Let $A$ be a semi-abelian variety over a field $k$($char\, k=p$). Namely, there is an exact sequence of group schemes $$0\to T\to A\to B\to 0$$ where $T$ is a torus, $B$ an abelian variety. Assume that $A$ lifts to $W(k)$.

When it is possible to lift the Frobenius endomorphism of $A$ to $W(k)$?

Probably, $B$ has to be ordinary as only in this case Frobenius endomorphism of $B$ lifts. My problem is that I cannot use obstruction theory here as the existence of liftings on finite levels does not yield to a lifting on the whole $W(k)$.

Maybe some small examples can be done. The case $T=\mathbb{G}_m,\dim B=1$ is already very interesting.

  • $\begingroup$ I assume $k$ is perfect. Let $A'$ be the chosen lift of $A$, giving $B'$ and $T'$ lifting $B$ and $T$ respectively. Using the canonical link between such extensions and line bundles on $B'$ arising from the dual abelian scheme when $T = {\rm{GL}}_1$ (so we can regard $B'$ as primary and $A'$ as secondary), in general $A'$ corresponds to a $W(k)$-point $\xi$ of ${B'}^{\vee} \otimes {\rm{X}}^{\ast}(T)$ (using the dual abelian scheme). So the necessary and sufficient condition is that Frobenius of $B$ lifts to $B'$ and its dual map along with Frobenius for $T$ carries $\xi^{(p)}$ to $\xi$. $\endgroup$ – nfdc23 Mar 9 '16 at 4:24
  • $\begingroup$ What do you mean by `$A$ lifts to $W(k)$'? Do you want the lift to be a scheme or a formal scheme? $\endgroup$ – Piotr Achinger Mar 12 '16 at 14:10

My answer to your other question seems to show that if $B$ is ordinary, $A$ has a natural lift over $W$ as a scheme, together with a lift of the Frobenius. I don't see however why the group structure of $A$ should lift, and why it should be compatible with the lift of Frobenius.

EDIT. Regarding the group structure: say $B$ is a $T$-torsor over an abelian $A$, trivialized over $0\in A$. We want to give $B$ a group structure such that $p:B\to A$ is a homomorphism with kernel $T$. This should be a map $b:B\times B\to B\to B$ such that the square $\require{AMScd}$ \begin{CD} B\times B @>b>> B \\ @V{p\times p}VV @VVpV \\ A\times A @>a>> A \\ \end{CD} commutes. This map should factor through a map $b':B\times B\to B' = a^* B$ from a $T\times T$-torsor $B\times B$ over $A\times A$ to the $T$-torsor $B'$ over $A\times A$, equivariant with respect to the addition map $t:T\times T\to T$. It should be easy to write out explicitly in terms of line bundles corresponding to the torsor $B$ what this map is, and what it should satisfy to give $B$ a group structure, and most probably we will see that this structure is preserved by Teichmueller lifts, so that we get a group structure on the canonical lift.

EDIT 2. Some time ago I was pointed in a similar context to look up "bi-extensions" (SGA 7 I, Exp. VII-VIII), but I never did. Perhaps you will find an answer to your question there.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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