Library / Algebra / Chapter X / § 4
Theorem 1
Tag 0DVU, permanent. Cite it as [Bourbaki, Tag 0DVU].
Algebra, chapter X, § 4. Produits de torsion
The left-infinite sequence of k k k
⋯ ⟶ T o r n A ( M , N ′ ) → T o r n A ( 1 , u ) T o r n A ( M , N ) → T o r n A ( 1 , v ) T o r n A ( M , N ′ ′ )
\cdots \longrightarrow \mathrm{Tor}_n^A(M, N') \xrightarrow{\mathrm{Tor}_n^A(1, u)} \mathrm{Tor}_n^A(M, N) \xrightarrow{\mathrm{Tor}_n^A(1, v)} \mathrm{Tor}_n^A(M, N'')
⋯ ⟶ Tor n A ( M , N ′ ) Tor n A ( 1 , u ) Tor n A ( M , N ) Tor n A ( 1 , v ) Tor n A ( M , N ′′ ) → ∂ n ( M , E ) T o r n − 1 A ( M , N ′ ) → T o r n − 1 A ( 1 , u ) ⋯ → T o r 1 A ( 1 , v ) T o r 1 A ( M , N ′ ′ )
\xrightarrow{\partial_n(M, \mathscr{E})} \mathrm{Tor}_{n-1}^A(M, N') \xrightarrow{\mathrm{Tor}_{n-1}^A(1, u)} \cdots \xrightarrow{\mathrm{Tor}_1^A(1, v)} \mathrm{Tor}_1^A(M, N'')
∂ n ( M , E ) Tor n − 1 A ( M , N ′ ) Tor n − 1 A ( 1 , u ) ⋯ Tor 1 A ( 1 , v ) Tor 1 A ( M , N ′′ ) → c ^ 1 ( M , E ) M ⊗ A N ′ → 1 ⊗ u M ⊗ A N → 1 ⊗ v M ⊗ A N ′ ′ ⟶ 0
\xrightarrow{\hat{c}_1(M, \mathscr{E})} M \otimes_A N' \xrightarrow{1 \otimes u} M \otimes_A N \xrightarrow{1 \otimes v} M \otimes_A N'' \longrightarrow 0
c ^ 1 ( M , E ) M ⊗ A N ′ 1 ⊗ u M ⊗ A N 1 ⊗ v M ⊗ A N ′′ ⟶ 0 is exact.
To see this, consider the diagram
T o r ( M , N ′ ) → T o r ( 1 , u ) T o r ( M , N ) → T o r ( 1 , v ) T o r ( M , N ′ ′ ) → ∂ ( M , E ) T o r ( M , N ′ ) → T o r ( 1 , u ) T o r ( M , N ) ψ M ( N ′ ) ↓ ψ M ( N ) ↓ ψ M ( N ′ ′ ) ↓ ψ M ( N ′ ) ↓ ψ M ( N ′ ) ↓ H ( L ( M ) ⊗ N ′ ) → H ( 1 ⊗ u ) H ( L ( M ) ⊗ N ) → H ( 1 ⊗ v ) H ( L ( M ⊗ N ′ ′ ) → ∂ ( M M E ) H ( L ( M ) ⊗ N ′ ) → H ( 1 ⊗ u ) H ( L ( M ) ⊗ N )
\begin{array}{ccccccccc}
\mathrm{Tor}(M, N') & \xrightarrow{\mathrm{Tor}(1, u)} & \mathrm{Tor}(M, N) & \xrightarrow{\mathrm{Tor}(1, v)} & \mathrm{Tor}(M, N'') & \xrightarrow{\partial(M, \mathscr{E})} & \mathrm{Tor}(M, N') & \xrightarrow{\mathrm{Tor}(1, u)} & \mathrm{Tor}(M, N) \\
\psi_M(N') \downarrow & & \psi_M(N) \downarrow & & \psi_M(N'') \downarrow & & \psi_M(N') \downarrow & & \psi_M(N') \downarrow \\
H(L(M) \otimes N') & \xrightarrow{H(1 \otimes u)} & H(L(M) \otimes N) & \xrightarrow{H(1 \otimes v)} & H(L(M \otimes N'') & \xrightarrow{\partial^{(M\mathscr{ME})}} & H(L(M) \otimes N') & \xrightarrow{H(1 \otimes u)} & H(L(M) \otimes N)
\end{array}
Tor ( M , N ′ ) ψ M ( N ′ ) ↓ H ( L ( M ) ⊗ N ′ ) Tor ( 1 , u ) H ( 1 ⊗ u ) Tor ( M , N ) ψ M ( N ) ↓ H ( L ( M ) ⊗ N ) Tor ( 1 , v ) H ( 1 ⊗ v ) Tor ( M , N ′′ ) ψ M ( N ′′ ) ↓ H ( L ( M ⊗ N ′′ ) ∂ ( M , E ) ∂ ( M ME ) Tor ( M , N ′ ) ψ M ( N ′ ) ↓ H ( L ( M ) ⊗ N ′ ) Tor ( 1 , u ) H ( 1 ⊗ u ) Tor ( M , N ) ψ M ( N ′ ) ↓ H ( L ( M ) ⊗ N ) It commutes according to (X, p. 69, Remark 3) and Definition 2. Furthermore, the bottom row is exact (X, p. 30, Thm. 1), and the different ψ M \psi_M ψ M
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
Cited by
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Cites
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Translated with machine help, not yet reviewed by a person.