# When does the digital cohomological measurement come to be a mathematical stable?

In group cohomology concept, we understand that the cohomological measurement $cd(G)$ for a profinite team is a basic mathematical stable. We claim $G$ is of digital cohomological measurement $n$ (represent it by $vcd(G)$) if there exists an open subgroup $H$ such that $cd(H)=n$.

It appears that $vcd(G)$ is not constantly a dealt with integer and also my inquiry is:

Exists any kind of standard when $vcd(G)$ come to be dealt with, or claim, when $cd(G)=cd(H)$ for all open subgroup $H$ of $G$? If there are any kind of relevant referrals please allow me recognize, thx!

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2019-05-04 16:51:55
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Although it is not noticeable from the definition, actually - - thinking that there exists at the very least one open subgroup of limited cohomological measurement, or else there is absolutely nothing to claim - - the digital cohomological measurement is constantly a dealt with integer. To put it simply, if there exists a solitary open subgroup $H_0$ of $G$ such that $\operatorname{cd}(H_0) < \infty$, after that for all open subgroups $H$ of $G$ with $\operatorname{cd}(H) < \infty$, we have
$\operatorname{cd}(H) = \operatorname{cd}(H_0)$.
(Moreover, this additionally holds for the $p$ - cohomological measurement at any kind of prime $p$.)
I think this outcome was first confirmed by Serre. Regardless, it adheres to conveniently from Proposition I. 14 in Serre's Galois Cohomology . Actually, a somewhat more powerful outcome is offered as Proposition I. 14$'$ : once more thinking that there exists at the very least one open subgroup $H_0$ of limited cohomological measurement, an open subgroup $H$ has boundless cohomological measurement if $H$ has nontrivial components of limited order and also cohomological measurement equivalent to $\operatorname{cd}(H_0)$ or else.