Theorem 2

Let $A$ be a regular ring, graded of type $\mathbf{N}$. The ring $A_0$ consisting of the elements of degree 0 in $A$ is regular; there exists a finitely generated projective $A_0$-module $P$ graded in degrees $> 0$ such that $A$ is isomorphic, as a graded $A_0$-algebra, to $S_{A_0}(P)$.

Let us denote by $P$ the graded $A_0$-module $A_+/A_+^2$. By Proposition 4 of No. 3, the ring $A_0$ is regular and the $A_0$-module $P$ is projective and finitely generated. The homogeneous components of $P$ are therefore projective, and there exists an $A_0$-linear section $\varphi : P \to A_+$, graded of degree 0 , of the canonical surjection $A_+ \to P$. Let $f : S_{A_0}(P) \longrightarrow A$ be the homomorphism of graded $A_0$-algebras extending $\varphi$. By Proposition 4, $f$ extends to an isomorphism of the separated completion of $S_{A_0}(P)$ for the $S_{A_0}(P)_+$-adic topology onto the separated completion of $A$ for the $A_+$-adic topology. Consequently, $f$ is injective and its image is dense in $A$ for the $A_+$-adic topology. But since the topologies induced on the homogeneous components of $A$ are discrete and the image of $f$ is a graded submodule, this implies that $f$ is bijective.