Étale splittings of certain Azumaya algebras on toric and hypertoric varieties in positive characteristic (Q1931844)

A lot of recent progress in characteristic \(p\) geometry stems from the realization [\textit{R. Bezrukavnikov} et al., Ann. Math. (2) 167, No. 3, 945--991 (2008; Zbl 1220.17009)] that the sheaf \(D_X\) of divided power differential operators on a smooth scheme \(V\) over a perfect field \(k\) is naturally an Azumaya algebra over the cotangent bundle \(T^*X^{(1)}\), where \(X^{(1)}=X \otimes_k k^{1/p}\) is the Frobenius twist of \(X\). There is a canonical flat cover trivializing \(D_X\), but in general it may be difficult to find a reasonably explicit trivializing étale cover. \textit{A. Ogus} and \textit{V. Vologodsky} [Publ. Math., Inst. Hautes Étud. Sci. 106, 1--138 (2007; Zbl 1140.14007)] showed that a splitting \(\Omega^1_{X^{(1)}/k}\to(F_{X/k})*\Omega^1_{X/k}\) of the Cartier operator naturally induces an étale ``fiberwise Artin-Schreier`` cover \(T^*X^{(1)}\to T^*X^{(1)}\) trivializing \(D_X\). The paper under review contains an independent construction of a trivializing étale cover in the case when \(X\) is a toric variety, making use of the action of the torus \(T\). The author shows that there is a canonical \(T\)-invariant such cover with a very explicit form, and compares it with the Ogus-Vologodsky construction. Because any toric variety has a canonical lift over \(W(k)\) together with a canonical lift of the Frobenius, there is a canonical splitting of the Cartier operator, hence the construction of Ogus and Vologodsky applies. The author notes that the two covers are the same. There are two interesting applications of this result. First, it is shown that \(D_X\) is trivial along the fibers of the moment map \(\mu:T^*X^{(1)} \to t^{*(1)}\). Second, the results are extended to certain Azumaya algebras on hypertoric varieties. The paper ends with a discussion of derived Beilinson-Bernstein localization in this setting.