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 $B=k[y]$.