On the Cartier duality of certain finite group schemes of order \(p^n\)
Let \(p\) be a prime number, let \(A\) be a commutative ring with unity, and let \(\lambda\) be a non-zero element of \(A\). Let \(X\) be an indeterminate. Then there are \(A\)-Hopf algebras \(A[X,(1+\lambda X)^{-1}]\); \(A[X,(1+\lambda^pX)^{-1}]\) and corresponding \(A\)-group schemes \({\mathcal G}^{(\lambda)}:= \text{Spec}\;A[X,(1+\lambda X)^{-1}]=\text{Hom}_{A-alg}(A[X,(1+\lambda X)^{-1}],-)\) and \({\mathcal G}^{(\lambda^p)}:= \text{Spec}\;A[X,(1+\lambda^pX)^{-1}]=\text{Hom}_{A-alg}(A[X,(1+\lambda^pX)^{-1}],-)\). There is a homomorphism of \(A\)-Hopf algebras \(\phi: A[X,(1+\lambda^pX)^{-1}]\rightarrow A[X,(1+\lambda X)^{-1}]\) defined as \(\phi(X)=\lambda^{-p}((1+\lambda X)^p-1)\). Consequently, there is a homomorphism of group schemes \(\psi: {\mathcal G}^{(\lambda)}\rightarrow {\mathcal G}^{(\lambda^p)}\) defined as follows: for an \(A\)-algebra \(B\), and element \(f\in {\mathcal G}^{(\lambda)}(B)\), one has \(\psi(f)(X)=f(\phi(X))\). For an affine group scheme \(G\), let \(\hat G\) denote the completion of \(G\) along the zero section. Then the homomorphism \(\psi\) above extends to a homomorphism \(\psi: \widehat{\mathcal G}^{(\lambda)}\rightarrow \widehat{\mathcal G}^{(\lambda^p)}\). For an integer \(l\geq 1\), the situation generalizes: there is a homomorphism \(\psi^{(l)}: \widehat{\mathcal G}^{(\lambda)}\rightarrow \widehat{\mathcal G}^{(\lambda^{p^l})}\), where \(\psi^{(l)}\) is defined through \(X\mapsto \lambda^{-p^l}((1+\lambda X)^{p^l}-1)\). Note that \(\psi=\psi^{(1)}\). Now assume that \(A\) has characteristic \(p\). For \(l\geq 1\), let \(N_l=\text{ker}(\psi^{(l)})\), and let \(N_l^D\) be its Cartier dual. The paper under review concerns the structure of \(N_l^D\). For the case \(l=1\), it has been shown that \(N_1^D\) is canonically isomorphic to the kernel of the homomorphism \(F-\lambda^{p-1}: {G}_{a,A}\rightarrow {G}_{a,A}\), where \(F: {G}_{a,A}\rightarrow {G}_{a,A}\) is the Frobenius endomorphism of the additive group scheme over \(A\), see \textit{Y. Tsuno} [Deformations of the Kummer Sequence (Preprint Series, CHUO MATH, No. 82) (2008)]. In the paper under review, the author generalizes Tsuno's result as follows. Let \(W_{l,A}\) be the Witt ring scheme of length \(l\) over \(A\). Note that \(W_{1,A}={G}_{a,A}\). Let \(F: W_{l,A}\rightarrow W_{l,A}\) denote the Frobenius endomorphism of \(W_{l,A}\) and let \([\lambda]\) denote the Teichmüller lifting of \(\lambda\). Then \(N_l^D\) is canonically isomorphic to the kernel of the homomorphism \(F-[\lambda^{p-1}]: W_{l,A}\rightarrow W_{l,A}\).
scientific article; zbMATH DE number 5776102
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