Proposition 2

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

Let $P$ be a complex zero on the right and $p:P\to M$ a morphism. For $(P,p)$ to be a left resolution of $M$ , it is necessary and sufficient that the sequence

(1)

\cdots\longrightarrow P_n\xrightarrow{d_P}P_{n-1}\longrightarrow\cdots\longrightarrow P_1\xrightarrow{d_P}P_0\xrightarrow{p}M\longrightarrow0

be exact.

In fact, to say that $p:P\to M$ is a homomorphism means that $H_i(P)=0$ for $i>0$ and that $p_0$ induces an isomorphism of $\operatorname{Coker}(d_P:P_1\to P_0)$ onto $M$ .

Similarly:

Proposition 2bis. — Let $E$ be a complex null on the left and $e:M\to E$ a morphism. In order that $(e,E)$ be a right resolution of $M$ , it is necessary and sufficient that the sequence

(1 bis)

0\longrightarrow M\xrightarrow{e_0}E^0\xrightarrow{d_E}E^1\longrightarrow\cdots\longrightarrow E^n\xrightarrow{d_E}E^{n+1}\longrightarrow\cdots

be exact.

By abuse of language, one often says that sequence (1) (resp. (1 bis)) is a left (resp. right) resolution of $M$ .

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

Cited by

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Cites

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