Corollary

Tag 0CO5, permanent. Cite it as [Bourbaki, Tag 0CO5].

Commutative Algebra, chapter VIII, § 4. Séries de Hilbert-Samuel

a) If $r = 0$ , then we have $\mathrm{long}_{H_0}(M) = c_M$ .

b) If $r = 1$ , then we have $\mathrm{long}_{H_0}(M_n) = c_M$ for $n$ sufficiently large.

c) If $r > 1$ , then we have $\mathrm{long}_{H_0}(M_n) = c_M \frac{n^{r-1}}{(r-1)!} + \rho_n n^{r-2}$ , where $\rho_n$ tends to a limit in $\mathbf{R}$ when $n$ increases indefinitely.

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

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