Proposition 10

Tag 0D4R, permanent. Cite it as [Bourbaki, Tag 0D4R].

Commutative Algebra, chapter X, § 3. Profondeur et dimension homologique

Let $A$ be a Gorenstein ring; then $A$ is a Macaulay ring, and satisfies $\mathrm{di}_A(A) = \dim(A)$ .

For every maximal ideal $m$ of $A$ , one has

\begin{align*} \dim(A_m) &\leq \mathrm{di}_{A_m}(A_m) & (\text{No. } 6, \text{Prop. } 8) \\ \mathrm{di}_{A_m}(A_m) &= \mathrm{prof}(A_m) & (\text{No. } 6, \text{Prop. } 9) \\ \mathrm{prof}(A_m) &\leq \dim(A_m) & (§ 1, \text{No. } 4, \text{Cor. } 2 \text{ of Th. } 2) ; \end{align*}

it follows that

A

is a Macaulay ring, and that

\mathrm{di}_A(A) = \dim(A)

by passing to the least upper bound (No. 2, Prop. 3).

Thus the noetherian rings $A$ such that $\mathrm{di}_A(A)$ is finite are the finite-dimensional Gorenstein rings (Prop. 3 of No. 2), and the noetherian rings such that the $A$ -module $A$ is injective are the artinian Gorenstein rings.

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

Cited by

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Cites

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