The Koszul Property for Graded Twisted Tensor Products

TL;DR

This paper investigates when a twisted tensor product of graded algebras retains the Koszul property, identifying key conditions like quadraticity and the unique extension property, with detailed analysis for polynomial algebras.

Contribution

It establishes criteria for the Koszul property in twisted tensor products, especially linking quadraticity and the unique extension property, and clarifies when Koszulness is preserved.

Findings

C is quadratic iff the twisting map has the unique extension property

A and B being Koszul does not necessarily imply C is Koszul

Provides conditions under which C is Koszul when A and B are

Abstract

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 $B=k[y]$.