lifting the isomorphisms between abelian schemes over PD thickenings - MathOverflow 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 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 Answer by ulrich for lifting the isomorphisms between abelian schemes over PD thickenings 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>