Let
(6)
0 → P → S m → S m − 1 → … → S 1 → λ N → 0
0 \to P \to S_m \to S_{m-1} \to \ldots \to S_1 \xrightarrow{\lambda} N \to 0
0 → P → S m → S m − 1 → … → S 1 λ N → 0 (7)
0 → N ↠ R n → R n − 1 → … → R 1 → M → 0
0 \to N \twoheadrightarrow R_n \to R_{n-1} \to \ldots \to R_1 \to M \to 0
0 → N ↠ R n → R n − 1 → … → R 1 → M → 0 be two exact sequences of left A-modules with respective classes θ ∈ Ext^m_A(N, P) and θ' ∈ Ext^n_A(M, N). The class in Ext^{m+n}_A(M, P) associated with the exact sequence (8)
0 → P → S m → … → S 1 → μ ∘ λ R n → … → R 1 → M → 0
0 \to P \to S_m \to \ldots \to S_1 \xrightarrow{\mu \circ \lambda} R_n \to \ldots \to R_1 \to M \to 0
0 → P → S m → … → S 1 μ ∘ λ R n → … → R 1 → M → 0 is the composition product θ ∘ θ'.
Choose commutative diagrams
0 ⟶ P ⟶ I 0 ( P ) ⟶ … ⟶ I m − 1 ( P ) → δ P m − 1 I m ( P ) l P ↑ w 0 ↑ w m − 1 ↑ w m ↑ 0 ⟶ P ⟶ S m ⟶ … ⟶ S 1 → λ N ⟶ 0 ,
\begin{array}{cccccccccccc}
0 & \longrightarrow & P & \longrightarrow & I^0(P) & \longrightarrow & \ldots \longrightarrow & I^{m-1}(P) & \xrightarrow{\delta_P^{m-1}} & I^m(P) \\
& & ^{l_P}\uparrow & & ^{w^0}\uparrow & & & ^{w^{m-1}}\uparrow & & ^{w^m}\uparrow \\
0 & \longrightarrow & P & \longrightarrow & S_m & \longrightarrow & \ldots \longrightarrow & S_1 & \xrightarrow{\lambda} & N & \longrightarrow & 0 ,
\end{array}
0 0 ⟶ ⟶ P l P ↑ P ⟶ ⟶ I 0 ( P ) w 0 ↑ S m ⟶ ⟶ … ⟶ … ⟶ I m − 1 ( P ) w m − 1 ↑ S 1 δ P m − 1 λ I m ( P ) w m ↑ N ⟶ 0 , 0 ⟶ N → e N I 0 ( N ) ⟶ … ⟶ I n − 1 ( N ) ⟶ I n ( N ) 1 N ↑ v 0 ↑ v n − 1 ↑ v n ↑ 0 ⟶ N → μ R n ⟶ … ⟶ R 1 ⟶ M ⟶ 0.
\begin{array}{cccccccccccc}
0 & \longrightarrow & N & \xrightarrow{e_N} & I^0(N) & \longrightarrow & \ldots \longrightarrow & I^{n-1}(N) & \longrightarrow & I^n(N) \\
& & ^{1_N}\uparrow & & ^{v^0}\uparrow & & & ^{v^{n-1}}\uparrow & & ^{v^n}\uparrow \\
0 & \longrightarrow & N & \xrightarrow{\mu} & R_n & \longrightarrow & \ldots \longrightarrow & R_1 & \longrightarrow & M & \longrightarrow & 0 .
\end{array}
0 0 ⟶ ⟶ N 1 N ↑ N e N μ I 0 ( N ) v 0 ↑ R n ⟶ ⟶ … ⟶ … ⟶ I n − 1 ( N ) v n − 1 ↑ R 1 ⟶ ⟶ I n ( N ) v n ↑ M ⟶ 0. Since I m ( P ) I^m(P) I m ( P ) h 0 : I 0 ( N ) → I m ( P ) h^0 \colon I^0(N) \rightarrow I^m(P) h 0 : I 0 ( N ) → I m ( P ) w m = h 0 ∘ e N w^m = h^0 \circ e_N w m = h 0 ∘ e N h 0 h^0 h 0 h : I ( N ) → I ( P ) ( − m ) h \colon I(N) \rightarrow I(P)\, (-m) h : I ( N ) → I ( P ) ( − m ) w m = h 0 ∘ e N = h 0 ∘ v 0 ∘ μ w^m = h^0 \circ e_N = h^0 \circ v^0 \circ \mu w m = h 0 ∘ e N = h 0 ∘ v 0 ∘ μ
δ I m − 1 ∘ w m − 1 = w m ∘ λ = h 0 ∘ v 0 ∘ ( μ ∘ λ ) ,
\delta_I^{m-1} \circ w^{m-1} = w^m \circ \lambda = h^0 \circ v^0 \circ (\mu \circ \lambda),
δ I m − 1 ∘ w m − 1 = w m ∘ λ = h 0 ∘ v 0 ∘ ( μ ∘ λ ) , and the following diagram is commutative:
0 ⟶ P ⟶ I 0 ( P ) ⟶ ⋯ ⟶ I m − 1 ( P ) ⟶ I m ( P ) ⟶ I m + 1 ( P ) ⟶ ⋯ ⟶ I m + n ( P ) ↑ i P ↑ w 0 ↑ w m − 1 ↑ t 0 ↑ t 1 ↑ t n 0 ⟶ P ⟶ S m ⟶ ⋯ ⟶ S 1 → μ ∘ λ R n ⟶ R n − 1 ⟶ ⋯ ⟶ M ⟶ 0
\begin{array}{ccccccccccccccccccc}
0&\longrightarrow&P&\longrightarrow&I^0(P)&\longrightarrow&\cdots&\longrightarrow&I^{m-1}(P)&\longrightarrow&I^m(P)&\longrightarrow&I^{m+1}(P)&\longrightarrow&\cdots&\longrightarrow&I^{m+n}(P)\\
&&\uparrow^{i_P}&&\uparrow^{w^0}&&&&\uparrow^{w^{m-1}}&&\uparrow^{t^0}&&\uparrow^{t^1}&&&&\uparrow^{t^n}\\
0&\longrightarrow&P&\longrightarrow&S_m&\longrightarrow&\cdots&\longrightarrow&S_1&\xrightarrow[\mu\circ\lambda]{}&R_n&\longrightarrow&R_{n-1}&\longrightarrow&\cdots&\longrightarrow&M&\longrightarrow&0
\end{array}
0 0 ⟶ ⟶ P ↑ i P P ⟶ ⟶ I 0 ( P ) ↑ w 0 S m ⟶ ⟶ ⋯ ⋯ ⟶ ⟶ I m − 1 ( P ) ↑ w m − 1 S 1 ⟶ μ ∘ λ I m ( P ) ↑ t 0 R n ⟶ ⟶ I m + 1 ( P ) ↑ t 1 R n − 1 ⟶ ⟶ ⋯ ⋯ ⟶ ⟶ I m + n ( P ) ↑ t n M ⟶ 0 where t 0 = h 0 ∘ v 0 t^0=h^0\circ v^0 t 0 = h 0 ∘ v 0 t 1 = ( − 1 ) m h 1 ∘ v 1 t^1=(-1)^m h^1\circ v^1 t 1 = ( − 1 ) m h 1 ∘ v 1 … \ldots … t i = ( − 1 ) m i h i ∘ v i t^i=(-1)^{mi}h^i\circ v^i t i = ( − 1 ) mi h i ∘ v i … \ldots … t n = ( − 1 ) m n h n ∘ v n t^n=(-1)^{mn}h^n\circ v^n t n = ( − 1 ) mn h n ∘ v n
The class θ \theta θ ( − 1 ) m ( m + 1 ) / 2 w m ∈ Homgr A m ( N , I ( P ) ) (-1)^{m(m+1)/2}w^m\in\operatorname{Homgr}_A^m(N,I(P)) ( − 1 ) m ( m + 1 ) /2 w m ∈ Homgr A m ( N , I ( P )) α N , P \alpha_{N,P} α N , P ( − 1 ) m ( m + 1 ) / 2 h ∈ Homgr A m ( I ( N ) , I ( P ) ) (-1)^{m(m+1)/2}h\in\operatorname{Homgr}_A^m(I(N),I(P)) ( − 1 ) m ( m + 1 ) /2 h ∈ Homgr A m ( I ( N ) , I ( P )) θ ′ \theta' θ ′ ( − 1 ) n ( n + 1 ) / 2 v n ∈ Homgr n ( M , I ( N ) ) (-1)^{n(n+1)/2}v^n\in\operatorname{Homgr}^n(M,I(N)) ( − 1 ) n ( n + 1 ) /2 v n ∈ Homgr n ( M , I ( N )) ( − 1 ) ( m + n ) ( m + n + 1 ) / 2 t n ∈ Homgr m + n ( M , I ( P ) ) (-1)^{(m+n)(m+n+1)/2}t^n\in\operatorname{Homgr}^{m+n}(M,I(P)) ( − 1 ) ( m + n ) ( m + n + 1 ) /2 t n ∈ Homgr m + n ( M , I ( P ))
m ( m + 1 ) / 2 + n ( n + 1 ) / 2 = ( m + n ) ( m + n + 1 ) / 2 − m n .
m(m+1)/2+n(n+1)/2=(m+n)(m+n+1)/2-mn.
m ( m + 1 ) /2 + n ( n + 1 ) /2 = ( m + n ) ( m + n + 1 ) /2 − mn . 4. — Considérons un diagramme commutatif de A A A
0 ⟶ N ⟶ R n ⟶ R n − 1 ⟶ ⋯ ⟶ R 1 ⟶ M ⟶ 0 ↓ g ↓ ↓ ↓ ↓ f 0 ⟶ N ′ ⟶ R n ′ ⟶ R n − 1 ′ ⟶ ⋯ ⟶ R 1 ′ ⟶ M ′ ⟶ 0.
\begin{array}{ccccccccccccccc}
0&\longrightarrow&N&\longrightarrow&R_n&\longrightarrow&R_{n-1}&\longrightarrow&\cdots&\longrightarrow&R_1&\longrightarrow&M&\longrightarrow&0\\
&&\downarrow^{g}&&\downarrow&&\downarrow&&&&\downarrow&&\downarrow^{f}&&\\
0&\longrightarrow&N'&\longrightarrow&R'_n&\longrightarrow&R'_{n-1}&\longrightarrow&\cdots&\longrightarrow&R'_1&\longrightarrow&M'&\longrightarrow&0.
\end{array}
0 0 ⟶ ⟶ N ↓ g N ′ ⟶ ⟶ R n ↓ R n ′ ⟶ ⟶ R n − 1 ↓ R n − 1 ′ ⟶ ⟶ ⋯ ⋯ ⟶ ⟶ R 1 ↓ R 1 ′ ⟶ ⟶ M ↓ f M ′ ⟶ ⟶ 0 0. Let θ \theta θ θ ′ \theta' θ ′
Ext A n ( M , N ) (resp. Ext A n ( M ′ , N ′ ) ) .
\operatorname{Ext}_A^n(M,N)\quad\text{(resp. }\operatorname{Ext}_A^n(M',N')\text{)}.
Ext A n ( M , N ) (resp. Ext A n ( M ′ , N ′ ) ) . In Ext A n ( M , N ′ ) \operatorname{Ext}_A^n(M,N') Ext A n ( M , N ′ ) θ ′ ∘ f = g ∘ θ \theta'\circ f=g\circ\theta θ ′ ∘ f = g ∘ θ
Indeed, consider a commutative diagram
L n ( M ) → d n L n − 1 ( M ) ⟶ ⋯ ⟶ L 0 ( M ) → p M M ⟶ 0 ↓ u n ↓ ↓ ↓ 1 0 ⟶ N ⟶ R n ⟶ ⋯ ⟶ R 1 ⟶ M ⟶ 0 ↓ g ↓ ↓ ↓ f 0 ⟶ N ′ ⟶ R n ′ ⟶ ⋯ ⟶ R 1 ′ ⟶ M ′ ⟶ 0 ↓ ↓ ↓ ↓ v n 0 ⟶ N ′ → ϵ N ′ I 0 ( N ′ ) → δ 0 ⋯ ⟶ I n − 1 ( N ′ ) → δ n − 1 I n ( N ′ )
\begin{array}{ccccccccccccc}
L_n(M)&\xrightarrow{d_n}&L_{n-1}(M)&\longrightarrow&\cdots&\longrightarrow&L_0(M)&\xrightarrow{p_M}&M&\longrightarrow&0\\
\downarrow^{u_n}&&\downarrow&&&&\downarrow&&\downarrow^{1}\\
0&\longrightarrow&N&\longrightarrow&R_n&\longrightarrow&\cdots&\longrightarrow&R_1&\longrightarrow&M&\longrightarrow&0\\
&&\downarrow^{g}&&\downarrow&&&&\downarrow&&\downarrow^{f}\\
0&\longrightarrow&N'&\longrightarrow&R'_n&\longrightarrow&\cdots&\longrightarrow&R'_1&\longrightarrow&M'&\longrightarrow&0\\
&&\downarrow&&\downarrow&&&&\downarrow&&\downarrow^{v^n}\\
0&\longrightarrow&N'&\xrightarrow{\epsilon_{N'}}&I^0(N')&\xrightarrow{\delta^0}&\cdots&\longrightarrow&I^{n-1}(N')&\xrightarrow{\delta^{n-1}}&I^n(N')
\end{array}
L n ( M ) ↓ u n 0 0 0 d n ⟶ ⟶ ⟶ L n − 1 ( M ) ↓ N ↓ g N ′ ↓ N ′ ⟶ ⟶ ⟶ ϵ N ′ ⋯ R n ↓ R n ′ ↓ I 0 ( N ′ ) ⟶ ⟶ ⟶ δ 0 L 0 ( M ) ↓ ⋯ ⋯ ⋯ p M ⟶ ⟶ ⟶ M ↓ 1 R 1 ↓ R 1 ′ ↓ I n − 1 ( N ′ ) ⟶ ⟶ ⟶ δ n − 1 0 M ↓ f M ′ ↓ v n I n ( N ′ ) ⟶ ⟶ 0 0 By definition, θ ′ ∘ f \theta'\circ f θ ′ ∘ f ( − 1 ) n ( n + 1 ) / 2 v n ∘ f ∘ p M ∈ Homgr n ( L ( M ) , I ( N ′ ) ) (-1)^{n(n+1)/2}v^n\circ f\circ p_M\in\operatorname{Homgr}^n(L(M),I(N')) ( − 1 ) n ( n + 1 ) /2 v n ∘ f ∘ p M ∈ Homgr n ( L ( M ) , I ( N ′ )) g ∘ θ g\circ\theta g ∘ θ ( − 1 ) n ( n + 1 ) / 2 ϵ N ′ ∘ g ∘ u n (-1)^{n(n+1)/2}\epsilon_{N'}\circ g\circ u_n ( − 1 ) n ( n + 1 ) /2 ϵ N ′ ∘ g ∘ u n
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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