Growth, relations and prime spectrum of monomial algebras

Abstract

We show that monomial algebras defined by $r_n$ relations of each degree can (under mild assumptions) be mapped onto prime monomial algebras with prescribed growth functions tightly bounded by any function growing slower than $r_n$.

Using the same method, we are able to construct finitely generated, prime monomial algebras with GK-dimension 2 admitting arbitrarily long chains of prime ideals, providing an answer to a question of Bergman from 1989. Unlike the previously known examples, our algebras are computable, concretely constructed and concrete upper bounds of the form $n^{2}f(n)$ can be put on their growth rates, for arbitrarily slow unbounded non-decreasing function $f$.