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.

@MarianoSuárez-Alvarez Ah, yes, of course. That even fits my analogy better. And Serre writes $\Lambda$ for $\mathbb{Z}[G]$, making the typo ($\Lambda$ instead of $A$) quite subtle visually, probably why it wasn't detected before printing. — Brett Frankel, May 14 '13 at 19:19

score 5 · Accepted Answer · answered May 14 '13 at 19:17

5

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.

answered May 14 '13 at 19:17

Henry T. Horton

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score 2 · Answer 2 · answered Nov 17 '13 at 18:23

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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.

answered Nov 17 '13 at 18:23

Bruno Joyal

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Isn't it precisely the other way round? (Moral of the story: always define the induced module when writing about it.) – darij grinberg Aug 06 '15 at 13:29

score 0 · Answer 3 · answered Aug 06 '15 at 13:11

Certainly, you confused $A$, with $\Lambda$. In the $14$-th line in the begining of the chapter one finds

", if $\Lambda$ denotes the algebra $\mathbb{Z}[G]$..."

Later, He says

"The $G$-module $A$ is said to be induced if it has the form $\Lambda \otimes X$, where $X$ is an abelian group..."

Definition of Induced Module - Typo in Corps Locaux?

3 Answers3