Proposition 1

Tag 0D5M, permanent. Cite it as [Bourbaki, Tag 0D5M].

Commutative Algebra, chapter X, § 4. Anneaux réguliers

Let $A$ be a regular noetherian local ring and $n$ its dimension. Then $\mathrm{dh}(A) = n$ and, for every integer $i \geqslant 0$ ,

[\mathrm{Ext}_A^i(\kappa_A, \kappa_A) : \kappa_A] = [\mathrm{Tor}_i^A(\kappa_A, \kappa_A) : \kappa_A] = \binom{n}{i}.

Let $x = (x_1, \ldots, x_n)$ be a system of coordinates of $A$ (VIII, § 5, No. 1, def. 1). The sequence $x$ generates $m_A$ and is completely secant for $A$ (loc. cit., No. 2, th. 1). The Koszul complex $K_\bullet(x, A)$ is a free resolution of $\kappa_A$ (A, X, p. 159, remark 3), whose differential is zero modulo $m_A$ . Hence, for every integer $i \geqslant 0$ , one has (§ 3, No. 3, formula (1))

[\mathrm{Ext}_A^i(\kappa_A, \kappa_A) : \kappa_A] = [\mathrm{Tor}_i^A(\kappa_A, \kappa_A) : \kappa_A] = \mathrm{rg}_A(K_i(x, A)) = \binom{n}{i}.

It then follows from cor. 1 of prop. 4 of No. 3, § 3 that $\mathrm{dh}(A) = n$ .

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

Cited by

Nothing.

Cites

Nothing.