most recent 30 from http://mathoverflow.net 2013-05-19T15:49:40Z http://mathoverflow.net/feeds/question/76126 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/76126/lifting-the-isomorphisms-between-abelian-schemes-over-pd-thickenings Jack 2011-09-22T14:39:11Z 2011-09-22T15:34:09Z

<p>Assume that X and Y are abelian schemes (or even abelian varieties) over a base T. If $\ S --> T$ is a PD nilpotent thickening (i.e. the ideal of $\ S$ in $ \ T$ is a nilpotent divided power ideal) and S is of characteristic $\ p>0$ . If $\ X_{0}$ and $ \ Y_{0}$ are reductions of $\ X$ and $\ Y$ to $\ S$. If $\ f_{0} : X_{0} ---> Y_{0} $ is an isomorphism over S, is this true that this isomorphism (if it can be lifted!) lifts to an isomorphism $\ f : X ---> Y$ ? one may also assume that $ \ H¹_ {cris} (f_{0})$ preserves the Hodge filtration. </p>

http://mathoverflow.net/questions/76126/lifting-the-isomorphisms-between-abelian-schemes-over-pd-thickenings/76132#76132 ulrich 2011-09-22T15:34:09Z 2011-09-22T15:34:09Z

<p>No, this is very far from being true.</p> <p>For a counterexample, let $S = Spec(\mathbb{Z}/p)$ and $T = Spec(\mathbb{Z}/p^2)$. The versal deformation space of an elliptic curve $E$ over $S$ is isomorphic to <code>$Spec(\mathbb{Z}_p[[x]])$</code> so lifts of $E$ to $T$ are parametrized by the set of homomorphisms of local algebras $Hom(\mathbb{Z}_p[[x]], \mathbb{Z}/p^2) = p\mathbb{Z}/p^2$. So there do exist lifts for which the identity map of $E$ does not lift to an isomorphism.</p>