局部Ogus-Vologodsky对应的推广
Notes on generalizations of local Ogus-Vologodsky correspondence
摘要：
给定$\Z/p^n\Z$上的一个光滑概形，其相对提升为Frobenius到$\Z/p^{n+1}\Z$，我们构造了一个从Higgs模范畴到具有可积联系的模范畴的函子，作为$1 \leq m \leq n$的从具有可积$p^{m}$ -联系的模范畴到具有可积$p^{m-1}$ -联系的模范畴的水平提升逆像函子的合成.在$m=1$情形下，我们证明了当限制于拟幂零对象时，水平提升逆像函子是等价的，推广了Ogus-Vologodsky的一个局部结果.我们还证明了上述水平提高逆像函子的光滑$p$ -进形式计划诱导等价的$\Q$ -线性化范畴一般$m$时，限制到幂零对象（在强意义上），在Frobenius提升的强条件下。对于具有可积$p^{m}$ -Witt-联络的模范畴，我们也证明了类似的结果.
英文摘要：
Given a smooth scheme over $\Z/p^n\Z$ with a lift of relative Frobenius to $\Z/p^{n+1}\Z$, we construct a functor from the category of Higgs modules to that of modules with integrable connections as the composite of the level raising inverse image functors from the category of modules with integrable $p^{m}$-connections to that of modules with integrable $p^{m-1}$-connections for $1 \leq m \leq n$. In the case $m=1$, we prove that the level raising inverse image functor is an equivalence when restricted to quasi-nilpotent objects, which generalizes a local result of Ogus-Vologodsky. We also prove that the above level raising inverse image functor for a smooth $p$-adic formal scheme induces an equivalence of $\Q$-linearized categories for general $m$ when restricted to nilpotent objects (in strong sense), under a strong condition on Frobenius lift. We also prove a similar result for the category of modules with integrable $p^{m}$-Witt-connections.
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