The Gel'fand–Kirillov dimension of relatively free associative algebras

In this paper the Gel'fand–Kirillov dimension $\operatorname{GKdim}(A)$ is calculated for a relatively free associative algebra $A$ over an arbitrary ground field. This dimension is determined by the complexity type of the algebra $A$ or by the set of semidirect products of matrix algebras over a polynomial ring contained in the variety $\operatorname{Var}(A)$. The proof is comparatively elementary and does not use the local representability of relatively free algebras.