Invariants of algebraic groups and retract rationality of classifying spaces
Invariants of algebraic groups and retract rationality of classifying spaces
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代数群的不变量和收回分类空间的合理性
DOI：
10.1090/pspum/094/06
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发表时间：
2017
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影响因子：
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通讯作者：
A. Merkurjev
中科院分区：
文献类型：
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作者：
A. Merkurjev
设G是域F上的代数群，V是G的一般自由表示（即，V中一般点的稳定子是平凡的）和U ∈ V是G等变开子集使得存在G-挠函数f：U −→ U/G。这是一个U/G-扭子，特别地，在一个域扩张K/F上的每个G-扭子都同构于f在U/G的一个K-点上的纤维。因此，U/G的K-点是Spec（K）上所有G-挠量的参数。我们认为U/G是分类空间BG的近似。U/G的稳定双有理等价类和收缩有理等价类与V和U的选择无关。我们简单地说BG是稳定有理的（相应地，收缩有理的），如果U/G是稳定有理的。在这些情况下，F的域扩张上的所有G-挠量都可以用代数独立变量参数化。BG的稳定（收缩）非理性可以通过J. - P. Serre in [23].域F上的代数群G的上同调不变量，其系数在F上的伽罗瓦模M中，自然地赋予域扩张K/F上的每个G-torsor一个系数在M中的K的伽罗瓦上同调类。一个上同调不变量I称为非分歧的，如果I在域扩张K/F上的所有值关于K在F上的所有离散赋值都是非分歧的。BG的收缩有理数与非分歧不变量之间的关系如下：如果G允许一个非常数的非分歧上同调不变量，则分类空间BG不是收缩有理数的.例如，D.索特曼谁反驳诺特猜想的合理性分类空间的有限团体，表明某些有限团体承认一个非常数度2 unramified上同调不变。代数闭域上是否存在一个连通代数群G，使得BG不是收缩有理数群，这是一个未解决的问题。本文对分类空间和上同调不变量性质的一些经典结果进行了回顾和改进。我们不对地面场F施加任何限制。如果p = char（F）> 0，则系数模Q/Z（j）包含非平凡p-准素分量。特别地，具有Q/Z（j）中的值的伽罗瓦上同调群不形成循环模
LetG be an algebraic group over a field F , V a generically free representation of G (i.e., the stabilizer of the generic point in V is trivial) and U ⊂ V a Gequivariant open subset such that there is a G-torsor f : U −→ U/G. This is a versal G-torsor, in particular, every G-torsor over a field extension K/F with K infinite is isomorphic to the fiber of f over a K-point of U/G. Thus, the K-points of U/G parameterize all G-torsors over Spec(K). We think of U/G as an approximation of the classifying space BG. The stable birational and retract rational equivalence classes of U/G are independent of the choice of V and U . We simply say that BG is stably rational (respectively, retract rational) if so is U/G. In these cases all the G-torsors over field extensions of F can be parameterized by algebraically independent variables. The stable (retract) non-rationality of BG can be detected by cohomological invariants which were introduced by J.-P. Serre in [23]. A cohomological invariant of an algebraic group G over a field F with coefficients in a Galois module M over F assigns naturally to every G-torsor over a field extension K/F a Galois cohomology class of K with coefficients in M . A cohomological invariant I is called unramified if all values of I over a field extension K/F are unramified with respect to all discrete valuations of K over F . A relation between retract rationality of BG and unramified invariants is given by the following statement: If G admits a non-constant unramified cohomological invariant, then the classifying space BG is not retract rational. For example, this was used by D. Saltman who disproved Noether’s Conjecture on the rationality of classifying spaces of finite groups by showing that certain finite groups admit a non-constant degree 2 unramified cohomological invariant. It is still an open problem whether there exists a connected algebraic group G over an algebraically closed field such that BG is not retract rational. In the present paper we review and slightly improve some classical results on the properties of classifying spaces and cohomological invariants. We don’t impose any restrictions on the ground field F . The coefficient module Q/Z(j) includes nontrivial p-primary component if p = char(F ) > 0. In particular, Galois cohomology groups with values in Q/Z(j) do not form a cycle module