Lemma 4

$^2$ Let $A =\bigoplus_{n\geq 0}A^n$ be a graded $k$-algebra, $k'$ an extension of $k$, and $A'= A\otimes_kk'$. Assume that $A'$ is a graded polynomial algebra over $k'$. Then A is a graded polynomial algebra over $k$.

We have ${A'}^0=k'$, so $A^0=k$. Put $A_+=\bigoplus_{n\geq 1}A^n$ and $P = A_+/A^2_+$. Then P is a graded vector space, and there is a graded linear map $f: P\rightarrow A_+$ of

$^2$ In Lemmas 4, 5 and $6,k$ can be any (commutative) field. degree zero such that the composite with the canonical projection $A_+\rightarrow P$ is the identity on P. Give $\mathbf{S}(P)$ the graded structure induced by that of P (Algebra, Chap. III, p. 506). The homomorphism of $k$-algebras $g:\mathbf{S}(P)\rightarrow A$ that extends $f$ (Algebra, Chap. III, p. 497) is a graded homomorphism of degree 0; an immediate induction on the degree shows that $g$ is surjective.