Proposition 19

Tag 0677, permanent. Cite it as [Bourbaki, Tag 0677].

Let

\begin{array}{ccc} K & \xrightarrow{\rho} & K' \\ \downarrow \eta & & \downarrow \eta' \\ A & \xrightarrow{u} & A' \end{array}

be a commutative diagram of commutative ring homomorphisms, $\eta$ (resp. $\eta'$ ) making

A (resp. $A'$ ) into a $K$ -algebra (resp. $K'$ -algebra). There exists one and only one $A$ -linear mapping $v : \Omega_K(A) \to \Omega_{K'}(A')$ rendering commutative the diagram

\begin{array}{ccc} A & \xrightarrow{u} & A' \\ d_{A/K} \downarrow & & d_{A'/K'} \downarrow \\ \Omega_K(A) & \xrightarrow{v} & \Omega_{K'}(A') \end{array}

$d_{A'/K'} \circ u$ is a $K$ -derivation of $A$ with values in the $A$ -module $\Omega_{K'}(A')$ ; the existence and uniqueness of $v$ then follow from Proposition 18 of no. 11.

The mapping $v$ of Proposition 19 will be denoted by $\Omega(u)$ ; if there is a commutative diagram of commutative ring homomorphisms

\begin{array}{cccccc} K & \xrightarrow{\sigma} & K' & \xrightarrow{\sigma'} & K'' \\ \eta \downarrow & & \eta' \downarrow & & \eta'' \downarrow \\ A & \xrightarrow{u} & A' & \xrightarrow{u'} & A'' \end{array}

it follows immediately from the uniqueness property of Proposition 19 that

\Omega(u' \circ u) = \Omega(u') \circ \Omega(u).

Since $\Omega_{K'}(A')$ is an $A'$ -module, from $\Omega(u)$ we derive canonically an $A'$ -linear mapping

(41)

\Omega_0(u) : \Omega_K(A) \otimes_A A' \to \Omega_{K'}(A')

such that $\Omega(u)$ is the composition of $\Omega_0(u)$ and the canonical homomorphism $i_A : \Omega_K(A) \to \Omega_K(A) \otimes_A A'$ . For every $A'$ -module $E'$ , there is a commutative diagram

\begin{array}{ccc} \operatorname{Hom}_{A'}(\Omega_{K'}(A'), E') & \xrightarrow{\operatorname{Hom}(\Omega_0(u), 1_{E'})} & \operatorname{Hom}_{A'}(\Omega_K(A) \otimes_A A', E') \\ \phi_{A'} \downarrow & & \phi_A \circ r_A \downarrow \\ D_{K'}(A', E') & \xrightarrow{C(u)} & D_K(A, E) \end{array}

where $C(u)$ is the mapping $D \mapsto D \circ u$ (no. 7, Proposition 9) and $r_A$ is the canonical isomorphism

\operatorname{Hom}(i_A, 1_{E'}) : \operatorname{Hom}_{A'}(\Omega_K(A) \otimes_A A', E') \to \operatorname{Hom}_A(\Omega_K(A), E');

this follows immediately from Proposition 19 and the definition of the isomorphisms $\phi_A$ and $\phi_{A'}$ .

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

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Nothing.

Proposition 19

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