Koszul pairs, including a generalization of Fröberg Theorem [12], are discussed in the last part of the paper.

*Keywords: *Koszul rings, Koszul pairs, Hochschild (co)homology, twisted tensor products

Jara Pascual, López Peña Javier, Ştefan Dragoş: Koszul pairs and applications. *J. Noncommut. Geom.* 11 (2017), 1289-1350. doi: 10.4171/JNCG/11-4-3