On the cohomologies of commutative affine group schemes
On the cohomologies of commutative affine group schemes
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关于交换仿射群方案的上同调
DOI：
10.1215/kjm/1250524175
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发表时间：
1968
影响因子：
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通讯作者：
M. Miyanishi
中科院分区：
文献类型：
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作者：
M. Miyanishi
设k是具有正特征p的代数闭域，x是有限型真积分k-概型，G B e是交换仿射k-群概型.对于k-预拟阵T，X ~ T上的主纤维空间Y与群G的同构类构成一个阿贝尔群，其乘法是众所周知的.我们用PH（G，X / k）（T）表示这个阿贝尔群。则函子X / k）（T）是从k-预方案范畴（Sch/k）到交换群范畴（Ab）的逆变函子。PH（G，X / k）关于（Sch/k）的（fpqc）拓扑的相关层记为PH（G，X/k）.如果G是乘法群Gm，PH（G，n，X/ k）与X的Picard函子Pic（X/k）重合，且Pic（X / k）可表示为k上局部有限型的交换k-群概型.本文的目的是研究函子PH（G，X /k）对任意有限型可换仿射k-群概型的可表示性。若G是加群Gn，则PH（Gn X/k）可表示为Lie（Pic（X / k）），它同构于G的直积.若G是一个简单的有限k-群概型（即G=trp，Pp>（Z/ PZ）k且（Z / qZ），; q：素数，（p，q）= 1），则PH（G，X / k）为 *）。
Let k be an algebraically closed field of positive characteristic p, x a proper integral k-scheme o f finite type and G b e a commutative affine k-group scheme. For a k-prescheme T , the isomorphism classes o f principal fibre spaces Y over X T with group G form an abelian group with the well-known multiplication. We shall denote this abelian group by PH(G, X / k ) ( T ) . Then the functor X / k)(T ) is a contravariant functor from the category o f k-preschemes (Sch/k) to the category o f abelian groups (Ab). The associated sheaf o f PH(G, X / k ) with respect to th e (fpqc)topology of (Sch/k) is denoted by PH(G, X/k). If G is the multiplicative group Gm ., PH(G,„, X/ k ) coincides with the Picard functor Pic(X/k) o f X, and P ic (X / k ) is representable by a commutative k-group scheme, locally of finite type over k. The purpose o f this paper is to study the representability of the functor PH(G, X /k ) for an arbitrary commutative affine k-group scheme of finite type. If G is the additive group G „, PH(G„ X/k) is representable by L ie(P ic(X / k)) which is isomorphic to a direct product of G . If G is a simple finite k-group scheme (i.e. G=trp, P p > (Z/ PZ)k and (Z / qZ ),; q : prime, (p , q )= 1 ), P H (G , X / k ) is * ) This article was presented as a d octoral thesis to the Faculty of Science.