2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/58744We propose the notion of the {\em crystalline sub-representation functor} defined on $p$-adic representations of the Galois groups of finite extensions of $\Qp$, with certain restrictions in the case of integral representations. By studying its right-derived functors, we find a natural extension of a formula of Grothendieck expressing the group of connected components of a Neron model of an abelian variety in terms of Galois cohomology.Algebraic GeometryNumber Theory11G10text