Example 3

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

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

Suppose again $H_0$ artinian; let M be a graded H-module, and suppose that there exists an exact sequence of graded H-modules and homomorphisms of degree 0

0 \to L_n \to L_{n-1} \to \cdots \to L_0 \to M \to 0,

where, for

k = 0, 1, ..., n

L_k

is a free graded H-module generated by linearly independent homogeneous elements, of respective degrees

\delta_{k,1}, ..., \delta_{k,m(k)}

. Then, by remark 2 and example 2, we have

Q_M = \mathrm{long}(H_0).\sum_{0 \leq k \leq n} \sum_{1 \leq j \leq m(k)} (-1)^k T^{\delta_{k,j}},

c_M = \mathrm{long}(H_0).\sum_{0 \leq k \leq n} (-1)^k m(k).

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

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