If proper, smoothscheme over of dimension , then all with the action of pullback by Frobenius is a Cartier module when <.
Consider the Cartier module . Let be the fraction field of . Define the finite dimensional vector space . Extend linearly to . Note that preserves the -lattice inside by construction.
Define to be the noncommutative polynomial ring with commutative relation . This allows us to define a left -action on by .
Define to be the left -module . This is the canonical -module of pure slope and multiplicity . It is a -vector space of dimension .
When preserves the lattice . We have that is simple if and only if . It is a theorem of Dieudonne and Manin that when is algebraically closed there is a unique choice of integers with such that < < < where decomposes as a direct sum where is noncanonically isomorphic as an -module to . This is called the slope decomposition of .
The are called the slopes of with multiplicity . Up to noncanonical isomorphism is completely determined by knowledge of the slopes and multiplicities.
Let with . Given a Cartier module , the slopes of are the -adic valuations (chosen so ) of the eigenvalues of the linear endomorphism of , and the multiplicity is the (algebraic) multiplicity of this eigenvalue.