Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

This question arises from reading the note Hodge cycles on abelian varieties by P. Deligne (notes by J.S. Milne). Suppose we are given a group $G$ (for example, either a fundamental group $\pi_1(S, s_0)$ or a Galois group $Gal(k/k_0)$) acting on a finite dimensional $\mathbb{Q}$-vectors space. It seems to me that the author concludes that the action factors through a finite subgroup once it is known there is a symmetric positive definite $\mathbb{Q}$-valued bilinear form on $V$. Of course, this is note true in general (Take $V=\mathbb{Q}^2$ with the standard inner product, for example.) So I am wondering what is the missing link in my understanding of the proofs. The proofs in question are Theorem 2.15 and (iii) of Propositin 2.9(b). Why does the action factors through finite subgroups in those proofs?

share|cite|improve this question

In the first case, the image of the group is obviously a discrete subgroup of $Aut(V\otimes \mathbb{R})$, hence discrete and compact. In the second case, the group preserves a lattice in $V$, and so the image is again discrete and compact.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.