nLab Cartier module

Definition

Let $k$ be a perfect field of characteristic $p\neq 0$. Let $W$ be the ring of Witt vectors over $k$. A Cartier module is a pair $(M, f)$ where $M$ is a free $W$-module of finite rank and $f:M\to M$ is a semi-linear endomorphism in the following sense: $f(a\cdot m)=\phi(a)f(a)$ where $\phi$ is the Frobenius map.

Cartier modules form a category by taking morphisms to be in the category of W-modules that also respect the extra $f$ data.

Examples

• $(W, \phi)$ is a Cartier module

• If $G$ is a p-divisible group of height $h$, then the Dieudonne module $D(G)$ is a free $W$-module of rank $h$. The natural action of Frobenius turns $D(G)$ into a Cartier module.

• If $X$ proper, smooth scheme over $k$ of dimension $n$, then all $H^m_{crys}(X/W)/torsion$ with the action of pullback by Frobenius $F^*$ is a Cartier module when $m$<$n$.

Slope Decomposition

Consider the Cartier module $(M, f)$. Let $K$ be the fraction field of $W$. Define the finite dimensional vector space $V=M\otimes_W K$. Extend $f$ linearly to $V$. Note that $f$ preserves the $W$-lattice $M$ inside $V$ by construction.

Define $A=K[T]$ to be the noncommutative polynomial ring with commutative relation $Ta=\phi(a)T$. This allows us to define a left $A$-action on $V$ by $T\cdot v=f(v)$.

Define $U_{r,s}$ to be the left $A$-module $A/A\cdot (T^s-p^r)$. This is the canonical $A$-module of pure slope $r/s$ and multiplicity $s$. It is a $K$-vector space of dimension $s$.

When $r\geq 0$ $T$ preserves the $W$ lattice $W[t]/W[t]\cdot(T^s-p^r)\subset U_{r,s}$. We have that $U_{r,s}$ is simple if and only if $(r,s)=1$. It is a theorem of Dieudonne and Manin that when $k$ is algebraically closed there is a unique choice of integers $r_i, s_i$ with $s_i\geq 1$ such that $r_1/s_1$ < $r_2/s_2$ < $\cdots$ < $r_i/s_i$ where $V$ decomposes as a direct sum $\bigoplus_{i=1}^t V_{r_i/s_i}$ where $V_{r_i/s_i}$ is noncanonically isomorphic as an $A$-module to $U_{r_i, s_i}$. This is called the slope decomposition of $V$.

The $r_i/s_i$ are called the slopes of $V$ with multiplicity $s_i$. Up to noncanonical isomorphism $V$ is completely determined by knowledge of the slopes and multiplicities.

Examples

• Let $k=\mathbb{F}_q$ with $q=p^a$. Given a Cartier module $(M, F)$, the slopes of $(M\otimes_{W(k)}W(\overline{k}), F)$ are the $p$-adic valuations (chosen so $\nu(q)=1$) of the eigenvalues of the linear endomorphism $F^a$ of $M$, and the multiplicity is the (algebraic) multiplicity of this eigenvalue.

• In the second example above, if $G$ a p-divisible group, then $(D(G), F)$ has all slopes in $[0,1)$.

• In the third example above if $X$ is projective, then since $F_*\circ F^* = p^n$, all the slopes of $H^m_{crys}(X/W)/torsion$ lie in $[0,n]$.

References

• Michael Artin, Barry Mazur, Formal Groups Arising from Algebraic Varieties, numdam, MR56:15663

• Pierre Berthelot, Slopes of Frobenius in Crystalline Cohomology, Proceedings of Symposia in Pure Mathematics Vol 29, 1975.