Corollary 2

Tag 0CRN, permanent. Cite it as [Bourbaki, Tag 0CRN].

Commutative Algebra, chapter VIII, § 6. Dimension des anneaux gradués

Let $A$ be a noetherian local ring and $n \in \mathbf{N}$ . In order that one may have $\dim(A) \geq n$ , it is necessary and sufficient that for every integer $r \geq 0$ , one have

[m_A^r/m_A^{r+1} : \kappa_A] \geq \binom{n+r-1}{n-1}, \quad \left( \text{resp. } \mathrm{long}_A(A/m_A^{r+1}) \geq \binom{n+r}{n} \right);

one has equality for every $r$ if and only if $A$ is regular of dimension $n$ .

The condition is sufficient (§ 4, No. 4, Theorem 3 and § 5, No. 2, Theorem 1). Let us show that it is necessary. Consider the graded ring gr(A) = gr_{m_A}(A); let k be an infinite extension of the field $\kappa_A$ , and put $H = k \otimes_{\kappa_A} \mathrm{gr}(A)$ . The ring H is of dimension $\geq n$ (Prop. 5 and its corollary); one therefore deduces from Corollary 1 the existence of an injective graded homomorphism of graded k-algebras $\varphi : H_0[X_1, ..., X_n] \to H$ . Consequently one has, for every integer $r \geq 0$ ,

[\mathrm{gr}_r(A) : \kappa_A] = [H_r : H_0] \geq \binom{n + r - 1}{n - 1},

and equality for every $r$ implies the bijectivity of $\varphi$ , hence the regularity of A (§ 5, No. 2, Theorem 1).

The equalities

\mathrm{long}_A(A/m_A^{r+1}) = \sum_{i=0}^r [\mathrm{gr}_i(A) : \kappa_A]

and

\binom{n + r}{n} = \sum_{i=0}^r \binom{n + i - 1}{n - 1}

then imply the analogous assertions for the function $r \mapsto \mathrm{long}_A(A/m_A^{r+1})$ .

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

Nothing.