Xxxxxxxx-Laffaille Theory. → For this section we shall not work with coefficients. Following [12] we write MFk for the category whose objects are triples (M, Mi, ϕi) where M is a k-vector space, Mi is a filtration on M and ϕi : Mi M is a ϕ-semilinear map such that ϕi(Mi+1) = 0. Construction 6.1.1. If M ∈ ModBK is such that Weight(M ) ⊂ [0, p − 1] then we associate to M a triple (Mk, FiMk, ϕi) in MFk as follows. The vector space Mk and the filtration on Mk are as defined in Lemma 4.3.2. The ϕi are semilinear maps FiMk → Mk whose ui construction we now explain. For i ∈ Z we have maps ę : FiM → M which fit into commutative diagrams Fi—pM u ϕ ui−p FiM ϕ ui M M i−pu If i ∈ [0, p − 1] then the image of ę is contained in uM . Therefore the ui map ę descends to a semilinear ϕi : FiMk = FiM/u(Fi—pM ) → Mk. Note that ϕi(Fi+1Mk) = 0 and so M '→ (Mk, FiMk, ϕi) describes a functor taking values in MFk. Observe that this construction only makes sense if Weight(M ) ⊂ [0, p − 1]. Lemma 6.1.2. With M as above, M is strongly divisible as a Breuil– Σ Kisin module if and only if Mk is a strongly divisible object of MFk (i.e. ϕi(FiMk) = Mk). 80 6. COMPARISON WITH XXXXXXXX–LAFFAILLE THEORY ΣSD i iThus if M ∈ Mod then ϕ (F M ) = M .k k Proof. Let (fi) be a k[[u]]-basis of M such that (urifi) is a k[[u]]-basis of ϕ(M ), as in Lemma 4.3.4. Let gi ∈ M be such that ϕ(gi) = urifi. Letting gi, fi denote the respective images in Mk, then gi ∈ Fri Mk and ϕri (gi) = fi.
