Library / Algebra / Chapter X / § 2
Theorem 1
Tag 0DRJ, permanent. Cite it as [Bourbaki, Tag 0DRJ].
Algebra, chapter X, § 2. Complexes de A-modules
The unlimited sequence of homomorphisms of A A A
⋯ ⟶ H n + 1 ( C ′ ′ ) → ∂ n + 1 ( u , v ) H n ( C ′ ) → H n ( u ) H n ( C ) → H n ( v ) H n ( C ′ ′ )
\cdots \longrightarrow H_{n+1}(C'') \xrightarrow{\partial_{n+1}(u,v)} H_n(C') \xrightarrow{H_n(u)} H_n(C) \xrightarrow{H_n(v)} H_n(C'')
⋯ ⟶ H n + 1 ( C ′′ ) ∂ n + 1 ( u , v ) H n ( C ′ ) H n ( u ) H n ( C ) H n ( v ) H n ( C ′′ ) → ∂ n ( u , v ) H n − 1 ( C ′ ) → H n − 1 ( u ) H n − 1 ( C ) → H n − 1 ( v ) H n − 1 ( C ′ ′ ) → ∂ n − 1 ( u , v ) H n − 2 ( C ′ ) ⟶ ⋯
\xrightarrow{\partial_n(u,v)} H_{n-1}(C') \xrightarrow{H_{n-1}(u)} H_{n-1}(C) \xrightarrow{H_{n-1}(v)} H_{n-1}(C'') \xrightarrow{\partial_{n-1}(u,v)} H_{n-2}(C') \longrightarrow \cdots
∂ n ( u , v ) H n − 1 ( C ′ ) H n − 1 ( u ) H n − 1 ( C ) H n − 1 ( v ) H n − 1 ( C ′′ ) ∂ n − 1 ( u , v ) H n − 2 ( C ′ ) ⟶ ⋯ is exact.
This sequence is called the exact homology sequence associated with the exact sequence ( u , v ) (u,v) ( u , v ) exact triangle of A A A
H ( C ) ↗ H ( u ) ↘ H ( v ) H ( C ′ ) ← ∂ ( u , v ) H ( C ′ ′ ) .
\begin{array}{ccc}
&&H(C)\\
&\nearrow^{H(u)}&\searrow^{H(v)}\\
H(C')&\xleftarrow{\partial(u,v)}&H(C'') .
\end{array}
H ( C ′ ) ↗ H ( u ) ∂ ( u , v ) H ( C ) ↘ H ( v ) H ( C ′′ ) . Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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