2
$\begingroup$

Let $T$ be the torus over $\mathbb{Q}$ defined by the equation $x^2+y^2=1$, then it becomes isomorphic to $\mathbb{G}_m$ over field $\mathbb{Q}[i]$. Let $X$ denote the character group of $T$, regarded as a locally constant sheaf for the etale topology or the fppf topology over $\mathrm{Spec}\mathbb{Q}$.

Now let's work in the fppf site of $\mathrm{Spec}\mathbb{Q}$, consider the Hom sheaf $\mathscr{H}om(X,\mathbb{G}_a)$, it seems for me that:

  1. this sheaf has trivial global sections;

  2. it becomes isomorphic to $\mathbb{G}_a$ over $\mathrm{Spec}\mathbb{Q[i]}$,

My questions are 1. Is this sheaf represented by a group scheme?

  1. If the answer to 1 is no, is it representable by an algebraic space (something like $\mathbb{G}_a/(\mathbb{Z}/2)$)?

  2. If it is indeed something like $\mathbb{G}_a/(\mathbb{Z}/2)$ ,

how to compute the fppf-cohomology of the algebraic space associated to an action of a finite group on a group scheme? e.g. $H^1_{\mathrm{fppf}}(\mathbb{Q},\mathscr{H}om(X,\mathbb{G}_a))$

$\endgroup$
3
  • 1
    $\begingroup$ It is not a group scheme since there are no nontrivial forms of G_a. For algebraic spaces, see mathoverflow.net/questions/8918/… $\endgroup$
    – user2035
    Sep 16, 2011 at 15:43
  • $\begingroup$ Thanks! Is the sheaf $\mathscr{H}om(X,\mathbb{G}_a)$ isomorphic to the algebraic space associated to the action of Z/2Z on $\mathbb{G}_a$? Do you have any idea for computing the first cohomology group of this sheaf (e.g. torsioness)? $\endgroup$
    – Heer
    Sep 17, 2011 at 11:53
  • $\begingroup$ Didn't I already answer the first question? For the cohomology, use the cover $\mathrm{Spec}(\mathbb Q(i))\to\mathrm{Spec}(\mathbb Q)$. $\endgroup$
    – user2035
    Sep 20, 2011 at 7:32

0

Your Answer

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

Browse other questions tagged or ask your own question.