nLab
Cartier module

Contents

Definition

Let kk be a perfect field of characteristic p0p\neq 0. Let WW be the ring of Witt vectors over kk. A Cartier module is a pair (M,f)(M, f) where MM is a free WW-module of finite rank and f:MMf:M\to M is a semi-linear endomorphism in the following sense: f(am)=ϕ(a)f(a)f(a\cdot m)=\phi(a)f(a) where ϕ\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 ff data.

Examples

  • (W,ϕ)(W, \phi) is a Cartier module

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

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

Slope Decomposition

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

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

Define U r,sU_{r,s} to be the left AA-module A/A(T sp r)A/A\cdot (T^s-p^r). This is the canonical AA-module of pure slope r/sr/s and multiplicity ss. It is a KK-vector space of dimension ss.

When r0r\geq 0 TT preserves the WW lattice W[t]/W[t](T sp r)U r,sW[t]/W[t]\cdot(T^s-p^r)\subset U_{r,s}. We have that U r,sU_{r,s} is simple if and only if (r,s)=1(r,s)=1. It is a theorem of Dieudonne and Manin that when kk is algebraically closed there is a unique choice of integers r i,s ir_i, s_i with s i1s_i\geq 1 such that r 1/s 1r_1/s_1 < r 2/s 2r_2/s_2 < \cdots < r i/s ir_i/s_i where VV decomposes as a direct sum i=1 tV r i/s i\bigoplus_{i=1}^t V_{r_i/s_i} where V r i/s iV_{r_i/s_i} is noncanonically isomorphic as an AA-module to U r i,s iU_{r_i, s_i}. This is called the slope decomposition of VV.

The r i/s ir_i/s_i are called the slopes of VV with multiplicity s is_i. Up to noncanonical isomorphism VV is completely determined by knowledge of the slopes and multiplicities.

Examples

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

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

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

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