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

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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.