Gelfand-Kirillov dimension under base field extension (Q1182819)

Let \(K\) be an extension field of \(F\), and let \(A\) be a \(K\)-algebra. It is shown that the Gelfand-Kirillov dimensions of \(A_ F\) and \(A_ K\) differ by at least the transcendence degree of \(K\) over \(F\), and that \(\text{GK}\dim_ F(A)=\text{GK}\dim_ K(A)+\text{tr}\deg_ F(K)\) in case \(A\) is commutative or noetherian PI. The author gives an example to show that this equality does not hold in general. Furthermore, it can happen that \(A_ K\) has finite Gelfand-Kirillov dimension, whereas that of \(A_ F\) is infinite, even when \(K\) is an algebraic extension of \(F\). Both examples are homomorphic images of the free algebra on two generators.