A motivic conjecture of Milne (Q2872167)

This article deals with integral \(p\)-adic Hodge theory of \(p\)-divisible groups. Let \(k\) be an algebraically closed field of characteristic \(p>0\) and let \(D\) be a \(p\)-divisible group over the ring \(W(k)\) of Witt vectors of \(k\). Denote by \(B(k)\) the fraction field of \(W(k)\) and choose an algebraic closure \(\bar{B(k)}\). One associates to \(D\) the Galois module \(H^1(D) = T_p D^t(\bar{B(k)})(-1)\) (\(D^t\) is the Cartier dual of \(D\)), which is a finite rank free \(\mathbb Z_p\) module with an action of \(\text{Gal}(\bar{B(k)}/B(k))\), and the filtered module \(M\), the contravariant Dieudonné module, which is a finite rank free \(W(k)\)-module endowed with a semilinear Frobenius and a filtration. These two invariants are related by the Fontaine's comparison theory: there exists a \(W(k)\)-algebra \(B^+\), endowed with a Galois action, a Frobenius, and a filtration, and an isomorphism NEWLINE\[NEWLINE B^+ \otimes_{W(k)} M \overset{\sim}{\to} B^+ \otimes_{\mathbb Z_p} H^1(D) NEWLINE\]NEWLINE suitably compatible with the three structures. If \(D\) is the \(p\)-divisible group of an abelian scheme \(A\), then \(M = H^1(A_{\bar K}, \mathbb Z_p)\), \(H^1(D) = H^1_{\text{cris}}(A/W(k))\), and the above isomorphism is an integral version of the crystalline period isomorphism.NEWLINENEWLINEThe goal of this article is to identify \(H^1(D)\otimes_{\mathbb Z_p} W(k)\) with \(M\), compatibly with families of tensors. The structure of a filtered module on \(M\) induces a Frobenius and a filtration on the ``essential tensor algebra'' \(\mathcal{T}(M) = \bigoplus_{s, t} M^{\otimes s}\otimes (M^*)^{\otimes t}\). Similarly, the Galois action on \(H^1(D)\) induces a Galois action on \(\mathcal{T}(H^1(D))\). Let \((t_\alpha)_{\alpha \in \mathcal{J}}\) be a family of elements of \((F^0 \mathcal{T}(M)[\frac{1}{p}])^{\phi=\text{id}}\). Then by Fontaine's comparison theory, one obtains Galois-invariant tensors \((v_\alpha)_{\alpha\in \mathcal{J}}\) in \(\mathcal{T}(H^1(D))\). The main result (Theorem~1.2) is the construction of an isomorphism \(M\overset{\sim}{\to} H^1(D)\otimes_{\mathbb Z_p} W(k)\) such that the induced isomorphism \(\mathcal{T}(M)\overset{\sim}{\to} \mathcal{T}(M)\otimes_{\mathbb Z_p} W(k)\) sends \(t_\alpha\) to \(v_\alpha\) (if \(p=2\), one has to impose an additional condition on \(D\)). A special case of this result has been conjectured by Milne in relation to abelian varieties.