Kummer对数拓扑与经典拓扑II的比较
Comparison of Kummer logarithmic topologies with classical topologies II
摘要：
我们证明了光滑交换群格式从Kummer对数平坦点到经典平坦点的高阶直象是挠的。对于（1）具有几何连通纤维的光滑仿射交换格式，（2）有限平坦群格式，（3）阿贝尔格式的环面扩张，我们给出了第二高阶直象的显式描述。如果对数结构在基的任意几何点上的秩至多为1，我们证明了在情形（1）、情形（3）和情形（2）的某些子情形下，群概型的第二高阶直像为零.如果基的基础概型是在$\mathbb{Q}$上的或具有特征$p>0$，我们还可以给出情形（1）、情形（3）和情形（3）的某些子情形下群概型的第二高阶直象的更明确的描述.在有限剩余域上的标准Hensel对数特征上，分别计算了情形（1）、情形（3）和情形（3）的某些子情形下的带系数的第一和第二Kummer对数平坦上同调群.
英文摘要：
We show that the higher direct images of smooth commutative group schemes from the Kummer log flat site to the classical flat site are torsion. For (1) smooth affine commutative schemes with geometrically connected fibers, (2) finite flat group schemes, (3) extensions of abelian schemes by tori, we give explicit description of the second higher direct image. If the rank of the log structure at any geometric point of the base is at most one, we show that the second higher direct image is zero for group schemes in case (1), case (3), and certain subcase of case (2). If the underlying scheme of the base is over $\mathbb{Q}$ or of characteristic $p>0$, we can also give more explicit description of the second higher direct image of group schemes in case (1), case (3), and certain subcase of case (3). Over standard Henselian log traits with finite residue field, we compute the first and the second Kummer log flat cohomology group with coefficients in group schemes in case (1), case (3), and certain subcase of case (3).
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