Proposition 8

a) One has $\dim(H) \leq [H_+/H_+^2 : H_0]$. b) The following conditions are equivalent: (i) $\dim(H) = [H_+/H_+^2 : H_0]$; (ii) the noetherian local ring $S^{-1}H$ is regular; (iii) $H$ is generated as an $H_0$-algebra by a family of homogeneous elements of degrees $> 0$, algebraically independent over $H_0$. c) Assume the conditions of b) satisfied, and let $a_1, ..., a_n \in H$ be homogeneous elements of degrees $> 0$. The following conditions are equivalent: (i) the classes of the $a_i$ form a basis of the $H_0$-vector space $H_+/H_+^2$; (ii) the $a_i/1$ form a system of coordinates of the regular noetherian local ring $S^{-1}H$;

(iii) the $a_i$ are algebraically independent over $H_0$ and generate $H$ as an $H_0$-algebra.

One has $\dim(H) = \dim(S^{-1}H)$ (No. 2, th. 1), and $[H_+/H_+^2 : H_0] = [(S^{-1}H_+)/ (S^{-1}H_+)^2 : H_0]$ (II, § 3, No. 3, prop. 9); after § 5, No. 1, this implies $a$ and the equivalences (i) $\Leftrightarrow$ (ii) in $b$ and $c$. The two implications (iii) $\Rightarrow$ (i) in $b$ and $c$ are trivial. Let us prove the implications (i) $\Rightarrow$ (iii). Assume therefore that one has $\dim(H) = [H_+/H_+^2 : H_0]$ and let $a_1, ..., a_n$ be homogeneous elements of $H$, of degrees $> 0$, whose classes form a basis of $H_+/H_+^2$; consider the homomorphism of graded algebras $\varphi : H_0[X_1, ..., X_n] \to H$ which sends $X_i$ to $a_i$. The ideal $H_+$ of $H$ is generated by the $a_i$ (A, II, p. 171, cor. 2 and remark); consequently the homomorphism $\varphi$ is surjective (III, § 1, No. 2, prop. 1). But one has

(§ 2, No. 4, cor. 1 of th. 3), therefore $\mathrm{Ker}\, \varphi = 0$ since $H_0[X_1, ..., X_n]$ is an integral domain. This implies assertions (iii).

Under the hypotheses of $b$, one says that $H$ is a regular graded $H_0$-algebra, or a polynomial graded $H_0$-algebra. Under the hypotheses of $c$, one says that the $a_i$ form a graded system of coordinates of $H$.