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In group cohomology theory, we know that the cohomological dimension $cd(G)$ for a profinite group is a fundamental numerical invariant. We say $G$ is of virtual cohomological dimension $n$ (denote it by $vcd(G)$) if there exists an open subgroup $H$ such that $cd(H)=n$.

It seems that $vcd(G)$ is not always a fixed integer and my question is:

Is there any criterion when $vcd(G)$ become fixed ,or say, when $cd(G)=cd(H)$ for all open subgroup $H$ of $G$ ? If there are any related references please let me know, thx !

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up vote 5 down vote accepted

Although it is not obvious from the definition, in fact -- assuming that there exists at least one open subgroup of finite cohomological dimension, otherwise there is nothing to say -- the virtual cohomological dimension is always a fixed integer. In other words, if there exists a single open subgroup $H_0$ of $G$ such that $\operatorname{cd}(H_0) < \infty$, then for all open subgroups $H$ of $G$ with $\operatorname{cd}(H) < \infty$, we have

$\operatorname{cd}(H) = \operatorname{cd}(H_0)$.

(Moreover, this also holds for the $p$-cohomological dimension at any prime $p$.)

I believe this result was first proved by Serre. In any case, it follows easily from Proposition I.14 in Serre's Galois Cohomology. In fact, a slightly stronger result is given as Proposition I.14$'$: again assuming that there exists at least one open subgroup $H_0$ of finite cohomological dimension, an open subgroup $H$ has infinite cohomological dimension if $H$ has nontrivial elements of finite order and cohomological dimension equal to $\operatorname{cd}(H_0)$ otherwise.

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I see, so vcd(G) is of additional interest only if cd(G) is infinite – Ch Zh Aug 6 '10 at 10:11
@Z: yes, that's exactly right. – Pete L. Clark Aug 6 '10 at 10:21

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