# nLab Cartier module

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Definition

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

• $\left(W,\varphi \right)$ is a Cartier module

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

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

## Slope Decomposition

Consider the Cartier module $\left(M,f\right)$. 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 $A=K\left[T\right]$ to be the noncommutative polynomial ring with commutative relation $\mathrm{Ta}=\varphi \left(a\right)T$. This allows us to define a left $A$-action on $V$ by $T\cdot v=f\left(v\right)$.

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

When $r\ge 0$ $T$ preserves the $W$ lattice $W\left[t\right]/W\left[t\right]\cdot \left({T}^{s}-{p}^{r}\right)\subset {U}_{r,s}$. We have that ${U}_{r,s}$ is simple if and only if $\left(r,s\right)=1$. 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 ${s}_{i}\ge 1$ such that ${r}_{1}/{s}_{1}$ < ${r}_{2}/{s}_{2}$ < $\cdots$ < ${r}_{i}/{s}_{i}$ where $V$ decomposes as a direct sum ${⨁}_{i=1}^{t}{V}_{{r}_{i}/{s}_{i}}$ 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 $k={𝔽}_{q}$ with $q={p}^{a}$. Given a Cartier module $\left(M,F\right)$, the slopes of $\left(M{\otimes }_{W\left(k\right)}W\left(\overline{k}\right),F\right)$ are the $p$-adic valuations (chosen so $\nu \left(q\right)=1$) 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 $\left(D\left(G\right),F\right)$ has all slopes in $\left[0,1\right)$.

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

## 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.

Revised on July 29, 2011 16:09:59 by hilbertthm90 (128.95.224.57)