The Koszul property for graded twisted tensor products

Let K be a field. Let A and B be connected N-graded K-algebras. Let C denote a twisted tensor product of A and B in the category of connected N-graded K-algebras. The purpose of this paper is to understand when C possesses the Koszul property, and related questions. We prove that if A and B are quadratic, then C is quadratic if and only if the associated graded twisting map has a property we call the unique extension property. We show that A and B being Koszul does not imply C is Koszul (or even quadratic), and we establish sufficient conditions under which C is Koszul whenever both A and B are. We analyze the unique extension property and the Koszul property in detail in the case where A=K[x] and B=K[y].