Free Subalgebras of Graded Algebras

Abstract

Let $k$ be a field and let $A=\bigoplus_{n\ge 1}A_n$ be a positively graded $k$-algebra. We recall that $A$ is graded nilpotent if for every $d\ge 1$, the subalgebra of $A$ generated by elements of degree $d$ is nilpotent. We give a method of producing grading nilpotent algebras and use this to prove that over any base field $k$ there exists a finitely generated graded nilpotent algebra that contains a free $k$-subalgebra on two generators.