Proposition 11

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

If M is a graded A-module bounded below (resp. if M is a graded A-module of finite type and if A is left noetherian), there exists an exact sequence of graded A-modules unbounded to the left

\cdots \longrightarrow L_n \xrightarrow{d_n} L_{n-1} \longrightarrow \cdots \longrightarrow L_1 \xrightarrow{d_1} L_0 \xrightarrow{d_0} M \longrightarrow 0

in which the $L_i$ are graded free and bounded below (resp. graded free and of finite type), and in which the $d_i$ are graded homomorphisms of degree 0.

If N is a graded A-module bounded below (resp. and of finite type over noetherian A), there exists a graded free A-module bounded below (resp. and of finite type) L and a surjective graded homomorphism $L \to N$ (II, p. 167, Remark 3).

This being so, suppose given an exact sequence of graded A-modules and graded homomorphisms of degree 0

L_n \xrightarrow{d_n} L_{n-1} \longrightarrow \cdots \longrightarrow L_0 \xrightarrow{d_0} M \longrightarrow 0,

where the $L_i, i = 0, \ldots, n$ , are graded free and bounded below (resp. graded free and of finite type). Then $N = \mathrm{Ker}\, d_n$ is bounded below (resp. of finite type); hence there exist a graded free a-module bounded below (resp. a graded free a-module of finite type) $L_{n+1}$ and a graded homomorphism $d_{n+1} : L_{n+1} \to L_n$ of degree 0, such that $\mathrm{Im}\, d_{n+1} = N$ ; the sequence

L_{n+1} \xrightarrow{d_{n+1}} L_n \xrightarrow{d_n} L_{n-1} \longrightarrow \cdots \longrightarrow L_0 \xrightarrow{d_0} M \longrightarrow 0

is then exact. The proposition then follows from the preceding by induction on $n$ .

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

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Cites

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