The Kummer-Artin-Schreier-Witt theory and the deformations of the Art theory

在赋值环A上，长度为ri的Witt向量的群方案W_n到环面G^n_的变形群W_n归纳为从W_1= G^ <(lambda)>开始的Ext^1（W_<n-1>, G^ <(lambda)>）的元素。扩展群Ext^1（W_<n-1>, G^n_）本质上与hm （W_<n-1>, G_<m， A>）同构，因此通过计算群hm （W_<n-1 b>, G_<m， A>）和Ext^1（W_<n-1>, G_<m， A>）来确定变形群W_n。在之前的研究中，我们展示了以下内容。设p是一个素数，a是一个Z(p)代数。然后构造Arti -Hasse指数级数的变形，利用这些变形的Artin-Hasse指数级数，我们可以证明t群hm （g^<(lambda)>, G_<m， A>）和Ext^1 （g^<(lambda)>, G_<m， A>）分别同构于扭曲Frobenius自同态F^<(lambda)> = F-[lambda^<P-1>]: W*W。在本研究中，我们尝试将这些同构推广到高维情况。事实上，对于给定的群方案W_n，将W_n变形为G^n_，我们可以构造由若干W的自同态组成的非自同态psi: W_n * W_n，其中W的自同态包含F^<(λ)>型的形变Frobenius自同态，我们可以得出群hm （W_n, G_<m, a >）和群Ext^1 （W_n, G_<m, a >）分别同构于的核和核。

The deformation groups W_n of the group scheme W_n of Witt vectors of length ri to the torus G^n_ over valuation ring A are given inductively as the elements of Ext^1(W_<n-1>, g^<(lambda)>) starting from W_1=g^<(lambda)>. The extension groups Ext^1(W_<n-1>, G^n_) are essentially isomorphic to Hom(W_<n-1>, G_<m, A>) and hence the deformation groups W_n are determined by calculating the groups Hom(W_<n-1>, G_<m, A>) and Ext^1(W_<n-1>, G_<m, A>).In the previous research, we showed the following.Let p be a prime integer, and A be a Z(p)-algebra. Then we can construct the deformations of the Arti Hasse exponential series, and using these deformed Artin-Hasse exponential series, we can show that t groups Hom(g^<(lambda)>, G_<m, A>) and Ext^1 (g^<(lambda)>, G_<m, A>) are isomorphic to the kernel and the cokernel of the twist Frobenius endomorphism F^<(lambda)> = F-[lambda^<P-1>] : W*W, respectively.In this research, we tried to generalize these isomorphy to higher dimensional cases. In fact, for a givgroup scheme W_n deforming W_nto G^n_, we can construct anendomorphism psi : W_n * W_n consisting some endomorphisms of W containing deformed Frobenius endomorphisms of type F^<(lambda)>, and we can sha that the groups Hom(W_n, G_<m, A>) nad Ext^1 (W_n, G_<m, A>) are isomorphic to the kernel and the cokernel of respectivly.