  Building on our previous work "Cartier modules: finiteness results" we start
in this manuscript an in depth study of the derived category of Cartier modules
and the cohomological operations which are defined on them. After localizing at
the sub-category of locally nilpotent objects we show that for a morphism
essentially of finite type $f$ the operations $Rf_*$ and $f^!$ are defined for
Cartier crystals. We show that, if $f$ is of finite type (but not necessarily
proper) $Rf_*$ preserves coherent cohomology (up to nilpotence) and that $f^!$
has bounded cohomological dimension. In a sequel we will explain how
Grothendieck-Serre Duality relates our theory of Cartier Crystals to the theory
of $\tau$-crystals as developed by Pink and the second author.
