Deformation theory of group schemes and Construction of extensions

通过给出离散赋值环上的滤子结构，完全确定了离散赋值环上将Witt群方案变形为环面的群方案.在这些群方案中，我们可以指定哪一个群方案是给出统一的Kummer-Artin-Shcreier-Witt理论的标准群方案，并且在低次情形下证明了这样的正规化标准群格式的唯一性，给出了离散赋值环上的群格式，它将Witt群格式变形为环面与群环概型的单位群概型有关系。实际上，我们可以分析这些群环格式的单位群格式的结构，在低维情况下，我们可以确定标准变形群格式与这些单位群格式之间的显式关系.我们需要计算Artin局部环上群概型的同态群和上同调群。当Artin局部环是F_p-代数时，可以完全计算这些群，今后的工作是构造一个离散赋值环上的群概型，它将Witt群概型变形为环面;当Artin局部环是Z/p^n-代数时，确定Artin局部环上群概型的同态群和上同调群。我们相信，治疗这些作品的基本方法已经给出。

The group scheme over a discrete valuation ring which gives a deformation of a Witt group scheme to a torus is completely determined by giving a filtered structure on it. Among those groups schemes, we can specify which group scheme is stangard as a group scheme which gives the unified Kummer-Artin-Shcreier-Witt theory, and morover in lower degree cases we can show the uniqueness of such a normalized stangard group scheme.The group scheme over a discrete valuation ring which gives a deformation of a Witt group scheme to a torus has a relationship with the unit group scheme of a group-ring scheme. In fact, we can analyze the structure of the unit group schemes of those group-ring schemes, and in a lower dimensional case we can decide the explicit relationship between the stangard deformation group schemes and such unit group schemes.To construct the group schemes over a discrete valuation ring which gives a deformation of a Witt group scheme to a torus, we need to compute the homomorphism groups and cohomology groups of group schemes over an Artin local ring. When the Artin local ring is F_p-algebra, we could compute completely those groups.In the future, the works which should be done are to conpactify the group schemes over a discrete valuation ring which gives a deformation of a Witt group scheme to a torus, and to decide the homomorphism groups and cohomology groups of group schemes over an Artin local ring when it is Z/p^n-algebra. We beleave that the fundamental methods of treating those works have been already given.