Proposition 9

Tag 0DVY, permanent. Cite it as [Bourbaki, Tag 0DVY].

Algebra, chapter X, § 4. Produits de torsion

Let f : M → M $_1$ be a homomorphism of right a-modules and

\begin{array}{ccccccccc} 0 & \longrightarrow & N' & \xrightarrow{u} & N & \xrightarrow{v} & N'' & \longrightarrow & 0 \\ & & g' \downarrow & & g \downarrow & & g'' \downarrow & & \\ 0 & \longrightarrow & N'_1 & \xrightarrow{u_1} & N_1 & \xrightarrow{v_1} & N''_1 & \longrightarrow & 0 \end{array}

(E $_1$ )

a commutative diagram with exact rows of homomorphisms of left a-modules. The diagram of k-modules

\begin{array}{ccc} \operatorname{Tor}^A(M, N'') & \xrightarrow{\partial(M, E)} & \operatorname{Tor}^A(M, N') \\ \operatorname{Tor}^A(f, g'') \downarrow & & \operatorname{Tor}^A(f, g') \downarrow \\ \operatorname{Tor}^A(M_1, N''_1) & \xrightarrow{\partial(M_1, E_1)} & \operatorname{Tor}^A(M_1, N'_1) \end{array}

is commutative.

This follows from X, p. 31, prop. 2, applied to the commutative diagram

\begin{array}{ccccccccc} 0 & \longrightarrow & L(M) \otimes_A N' & \xrightarrow{1 \otimes u} & L(M) \otimes_A N & \xrightarrow{1 \otimes v} & L(M) \otimes_A N'' & \longrightarrow & 0 \\ & & L(f) \otimes g' \downarrow & & L(f) \otimes g \downarrow & & L(f) \otimes g'' \downarrow & & \\ 0 & \longrightarrow & L(M_1) \otimes_A N'_1 & \xrightarrow{1 \otimes u_1} & L(M_1) \otimes_A N_1 & \xrightarrow{1 \otimes v_1} & L(M_1) \otimes_A N''_1 & \longrightarrow & 0 . \end{array}

In an analogous way, if N is a left a-module and

(F)

0 \to M' \xrightarrow{r} M \xrightarrow{s} M'' \to 0

an exact sequence of right a-modules, one defines linking homomorphisms

\partial(F, N) : \operatorname{Tor}^A(M'', N) \to \operatorname{Tor}^A(M', N)

\partial_n(F, N) : \operatorname{Tor}_n^A(M'', N) \to \operatorname{Tor}_{n-1}^A(M', N)

by $\partial(\mathscr F,\mathbf N)=\overline{\psi}_{\mathbf N}(M')^{-1}\circ\partial(\mathscr F^{\mathbf N})\circ\overline{\psi}_{\mathbf N}(M'')$ , where $\partial(\mathscr F^{\mathbf N})$ is the linking homomorphism of the exact sequence

(\mathscr F_{\mathbf N})\qquad 0\longrightarrow M'\otimes_A L(N)\longrightarrow M\otimes_A L(N)\longrightarrow M''\otimes_A L(N)\longrightarrow 0

deduced from $\mathscr F$ , and one has:

THEOREM 1 bis. — The left-unbounded sequence of homomorphisms of $k$ -modules

\longrightarrow \operatorname{Tor}_n^A(M',N)\xrightarrow{\operatorname{Tor}_n^A(r,1)}\operatorname{Tor}_n^A(M,N)\xrightarrow{\operatorname{Tor}_n^A(s,1)}\operatorname{Tor}_n^A(M'',N)\xrightarrow{\partial_n(\mathscr F,N)}\operatorname{Tor}_{n-1}^A(M',N)

\cdots\longrightarrow \operatorname{Tor}_1^A(M'',N)\xrightarrow{\partial_1(\mathscr F,N)}M'\otimes_A N\xrightarrow{r\otimes1}M\otimes_A N\xrightarrow{s\otimes1}M''\otimes_A N\longrightarrow0

is exact.

We leave to the reader the task of stating and proving the properties analogous to the corollaries of th. 1 and to prop. 9. Moreover:

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

Nothing.