Category of the smooth formal p-groups over a local ring - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T17:39:49Z http://mathoverflow.net/feeds/question/109619 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/109619/category-of-the-smooth-formal-p-groups-over-a-local-ring Category of the smooth formal p-groups over a local ring notengo 2012-10-14T15:19:23Z 2012-10-14T15:19:23Z <p>Fontaine showed in Asterisque 47-48 that the category of finite dimensional smooth formal \$p\$-groups over the ring \$A=W(k)\$ of the Witt vectors over a finite field \$k\$ is anti-equivalent to the category of triples \$(L,M,r)\$ where \$L\$ is a free \$A\$-module of finite rank, \$M\$ is a profinite \$A[F]\$-module such that \$F\$ is injective on \$M\$ and \$pM\subset FM\$, \$r\$ is an \$A\$-morphism from \$L\$ to \$M\$ which induces an isomorphism from \$L/pL\$ to \$M/FM\$. Moreover, the connected groups correspond under this anti-equivalence to the triples where the action of \$F\$ on \$M\$ is topologically nilpotent. In addition, Fontaine obtained similar results for finite dimensional smooth formal \$p\$-groups over the ring \$A’\$ of integers in a finite extension of \$Q_p\$ with the ramification index less than \$p-1\$.</p> <p>Do these results allow us to determine the categorical properties of these categories, i.e. if these categories admit kernels, cokernels, if they are semi-abelian etc.? I would appreciate if somebody could recommend me a reference where such questions are discussed.</p>