a) Let
(ε)
0 → P ′ → f P → g P ′ ′ → 0
0 \to P' \xrightarrow{f} P \xrightarrow{g} P'' \to 0
0 → P ′ f P g P ′′ → 0 be an exact sequence of right
A A A -modules,
θ ∈ E x t A 1 ( P ′ ′ , P ′ ) \theta \in \mathrm{Ext}_A^1(P'', P') θ ∈ Ext A 1 ( P ′′ , P ′ ) the associated class,
M M M a left
A A A -module. The connecting homomorphism
δ n ( E , M ) : T o r n A ( P ′ ′ , M ) → T o r n − 1 A ( P ′ , M ) is the mapping γ ↦ θ ∘ γ .
\delta_n(\mathscr{E}, M) : \mathrm{Tor}_n^A(P'', M) \to \mathrm{Tor}_{n-1}^A(P', M) \text{ is the mapping } \gamma \mapsto \theta \circ \gamma .
δ n ( E , M ) : Tor n A ( P ′′ , M ) → Tor n − 1 A ( P ′ , M ) is the mapping γ ↦ θ ∘ γ . b) Let
(ε₁)
0 → M ′ → M → M ′ ′ → 0
0 \to M' \to M \to M'' \to 0
0 → M ′ → M → M ′′ → 0 be an exact sequence of left
A A A -modules,
θ 1 ∈ E x t A 1 ( M ′ ′ , M ′ ) \theta_1 \in \mathrm{Ext}_A^1(M'', M') θ 1 ∈ Ext A 1 ( M ′′ , M ′ ) the associated class,
P P P a right
A A A -module. The connecting homomorphism
δ n ( P , E 1 ) : T o r n A ( P , M ′ ′ ) → T o r n − 1 A ( P , M ′ ) is the mapping γ ↦ θ 1 ∘ γ .
\delta_n(P, \mathscr{E}_1) : \mathrm{Tor}_n^A(P, M'') \to \mathrm{Tor}_{n-1}^A(P, M') \text{ is the mapping } \gamma \mapsto \theta_1 \circ \gamma .
δ n ( P , E 1 ) : Tor n A ( P , M ′′ ) → Tor n − 1 A ( P , M ′ ) is the mapping γ ↦ θ 1 ∘ γ . Let
γ ∈ T o r n A ( P ′ ′ , M ) \gamma \in \mathrm{Tor}_n^A(P'', M) γ ∈ Tor n A ( P ′′ , M ) be the class of a cycle
z ′ ′ ∈ Z ˙ n ( L ( P ′ ′ ) ⊗ A L ( M ) ) z'' \in \dot{Z}_n(L(P'') \otimes_A L(M)) z ′′ ∈ Z ˙ n ( L ( P ′′ ) ⊗ A L ( M )) , and let
0 ⟶ P ′ → f P → g P ′ ′ ⟶ 0 ↑ u 1 ↑ u 0 ↑ 1 L 1 ( P ′ ′ ) → d 1 L 0 ( P ′ ′ ) → p 0 ′ ′ P ′ ′ ⟶ 0
\begin{array}{ccccccccc}
0 & \longrightarrow & P' & \xrightarrow{f} & P & \xrightarrow{g} & P'' & \longrightarrow & 0 \\
& & \uparrow u_1 & & \uparrow u_0 & & \uparrow 1 & & \\
L_1(P'') & \xrightarrow{d_1} & L_0(P'') & \xrightarrow{p_0''} & P'' & \longrightarrow & 0
\end{array}
0 L 1 ( P ′′ ) ⟶ d 1 P ′ ↑ u 1 L 0 ( P ′′ ) f p 0 ′′ P ↑ u 0 P ′′ g ⟶ P ′′ ↑ 1 0 ⟶ 0 be a commutative diagram. We shall denote by
p ′ : L ( P ′ ) → P ′ p' : L(P') \to P' p ′ : L ( P ′ ) → P ′ and
p ′ ′ : L ( P ′ ′ ) → P ′ ′ p'' : L(P'') \to P'' p ′′ : L ( P ′′ ) → P ′′ the canonical morphisms of complexes. By definition,
δ ( γ ) ∈ T o r n − 1 A ( P ′ , M ) \delta(\gamma) \in \mathrm{Tor}_{n-1}^A(P', M) δ ( γ ) ∈ Tor n − 1 A ( P ′ , M ) is obtained as follows: one chooses
x ∈ P ⊗ L n ( M ) x \in P \otimes L_n(M) x ∈ P ⊗ L n ( M ) such that
( g ⊗ 1 ) ( x ) = ( p ′ ′ ⊗ 1 ) ( z ′ ′ ) (g \otimes 1)(x) = (p'' \otimes 1)(z'') ( g ⊗ 1 ) ( x ) = ( p ′′ ⊗ 1 ) ( z ′′ ) and
δ ( γ ) \delta(\gamma) δ ( γ ) is the class of cycles
z ′ ∈ Z n − 1 ( L ( P ′ ) ⊗ L ( M ) ) z' \in Z_{n-1}(L(P') \otimes L(M)) z ′ ∈ Z n − 1 ( L ( P ′ ) ⊗ L ( M )) such that
( f ⊗ 1 ) ( p ′ ⊗ 1 ) ( z ′ ) = ( 1 ⊗ d n ) ( x ) .
(f \otimes 1)(p' \otimes 1)(z') = (1 \otimes d_n)(x) .
( f ⊗ 1 ) ( p ′ ⊗ 1 ) ( z ′ ) = ( 1 ⊗ d n ) ( x ) . For
0 ≤ i ≤ n 0 \leq i \leq n 0 ≤ i ≤ n , let
z i ′ ′ z''_i z i ′′ be the component of
z ′ ′ z'' z ′′ in
L i ( P ′ ′ ) ⊗ L n − i ( M ) L_i(P'') \otimes L_{n-i}(M) L i ( P ′′ ) ⊗ L n − i ( M ) ; we have
0 = D z ′ ′ = ∑ i ( d i ⊗ 1 + ( − 1 ) i ⊗ d n − i ) ( z i ′ ′ ) ,
0 = Dz'' = \sum_i (d_i \otimes 1 + (-1)^i \otimes d_{n-i})(z''_i) ,
0 = D z ′′ = i ∑ ( d i ⊗ 1 + ( − 1 ) i ⊗ d n − i ) ( z i ′′ ) , hence
( d i ⊗ 1 ) ( z i ′ ′ ) = ( − 1 ) i ⊗ d n − i + 1 ( z i − 1 ′ ′ ) (d_i \otimes 1)(z''_i) = (-1)^i \otimes d_{n-i+1}(z''_{i-1}) ( d i ⊗ 1 ) ( z i ′′ ) = ( − 1 ) i ⊗ d n − i + 1 ( z i − 1 ′′ ) and in particular
( d 1 ⊗ 1 ) ( z 1 ′ ′ ) = − 1 ⊗ d n ( z 0 ′ ′ ) .
(d_1 \otimes 1)(z''_1) = -1 \otimes d_n(z''_0) .
( d 1 ⊗ 1 ) ( z 1 ′′ ) = − 1 ⊗ d n ( z 0 ′′ ) . Let us then choose
x = ( u 0 ⊗ 1 ) ( z 0 ′ ′ ) x = (u_0 \otimes 1)(z''_0) x = ( u 0 ⊗ 1 ) ( z 0 ′′ ) : we indeed have
( g ⊗ 1 ) ( x ) = ( p 0 ′ ′ ⊗ 1 ) ( z 0 ′ ′ ) = ( p ′ ′ ⊗ 1 ) ( z ′ ′ ) .
(g \otimes 1)(x) = (p_0'' \otimes 1)(z''_0) = (p'' \otimes 1)(z'') .
( g ⊗ 1 ) ( x ) = ( p 0 ′′ ⊗ 1 ) ( z 0 ′′ ) = ( p ′′ ⊗ 1 ) ( z ′′ ) . Since
( 1 ⊗ d n ) ( x ) = ( u 0 ⊗ 1 ) ( 1 ⊗ d n ) ( z 0 ′ ′ ) = − ( u 0 ⊗ 1 ) ( d 1 ⊗ 1 ) ( z 1 ′ ′ ) = − ( f ⊗ 1 ) ( u 1 ⊗ 1 ) ( z 1 ′ ′ ) ,
(1 \otimes d_n)(x) = (u_0 \otimes 1)(1 \otimes d_n)(z''_0) = - (u_0 \otimes 1)(d_1 \otimes 1)(z''_1)
= - (f \otimes 1)(u_1 \otimes 1)(z''_1) ,
( 1 ⊗ d n ) ( x ) = ( u 0 ⊗ 1 ) ( 1 ⊗ d n ) ( z 0 ′′ ) = − ( u 0 ⊗ 1 ) ( d 1 ⊗ 1 ) ( z 1 ′′ ) = − ( f ⊗ 1 ) ( u 1 ⊗ 1 ) ( z 1 ′′ ) , it follows that δ ( γ ) \delta(\gamma) δ ( γ ) z ′ ∈ Z n − 1 ( L ( P ′ ) ⊗ A L ( M ) ) z' \in Z_{n-1}(L(P') \otimes_A L(M)) z ′ ∈ Z n − 1 ( L ( P ′ ) ⊗ A L ( M )) ( p ′ ⊗ 1 ) ( z ′ ) = − ( u 1 ⊗ 1 ) ( z 1 ′ ′ ) (p' \otimes 1)(z') = - (u_1 \otimes 1)(z''_1) ( p ′ ⊗ 1 ) ( z ′ ) = − ( u 1 ⊗ 1 ) ( z 1 ′′ ) θ \theta θ E x t A 1 ( P ′ ′ , P ′ ) → H 1 ( H o m g r A ( L ( P ′ ′ ) , P ′ ) ) \mathrm{Ext}_A^1(P'', P') \to H^1(\mathrm{Homgr}_A(L(P''), P')) Ext A 1 ( P ′′ , P ′ ) → H 1 ( Homgr A ( L ( P ′′ ) , P ′ )) f : L ( P ′ ′ ) ( 1 ) → P ′ f : L(P'')(1) \to P' f : L ( P ′′ ) ( 1 ) → P ′ − u 1 - u_1 − u 1 θ ∘ γ \theta \circ \gamma θ ∘ γ
z ‾ ′ ∈ Z n − 1 ( L ( P ′ ) ⊗ A L ( M ) ) such that ( p ⊗ 1 ) ( z ‾ ′ ) = f ( z ′ ′ ) = − ( u 1 ⊗ 1 ) ( z 1 ′ ′ ) ,
\overline{z}' \in Z_{n-1}(L(P') \otimes_A L(M)) \quad \text{such that} \quad (p \otimes 1)(\overline{z}') = f(z'') = - (u_1 \otimes 1)(z''_1),
z ′ ∈ Z n − 1 ( L ( P ′ ) ⊗ A L ( M )) such that ( p ⊗ 1 ) ( z ′ ) = f ( z ′′ ) = − ( u 1 ⊗ 1 ) ( z 1 ′′ ) , which completes the proof of a). Assertion b) follows from a) by the commutation isomorphisms.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
Nothing.
Nothing.