A proper base change theorem for non-torsion sheaves in étale cohomology
A proper base change theorem for non-torsion sheaves in étale cohomology
复制标题
étale上同调中非扭轮的真基变定理
DOI：
10.1016/0022-4049(88)90102-8
复制
发表时间：
1988
影响因子：
0.8
通讯作者：
C. Deninger
中科院分区：
文献类型：
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作者：
C. Deninger
众所周知，上同调中的许多基本定理仅适用于扭力滑轮。特别是，对于真基变定理来说就是这种情况，如[l，XII，21中的示例所示。如果所讨论的层可以用平滑交换群方案表示，则在特殊情况下相关结果仍然可用[4，III，3.111。然而，在一些重要的情况下，滑轮既不扭转也不可表示。例如，在 Lichtenbaum 复合体理论 [3] 中就属于这种情况，其中出现了由代数 K 理论构造的 tale 滑轮。在本文中，我们将真基变定理的一个版本扩展到任意阿贝尔滑轮。回想一下，如果方案之间的态射与几何正常纤维平坦，则该态射被称为正常态射 [2,(6.8.1)]。对于优秀方案的定义我们参考[2,(7.8.5)]。我们的结果如下：
It is well known that many of the fundamental theorems in &ale cohomology are valid for torsion sheaves only. In particular this is the case for the proper base change theorem as is shown by the example in [l, XII, 21. If the sheaf in question is representable by a smooth commutative group scheme a related result is still available in a special case [4, III, 3.111. However there are important instances where the sheaves are neither torsion nor representable. Such is the case for example in the theory of Lichtenbaum’s complexes [3] where &tale sheaves constructed from algebraic K-theory occur. In this note we extend a version of the proper base change theorem to arbitrary abelian sheaves. Recall that a morphism between schemes is called normal if it is flat with geometrically normal fibres [2,(6.8. 1)]. For the definition of excellent schemes we refer to [2,(7.8. 5)]. Our result is the following: