On the extensions of \(\widehat{W}_n\) by \(\widehat{\mathcal G}^{(\mu)}\) over a \(\mathbb Z_{(p)}\)-algebra
Let \(p\) be prime, and let \(A\) be an algebra over the localized ring \(\mathbb{ Z}_{( p) }.\) Let \(W,\) \(W_{n},\,G_{a},\) and \(G_{m}\) (resp. \( \widehat{W},\,\widehat{W}_{n},\,\widehat{G}_{a},\) \(\widehat{G}_{m}\) ) be the group scheme of Witt vectors, the group schemes of Witt vectors of length \(n\), the additive group scheme, and the multiplicative group scheme (resp. the corresponding formal group schemes) over \(\mathbb{Z}\). The map \( W( A) \rightarrow W( A) \) given by multiplication by \( F^{n}\) provides us with a way of computing both \(\text{Hom}_{A}( \widehat{W}_{n,A},\widehat{G}_{m,A}) \) and \(H_{0}^{2}( \widehat{W} _{n,A},\widehat{G}_{m,A}) \) -- these are isomorphic to the kernel and cokernel of this map. Of course, this enables us to compute \(\text{Ext}_{A}^{1}( \widehat{W}_{n,A}, \widehat{G}_{m,A}) \). By reversing the roles of the group schemes (in the map) and the formal group schemes (in the cohomology groups) we can also compute \(\text{Ext}_{A}^{1}( W_{n,A},G_{m,A}) .\) These results are given in [\textit{T. Sekiguchi} and \textit{N. Suwa}, Tohoku Math. J. 49, 241--257 (1997; Zbl 0887.14019)]. These isomorphisms are obtained using deformations of the Artin-Hasse exponential series. Let \(\mu \in A,\) and define \(\mathcal{G}_{A}^{( \mu) }=\text{Spec}A[ T,T/( 1+\mu T)] \) with the Hopf algebra having coalgebra structure given by \(\Delta (T) =T\otimes 1+1\otimes T+\mu T\otimes T\). It can be readily seen that \(\mathcal{G} _{A}^{( 0) }=G_{a,A}\) and \(\mathcal{G}_{A}^{( 1) }\cong G_{m,A}\) and hence \(\mathcal{G}\) is a deformation between these two groups. The corresponding formal group scheme is denoted \(\widehat{\mathcal{G }}.\) In this work, the author studies \(\text{Ext}_{A}^{1}( W_{n,A},\mathcal{G} _{A}^{( \mu ) }) \) and \(( \widehat{W}_{n,A},\widehat{ \mathcal{G}}_{A}^{( \mu) }) .\) To accomplish this, the Witt vectors are replaced with the ``variants of Witt vectors''. The construction of the variant vectors are similar to the usual Witt vectors. The variant Witt vectors, denoted \(W^{(\mu) }\) have addition in the \(r\)th component given by \(S_{r}^{( \mu)}(\mathbb{X,Y})= \mu ^{-1}S_{r}( \mu \mathbb{X,} \mu \mathbb{Y}) ,\) where \(S_{r}\) is the \(r\)th component of the usual Witt vector addition and \(\mu \mathbb{X}\) (resp. \(\mu \mathbb{Y}\)) is multiplication of each entry of \(\mathbb{X}\) by \(\mu .\) (The usual constructions are also provided in this paper.) The product polynomials \( P_{r}^{(\mu) }\) and Frobenius polynomial \(F^{(\mu)}\) are analogous. The main theorem is completely analogous to the results of Sekiguchi-Suwa mentioned above. Here the map \(F^{(\mu) ^{n}}:W(A) \rightarrow W(A) \) is considered. It is proved that the kernel is isomorphic to \(\Hom_{A}( \widehat{W}_{n,A},\widehat{ \mathcal{G}}_{A}^{(\mu)}) \) and the cokernel is isomorphic to \(H_{0}^{2}( \widehat{W}_{n,A},\widehat{\mathcal{G}} _{A}) .\) If \(\mu \) is nilpotent, then if we reverse the roles of the group schemes and the formal group schemes we obtain the other analogous isomorphisms. The approach here is also to construct deformations of the Artin-Hasse exponential series. In the case where \(\mu \) us not nilpotent it is more difficult to compute \(\text{Ext}_{A}^{1} (W_{n,A},\mathcal{G}_{A}^{(\mu)})\). However, in the case where \(A\) is a discrete valuation ring with maximal ideal \(\mathfrak{m}\) it is shown that \(\text{Ext}_{A}^{1} (W_{n,A},\mathcal{G} _{A}^{( \mu)}) \) is isomorphic to the subgroups of \(H_{ \text{ét}}^{1}( W_{n,A},\mathcal{G}_{A}^{( \mu)})\) formed by the primitive elements. From this it is deduced that \[ \text{Ext}_{A}^{1}( W_{n,A},\mathcal{G}_{A}^{( \mu)}) \cong \Bigl\{ 1+\sum c_{i_{0},\dots i_{n-1}}T_{0}^{i_{0}}\cdots T_{n-1}^{i_{n-1}}\,| \,c_{i_{0},\dots i_{n-1}}\in \mathfrak{m} \Bigr\}. \]
scientific article; zbMATH DE number 5017534