Corollary 3 (Hilbert’s Syzygy Theorem)

Tag 0E1K, permanent. Cite it as [Bourbaki, Tag 0E1K].

Algebra, chapter X, § 8. Dimension homologique

Assume that $A_0$ is a commutative field and that $A$ is generated as an $A_0$ -algebra by $n$ homogeneous elements of degrees > 0 and algebraically independent. For every graded $a$ -module $M$ bounded below (resp. of finite type), there exists an exact sequence of graded $a$ -modules and of degree $0$ homogeneous maps

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

where the

L_i

are graded free and bounded below (resp. graded free and of finite type).

Indeed, $dh(A) = n$ by Theorem 1 of X, p. 143, and Corollary 2 is applied.

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

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