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

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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.