Definition of Induced Module - Typo in Corps Locaux?

Question

This is from the beginning of the section on group cohomology in Corps Locaux (English Edition).

Serre states that $A$ is an induced $G$-module if

(1) $A\cong A\otimes_\mathbb{Z}X$ for an abelian group $X$,

or, equivalently,

(2) $A=\bigoplus_{s\in G}s\cdot X$.

Is (1) a typo? This is a very strict condition not just on $A$, but also on $X$. It seems to me that the correct version of the first definition above should read $A\cong A\otimes_{\mathbb{Z}[X]}X$, which is in line with the usual corresponding notion for group representatinos over fields, but perhaps I'm just missing something simple.

Answer 1

Condition (1) says nothing about $G$. It should be something like

(1) $A \cong X \otimes_{\Bbb Z} \Bbb Z[G]$ for some abelian group $X$.

We assume that the $G$-action on $X \otimes_{\Bbb Z} \Bbb Z[G]$ acts only on the $\Bbb Z[G]$ factor.

Answer 2

I'd also like to point out that what Serre calls an "induced module" is now known as a "co-induced module", whereas an induced module is of the form $\hom_{\mathbf Z}(\Lambda, X)$, which is the "adjoint" notion. This can be a source of confusion when reading older stuff. When $G$ is finite, however, the notions coincide.