Example

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

If $C$ has zero differential, one has $Z(C) = C$ , $B(C) = 0$ and $H(C)$ identifies canonically with $C$ .

There are exact sequences, called canonical:

(I $_n$ )

0 \to Z_n(C) \to C_n \xrightarrow{\delta_n} B_{n-1}(C) \to 0

(II

_n

)

0 \to B_n(C) \to Z_n(C) \to H_n(C) \to 0

(III

_n

)

0 \to B_n(C) \to C_n \to C_n/B_n(C) \to 0

(IV

_n

)

0 \to H_n(C) \to C_n/B_n(C) \xrightarrow{\overline{\delta}_n} B_{n-1}(C) \to 0

where $\delta_n$ and $\overline{\delta}_n$ are deduced from $d_n$ . By combining (IV $_n$ ) and (II $_{n-1}$ ), one obtains the exact sequence

(V $_n$ )

0 \to H_n(C) \to C_n/B_n(C) \to Z_{n-1}(C) \to H_{n-1}(C) \to 0 ,

which is also written, changing $n$ into $-n$

(V $^n$ )

0 \to H^n(C) \to C^n/B^n(C) \to Z^{n+1}(C) \to H^{n+1}(C) \to 0 .

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

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