Lemma 5

A is a graded polynomial algebra if and only if P is finite dimensional and $g$ is bijective.

If P is finite dimensional, $\mathbf{S}(P)$ is clearly a graded polynomial algebra, and so is A if $g$ is bijective. Conversely, assume that A is generated by algebraically independent homogeneous elements $x_1, . . . , x_m$ of degrees $d_1, . . . , d_m$. Let $\overline{x}_i$ be the image of $x_i$ in P. It is immediate that the $\overline{x}_i$ form a basis of P; since $\overline{x}_i$ is of degree $d_i$, it follows that $\mathbf{S}(P)$ and A are isomorphic; in particular, dim $\mathbf{S}(P)^n=$ dim $A^n$ for all $n$. Since $g$ is surjective, it is necessarily bijective.

Lemma 4 is now immediate. Indeed, Lemma 5, applied to the $k'$-algebra $A'$, shows that $g\otimes 1 :\mathbf{S}(P)\otimes k'\rightarrow A\otimes k'$ is bijective, and hence so is $g$.