论文标题
在半光彩场上的边界共同体学课程
Bounding cohomology classes over semiglobal fields
论文作者
论文摘要
当$ h^i（f，μ__\ ell^{\ otimes i-1}）$中，我们为共同体学类的索引提供了统一的界限，当$ f $是一个半链球场字段（即，在一个完全值得离散价值的字段$ k $上）。该界限是根据$ K $的残基字段的类似数据及其最多有限生成的超越程度扩展的类似数据。我们还获得了共同体类别的类似界限。我们的结果为较高等级的函数字段提供了递归公式，在某些情况下，当已知残基领域的信息时，明确的界限。在此过程中，我们证明了在有限场上表面的表面上的第3度的共同学类别的分裂结果。
We provide a uniform bound for the index of cohomology classes in $H^i(F, μ_\ell^{\otimes i-1})$ when $F$ is a semiglobal field (i.e., a one-variable function field over a complete discretely valued field $K$). The bound is given in terms of the analogous data for the residue field of $K$ and its finitely generated extensions of transcendence degree at most one. We also obtain analogous bounds for collections of cohomology classes. Our results provide recursive formulas for function fields over higher rank complete discretely valued fields, and explicit bounds in some cases when the information on the residue field is known. In the process, we prove a splitting result for cohomology classes of degree 3 in the context of surfaces over finite fields.
