Corollary 1

Tag 0DQ2, permanent. Cite it as [Bourbaki, Tag 0DQ2].

The $A$ -module $E_A=\operatorname{Hom}_{\mathbf{Z}}(A,\mathbf{Q}/\mathbf{Z})$ is injective and a cogenerator.

We apply prop. 13 with $B=\mathbf{Z}$ , $F=\mathbf{Q}/\mathbf{Z}$ (example) and $P=A_d$ .

For every $A$ -module $M$ , let

I^0(M)=E_A^{\operatorname{Hom}(M,E_A)}

and let $e_M:M\to I^0(M)$ be the homomorphism which associates to $m\in M$ the element

(\varphi(m))_{\varphi\in\operatorname{Hom}(M,E_A)}\in I^0(M).

Then:

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

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