7
$\begingroup$

To motivate the question (and narrow it down if the one I asked is too broad), I'm doing readings from Manin's cubic forms book. A while back I was asked to compute the Galois cohomology $H^1(G, Pic(X_{\overline k}))$ of various Del Pezzo surfaces. My calculations seem to be fine, except I made the assumption that this cohomology group depends only on the image of $G$ in $Aut(Pic(X_{\overline k}))$. I know this isn't true in general. Manin indicates it holds in my case by Proposition 31.3 in his book, on page 179 - however, I've been informed that this proposition is probably not correct as stated, so I'd like some other reference, argument, general principle, or even heuristic that tells me when I'm on safe ground making this assumption (hopefully one as elementary as possible, since my understanding isn't extensive).

$\endgroup$
3
  • $\begingroup$ I have also posted this question here: math.stackexchange.com/questions/3516911/… $\endgroup$ Jan 28, 2020 at 21:02
  • $\begingroup$ Could you include the statement of the proposition in Manin's book? $\endgroup$ Jan 29, 2020 at 8:52
  • $\begingroup$ By inf-res, you computed a subgroup and its quotient is the kernel of $\operatorname{Hom}_{\bar G}(H,M) \to H^2(\bar G, M)$ where $\bar G$ is your image and $H$ the kernel. So for instance if $H$ is torsion and $M$ is torsionfree, you have everything. $\endgroup$ Jan 29, 2020 at 9:09

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.