Duality in the flat cohomology of curves
Duality in the flat cohomology of curves
复制标题
曲线平坦上同调的对偶性
DOI：
10.1007/bf01390135
复制
发表时间：
1976
影响因子：
3.1
通讯作者：
J. Milne
中科院分区：
文献类型：
--
作者：
M. Artin;J. Milne
设X是特征为p的代数闭域k上的光滑固有曲线。本文的目的是将X上的“Poincar6对偶性”推广到p-扭转群，通过允许X上的任意有限平面群方案a作为系数。设a D= Homgrp（a, I13,,）表示a的Cartier对偶。我们的结果（4.9）是自然对a | Po~在上同调上产生一个完美对偶。当人们考虑平坦系数时，会出现一个新现象，即它们的上同调群通常不是离散群。例如，如果X是一条超奇异椭圆曲线，则Hi(X, c~p),~Hl（X, Ox）。这是一个k向量空间，需要考虑它的自然代数结构。对偶可以在两种不同的情况下表述。第一部分遵循Grothendieck关于上同论在单幂准代数群的范畴中取值的建议。我们最初的建筑就是在这样的背景下进行的。设U k表示k上的可交换单幂代数群的范畴，设QU k表示拟代数群[17]的相应范畴。这是阿贝尔范畴uk与它的子范畴无穷小群的商。这是一个函子
Let X be a smooth proper curve over an algebraically closed field k of characteristic p. The purpose of this paper is to extend "Poincar6 duality" on X to p-torsion groups, by allowing as coefficients any finite flat group scheme A on X. Let A D= Homgrp(A, I13,,) denote the Cartier dual of A. Our result (4.9) is that the natural pairing A | Po~ gives rise to a perfect duality on cohomology. A new phenomenon which arises when one considers flat coefficients is that their cohomology groups are not, in general, discrete groups. For example, if X is a supersingular elliptic curve, then Hi(X, c~p),~Hl(X, Ox). This is a k-vector space, and its natural algebraic structure should be taken into account. There are two different contexts in which the duality can be formulated. The first follows a suggestion of Grothendieck that the cohomology theory should take its values in the category of unipotent quasi-algebraic groups. Our original construction was made in this context. Let U k denote the category of commutative unipotent algebraic groups over k, and let QU k be the corresponding category of quasi-algebraic groups [17]. This is the quotient of the abelian category U k by its subcategory of infinitesimal groups. There is a functor