On \(\pi \)-divisible \(\mathcal{O} \)-modules over fields of characteristic \(p\) (Q6099461)

In this paper, the author constructs a Dieudonné theory for \(\pi\)-divisible \(\mathcal{O}\)-modules over a perfect field of characteristic \(p\). The author classified \(\pi\)-divisible \(\mathcal{O}\)-modules over a perfect fields of characteristic \(p\), proved the existence of slope filtration of \(\pi\)-divisible \(\mathcal{O}\)-modules over an integral normal Noetherian base, explained the determination of minimal \(\pi\)-divisible \(\mathcal{O}\)-modules over algebraically closed fields of characteristic \(p\) by their \(\pi\)-torsion parts, and proved Traverso's isogeny conjecture in this context. The first result of the paper is a decomposition structure theorem for \(f\)-\(\mathcal{O}\)-isocrystal over \(k\). Theorem. Assume that \(k\) is perfect. Let \((N,V)\) be an \(f\)-\(\mathcal{O}\)-isocrystal over \(k\) with first Newton slope \(\lambda\). There exist uniquely determined sub-\(\mathcal{O}\)-isocrystals \((N_i,V_i)\) with \(\mathrm{Newton}(N_i,V_i) = \lambda_i\) such that \((N,V) = \bigoplus_{i=1}^r(N_i,V_i)\) and \(\lambda = \lambda_1 < \lambda_2 < \cdots < \lambda_r\). When \(k\) is algebraically closed, it is possible to further decompose each isoclinic \(f\)-\(\mathcal{O}\)-isocrystal \((N_i,V_i)\) as a direct sum of proper sub-\(\mathcal{O}\)-isocrystal \(N_{r_i,s_i}\). Using a classification of \(\pi\)-divisible \(\mathcal{O}\)-modules via \(\mathcal{O}\)-display, the author proves the following slope filtration theorem for \(\pi\)-divisible \(\mathcal{O}\)-modules. Theorem. A \(\pi\)-divisible \(\mathcal{O}\)-module over an integral, normal, Noetherian \(\mathbb{F}\)-scheme \(S\) with constant Newton polygon is isogenous to a \(\pi\)-divisible \(\mathcal{O}\)-module over \(S\) that admits a slope filtration. The author also generalizes of minimal \(p\)-divisible groups (in the sense of Oort) to minimal \(\pi\)-divisible \(\mathcal{O}\)-modules. For any Newton polygon \(\beta\) with slopes in \([0,1]\), a \(\pi\)-divisible \(\mathcal{O}\)-module \(H(\beta) = \bigoplus_i H_{m_i,m_i+n_i}^{r_i}\) where each \(H_{m_i,m_i+n_i}\) is defined using a combinatorial formula on a basis of the free \(W_{\mathcal{O}}(\mathbb{F})\)-module with Newton slope \(m_i/(m_i+n_i)\). A \(\pi\)-divisible \(\mathcal{O}\)-module \(X\) is called minimal if there exists a Newton polygon \(\beta\) and an isomorphism \(X_k \cong H(\beta)_k\). Theorem. Let \(k\) be an \(\mathcal{O}\)-algebra and an algebraically closed field of characteristic \(p\). Let \(X\) be a minimal \(\pi\)-divisible \(\mathcal{O}\)-module over \(k\). If \(Y\) is another \(\pi\)-divisible \(\mathcal{O}\)-module over \(k\) such that \(X[\pi]=Y[\pi]\), then \(X \cong Y\). If \(k\) is an \(\mathcal{O}\)-algebra and an algebraically closed field of characteristic \(p\), then the isomorphism classes of \(X\) is determined by a truncation \(X[\pi^{n_X}]\) of \(X\). This implies that there exists a minimal natural number \(b_X\) such that the isogeny classes of \(X\) is determined by \(X[\pi^{b_X}]\). This number \(b_X\) is called the isogeny cutoff of \(X\). The author proves the following Traverso's isogeny conjecture. Theorem. Let \(k\) be an \(\mathcal{O}\)-algebra and an algebraically closed field of characteristic \(p\). Let \(X\) be a \(\pi\)-divisible \(\mathcal{O}\)-module over \(k\). Assume that \(X\) has height \(h\) and dimension \(d\), then \(b_X \leq \lceil \frac{d(h-d)}{h} \rceil\).