On the description of \(\mathbb Z/p^2\mathbb Z\)-torsors by the Kummer-Artin-Schreier-Witt theory
Let \(p\) be a prime and let \(\zeta _{n}\) be a primitive \(p^{n}\)-th root of unity. For \(A=\mathbb Z_{\left( p\right) }\left[ \zeta _{n}\right] \) we have the Kummer-Artin-Schreier-Witt sequence \[ 0\rightarrow \left(\mathbb Z/p^{n} \mathbb Z\right) _{A}\rightarrow \mathcal{W}_{n}\rightarrow \mathcal{V}_{n}\rightarrow 0 \] where \(\mathcal{W}_{n}\) and \(\mathcal{V}_{n}\) are certain group schemes related to deformations of \(\mathbf{G}_{a}\) and \(\mathbf{G}_{m}\). This sequence gets its name from the fact that its special fiber is the Artin-Schreier-Witt sequence and its generic fiber is the Kummer sequence. Given a scheme \(X\) and a flat group scheme \(G\) which is locally of finite type over \(X\) one can consider the set PHS\(\left( G/X\right) \) of isomorphism classes of \(G\)-torsors over \(X\). This paper examines the case where \(G= \mathbb Z/p^{2}\mathbb Z\) and \(X\) is an \(A\)-scheme. In other words, a description of the unramified cyclic coverings of degree \(p^{2}\) over \(X\). The main result is as follows. Let \(\mathcal{U=}\left\{ U_{j}\right\} \) be an affine open covering on \(X\), and let \(\Psi ^{2}:\mathcal{W} _{2}\rightarrow \mathcal{V}_{2}\) be the map in the Kummer-Artin-Schreier-Witt sequence in the case where \(n=2\). (This is the only case in which the author explicitly describes these group schemes and this map, however it is also the only case used in this paper.) Let \(\mathbf{ f}_{ij}\in Z^{1}\left( \mathcal{U},\mathcal{W}_{2}\right) \) be such that \( \Psi ^{2}\left( [\mathbf{f}_{ij}]\right) =0.\) Then for each \(j\) there exists a \(\mathbf{b}_{j}\in \Gamma \left( U_{j},\mathcal{V}_{2}\right) \) such that \( \Psi ^{2}\left( \mathbf{f}_{ij}\right) =\left( \Lambda _{0}^{G}\left( \mathbf{b}_{j},I^{G}\left( \mathbf{b}_{i}\right) \right) ,\Lambda _{1}^{G}\left( \mathbf{b}_{j},I^{G}\left( \mathbf{b}_{i}\right) \right) \right) \) on \(U_{j}\cap U_{i}.\) For \(\mathbf{h\in \Gamma }\left( X,\mathcal{V }_{2}\right) \) a \(\mathbb Z/p^{2} \mathbb Z\)-torsor over \(X\) is described by \(\pi :X^{\prime }\rightarrow X\) given locally by the covering \(\Psi ^{2}\left( \mathbf{z}_{j}\right) =\left( \Lambda _{0}^{G}\left( \mathbf{b} _{j},\mathbf{h}\right) ,\Lambda _{1}^{G}\left( \mathbf{b}_{j},\mathbf{h} \right) \right) \) on Spec\(\left( \Gamma \left( U_{j},\mathcal{V}_{2}\right) \otimes _{A}A\left[ \mathbf{z}_{j}\right] \right) .\) In addition, the gluing is explicitly given as well as the \(\mathbb Z/p^{2}\mathbb Z\) on \( X^{\prime }.\) In the case where \(H^{1}\left( X,\mathbb Z/p^{2}\mathbf{Z}\right) \cong \text{coker}\left[ \Psi ^{2}:\Gamma \left( X,\mathcal{W}_{2}\right) \rightarrow \Gamma \left( X,\mathcal{V}_{2}\right) \right] \) it is shown that for \(C\) an unramified \(p^{2}\)-cyclic extension of \(B\) where \(X=\text{Spec}B\) there is an \(f:X\rightarrow \mathcal{V}_{2}\) such that the composition \(\text{Spec}C\rightarrow \mathcal{W}_{2}\rightarrow \mathcal{V }_{2}\) factors through \(X\). This allows for describing explicit conditions for when two \(\mathbb Z/p^{2} \mathbb Z\)-torsors over \(X\) are isomorphic. In the case where \(\ker \left[ \Psi ^{2}:H^{1}\left( X,\mathcal{W} _{2}\right) \rightarrow H^{1}\left( X,\mathcal{V}_{2}\right) \right] \) then \( H_{\text{fl}}^{1}\left( X,\text{coker}\left[ \Psi ^{2}:\Gamma \left( X, \mathcal{W}_{2}\right) \rightarrow \Gamma \left( X,\mathcal{V}_{2}\right) \right] \right) \) is isomorphic to a set of pairs of line bundles in Pic\( \left( X\right) \) satisfying certain conditions. In general, there is a correspondence between \(\mathbb Z/p^{2}\mathbb{Z}\) -torsors over \(X\) and \(\mu _{p^{2}}\)-torsors over \(X.\) In the final section the author shows how to describe a \(\mathbb{Z}/p^{2}\mathbb Z\)-torsor over \( X\) as a sequence of Néron blow-ups starting with a \(\mu _{p^{2}}\)-torsor.
scientific article; zbMATH DE number 2064100
scientific article