Degree three cohomology of function fields of surfaces
Degree three cohomology of function fields of surfaces
复制标题
曲面函数域的三阶上同调
DOI：
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发表时间：
2010
期刊：
影响因子：
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通讯作者：
V. Suresh
中科院分区：
文献类型：
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作者：
R. Parimala;V. Suresh
设F是有限域，l是不等于F的特征的素数.设K是F上曲面的函数场。假设K包含一个本原的1次单位根。本文利用H^2（K，{mu}_l）中的符号，证明了H^3（K，{mu}_l）中元素关于K的离散赋值的局部-整体原理.我们还证明了这个局部整体原理等价于有限域上某些非分歧的3重上同调群的消失。利用这个局部-整体原理，我们证明了H^3（F，{mu}_l）中的每个元素都是符号。非分歧上同调群的消失在积分Tate猜想和Brauer-Manin阻塞零圈存在性的研究中有重要意义。
Let F be a finite field and l a prime not equal to the characteristic of F. Let K be the function field of a surface over F. Assume that K contains a primitive lth root of unity. In the paper we prove a certain local-global principle for elements of H^3(K, {mu}_l) in terms of symbols in H^2(K, {mu}_l) with respect to discrete valuations of K. We also show that this local global principle is equivalent to the vanishing of certain unramified cohomology groups of 3-folds over finite fields. Using this local-global principle we show that every element in H^3(F, {mu}_l) is a symbol. The vanishing of the unramified cohomology groups has consequences in the study of integral Tate conjecture and Brauer-Manin obstruction for existence of zero-cycles.