The mapping σ ( C , C ′ ) : C ⊗ A C ′ → C ′ ∘ ⊗ A C ∘ \sigma(C, C') : C \otimes_A C' \to {C'}^\circ \otimes_A C^\circ σ ( C , C ′ ) : C ⊗ A C ′ → C ′ ∘ ⊗ A C ∘ k k k σ ( C ′ ∘ , C ∘ ) \sigma({C'}^\circ, C^\circ) σ ( C ′ ∘ , C ∘ )
Since the mappings σ ( C , C ′ ) \sigma(C, C') σ ( C , C ′ ) σ ( C ′ ∘ , C ∘ ) \sigma({C'}^\circ, C^\circ) σ ( C ′ ∘ , C ∘ ) σ ( C , C ′ ) \sigma(C, C') σ ( C , C ′ )
x ∈ C p = C p ∘ , x ′ ∈ C q ′ = C q ′ ∘ , p , q ∈ Z ,
x \in C_p = C_p^\circ, \quad x' \in C'_q = C_{q'}^\circ, \quad p, q \in \mathbf{Z},
x ∈ C p = C p ∘ , x ′ ∈ C q ′ = C q ′ ∘ , p , q ∈ Z , we have, denoting by
D ′ D' D ′ the differential of
C ′ ⊗ A ∘ C C' \otimes_{A^\circ} C C ′ ⊗ A ∘ C ,
σ ( C , C ′ ) ∘ D ( x ⊗ x ′ ) = σ ( C , C ′ ) ( d x ⊗ x ′ + ( − 1 ) p x ⊗ d ′ x ′ ) = = ( − 1 ) ( p + 1 ) q x ′ ⊗ d x + ( − 1 ) p + p ( q + 1 ) d ′ x ′ ⊗ x = ( − 1 ) p q d ′ x ′ ⊗ x + ( − 1 ) p q + q x ′ ⊗ d x = ( − 1 ) p q D ′ ( x ′ ⊗ x ) = D ∘ ∘ σ ( C , C ′ ) ( x ⊗ x ′ ) ;
\begin{align*}
\sigma(C, C') \circ D(x \otimes x') &= \sigma(C, C') (dx \otimes x' + (-1)^p x \otimes d'x') = \\
&= (-1)^{(p+1)q} x' \otimes dx + (-1)^{p+p(q+1)} d'x' \otimes x = (-1)^{pq} d'x' \otimes x + (-1)^{pq+q} x' \otimes dx \\
&= (-1)^{pq} D'(x' \otimes x) = D^\circ \circ \sigma(C, C') (x \otimes x');
\end{align*}
σ ( C , C ′ ) ∘ D ( x ⊗ x ′ ) = σ ( C , C ′ ) ( d x ⊗ x ′ + ( − 1 ) p x ⊗ d ′ x ′ ) = = ( − 1 ) ( p + 1 ) q x ′ ⊗ d x + ( − 1 ) p + p ( q + 1 ) d ′ x ′ ⊗ x = ( − 1 ) pq d ′ x ′ ⊗ x + ( − 1 ) pq + q x ′ ⊗ d x = ( − 1 ) pq D ′ ( x ′ ⊗ x ) = D ∘ ∘ σ ( C , C ′ ) ( x ⊗ x ′ ) ; which gives
σ ( C , C ′ ) ∘ D = D ∘ ∘ σ ( C , C ′ ) \sigma(C, C') \circ D = D^\circ \circ \sigma(C, C') σ ( C , C ′ ) ∘ D = D ∘ ∘ σ ( C , C ′ ) , whence the assertion sought.
The isomorphism σ ( C , C ′ ) : C ⊗ A C ′ → C ′ ∘ ⊗ A C ∘ \sigma(C, C') : C \otimes_A C' \to {C'}^\circ \otimes_A C^\circ σ ( C , C ′ ) : C ⊗ A C ′ → C ′ ∘ ⊗ A C ∘ commutation isomorphism of the tensor product of the complexes C C C C ′ C' C ′
If u : C → C 1 u : C \to C_1 u : C → C 1 v : C ′ → C 1 ′ v : C' \to C'_1 v : C ′ → C 1 ′
C ⊗ A C ′ → σ ( C , C ′ ) C ′ ∘ ⊗ A C ∘ u ⊗ u ′ ↓ ↓ u ′ ⊗ u C 1 ⊗ A C 1 ′ → σ ( C 1 , C 1 ′ ) C 1 ′ ∘ ⊗ A C 1 ∘ .
\begin{array}{ccc}
C \otimes_A C' & \xrightarrow{\sigma(C, C')} & {C'}^\circ \otimes_A C^\circ \\
u \otimes u' \downarrow & & \downarrow u' \otimes u \\
C_1 \otimes_A C'_1 & \xrightarrow{\sigma(C_1, C'_1)} & C'_1{}^\circ \otimes_A C_1^\circ.
\end{array}
C ⊗ A C ′ u ⊗ u ′ ↓ C 1 ⊗ A C 1 ′ σ ( C , C ′ ) σ ( C 1 , C 1 ′ ) C ′ ∘ ⊗ A C ∘ ↓ u ′ ⊗ u C 1 ′ ∘ ⊗ A C 1 ∘ . Suppose for the end of this number that the ring A A A commutative (cf. No. 9 for the general case).
Let C , C ′ , C ′ ′ C, C', C'' C , C ′ , C ′′ A A A A A A
φ : ( C ⊗ A C ′ ) ⊗ A C ′ ′ → C ⊗ A ( C ′ ⊗ A C ′ ′ )
\varphi : (C \otimes_A C') \otimes_A C'' \to C \otimes_A (C' \otimes_A C'')
φ : ( C ⊗ A C ′ ) ⊗ A C ′′ → C ⊗ A ( C ′ ⊗ A C ′′ ) is an
isomorphism of complexes , as is immediately verified using the definitions.
More generally, let ( C ( i ) , d ( i ) ) i ∈ I (C^{(i)}, d^{(i)})_{i \in I} ( C ( i ) , d ( i ) ) i ∈ I family of complexes of A A A I I I finite and totally ordered ; we shall identify I I I [ 1 , r ] [1, r] [ 1 , r ] N \mathbf{N} N A A A C = ⨂ i = 1 r C ( i ) C = \bigotimes_{i=1}^r C^{(i)} C = ⨂ i = 1 r C ( i )
C n = ∑ p 1 + p 2 + ⋯ + p r = n ( C ( 1 ) ) p 1 ⊗ ( C ( 2 ) ) p 2 ⊗ ⋯ ⊗ ( C ( r ) ) p r ,
C_n = \sum_{p_1 + p_2 + \cdots + p_r = n} (C^{(1)})_{p_1} \otimes (C^{(2)})_{p_2} \otimes \cdots \otimes (C^{(r)})_{p_r},
C n = p 1 + p 2 + ⋯ + p r = n ∑ ( C ( 1 ) ) p 1 ⊗ ( C ( 2 ) ) p 2 ⊗ ⋯ ⊗ ( C ( r ) ) p r , and define a graded
A A A -endomorphism of degree
( − 1 ) (-1) ( − 1 ) of
C C C by
D ( x 1 ⊗ … ⊗ x r ) = ∑ j = 1 r ( − 1 ) p 1 + ⋯ + p j − 1 x 1 ⊗ … ⊗ x j − 1 ⊗ d j x j ⊗ x j + 1 ⊗ … ⊗ x r
D(x_1 \otimes \ldots \otimes x_r) = \sum_{j=1}^r (-1)^{p_1 + \cdots + p_{j-1}} x_1 \otimes \ldots \otimes x_{j-1} \otimes d_j x_j \otimes x_{j+1} \otimes \ldots \otimes x_r
D ( x 1 ⊗ … ⊗ x r ) = j = 1 ∑ r ( − 1 ) p 1 + ⋯ + p j − 1 x 1 ⊗ … ⊗ x j − 1 ⊗ d j x j ⊗ x j + 1 ⊗ … ⊗ x r where
x i ∈ ( C ( i ) ) p i x_i \in (C^{(i)})_{p_i} x i ∈ ( C ( i ) ) p i for
i = 1 , … , n i = 1, \ldots, n i = 1 , … , n . Then
( C , D ) (C, D) ( C , D ) is a complex of
A A A -modules called the tensor product complex of the family
( C i , d i ) (C_i, d_i) ( C i , d i ) . For every strictly increasing sequence
r 0 , . . . , r k r_0, ..., r_k r 0 , ... , r k of
[ 0 , r ] [0, r] [ 0 , r ] such that
r 0 = 0 , r k = r r_0 = 0,\ r_k = r r 0 = 0 , r k = r , the canonical associativity isomorphism
⨂ j = 0 k − 1 ( ⨂ i = r j + 1 r j + 1 C ( i ) ) → ⨂ i = 1 r C ( i )
\bigotimes_{j=0}^{k-1} \left( \bigotimes_{i=r_j+1}^{r_{j+1}} C^{(i)} \right) \to \bigotimes_{i=1}^r C^{(i)}
j = 0 ⨂ k − 1 i = r j + 1 ⨂ r j + 1 C ( i ) → i = 1 ⨂ r C ( i ) is an isomorphism of complexes.
We define as above a graded homomorphism of degree 0
γ ( ( C ( i ) ) ) : ⨂ i ∈ I H ( C ( i ) ) → H ( ⨂ i ∈ I C ( i ) )
\gamma((C^{(i)})) : \bigotimes_{i \in I} H(C^{(i)}) \to H \left( \bigotimes_{i \in I} C^{(i)} \right)
γ (( C ( i ) )) : i ∈ I ⨂ H ( C ( i ) ) → H ( i ∈ I ⨂ C ( i ) ) Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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