On monomial algebras of finite global dimension

Abstract

Let G G be an associative monomial k {\mathbf {k}} -algebra. If G G is assumed to be finitely presented, then either G G contains a free subalgebra on two monomials or else G G has polynomial growth. If instead G G is assumed to have finite global dimension, then either G G contains a free subalgebra or else G G has a finite presentation and polynomial growth. Also, a graded Hopf algebra with generators in degree one and relations in degree two contains a free Hopf subalgebra if the number of relations is small enough.