On Koszulity of Twisted Tensor Products

Abstract

Let R be a semisimple ring. A pair (A,C) is called pre-Koszul if A is a connected graded R-ring and C is a compatible connected graded R-coring. To a pre-Koszul pair we associate several complexes and we show that one of them is exact if and only if all other are so. In this case we say that (A,C) is Koszul. We investigate the connection between Koszul pairs and the Koszulity of A and C. We use Koszul pairs to show that the twisted tensor product of two Koszul rings is Koszul.