Applications of the Kummer-Artin-Schreier-Witt theory to Number Theory and to Algebraic Geometry

关于在形式群的Cartier理论框架下对Kummer-Artin-Schreier-Witt理论的描述，我们得到了一些显著的结果。对Z[M]-代数A，我们引入了一个可加群W^(M) (A)，从而改写了经典的Witt向量理论。W^(M)(A)具有W(A)-模的结构，与经典情况一样，在W^(M)(A)上定义了Frobenius映射F^<(M)>和Verschiebung映射。映射（a_0, a_1,a_2，…）→（Ma_0,Ma_1,Ma_2，…）是W^(M)(a)到W(a)的W(a)-同态映射，兼容Frobenius映射和Verschiebung映射。另一方面，G^(M)_A表示的群格式在Kummer-Artin-Schreier-Witt理论中起着重要作用，G^(M)_A沿G^(M)_A的零截面的形式补全G^(M)_A就是具有形式群律f(X，Y) = X + Y + MXY的形式群Spf A[[T]]。给出了W^(M)(A)与G^(M)_A上p-典型曲线的Cartier模同构的第一个主要结果。第二个主要结果是将W^(M)(a)的自由分辨率描述为d_a模块。更进一步，如果ε是G^（Λ）_A除以G^(M)_A的扩展，则卡地亚模C（ε）是W^（Λ）(A)除以W^(M)(A)的扩展。引入形式幂级数，给出了C(^^ __ε) = Hom_< a -gr>(^^ __W,ε)的显式描述。我们已经把我们的结果写成文章<代数群和形式群的扩展注释，V>，这篇文章将发表在期刊上。

We have gotten some remarkable results concerning to a description of the Kummer-Artin-Schreier-Witt theory in the framework of the Cartier theory for formal groups. We intorduce an additive group W^(M) (A) for a Z[M]-algebra A, paraphrasing the classical theory of Witt vectors. W^(M)(A) has a structure of W(A)-module, and the Frobenius map F^<(M)> and the Verschiebung map are defined on W^(M)(A) as in the classical case. The map(a_0, a_1,a_2, … ) →(Ma_0,Ma_1,Ma_2, … ) is a W(A)-homomorphism of W^(M)(A) to W(A), com-patible with the Frobenius maps and the Verschiebung maps. On the other hand, the group scheme denoted by G^(M)_A plays an important role in the Kummer-Artin-Schreier-Witt thoery, and the formal completion G^(M)_A along the zero section of G^(M)_A is nothing but the formal group Spf A[[T]] with a formal group law f(X,Y) = X + Y + MXY. It is the first main result that W^(M)(A) is isomorphic to the Cartier module of p-typical curves on G^(M)_A. The second main result is a description of free resolution of W^(M)(A) as a D_A-module.Furhtermore, if ε is an extension of G^(Λ)_A by G^(M)_A, the Cartier module C(ε) is an extension of W^(Λ)(A) by W^(M)(A). Introducing a formal power serieswe give an explicit description of C(^^^__ε) = Hom_<A-gr>(^^^__W,ε). We have written up our results as the article <A note on extensions of algebraic and formal groups, V>, which will appear in a journal.