Let p be a prime number, k a perfect ﬁeld of characteristic p > 0, X a smooth k-scheme and _D_X(m) the algebra of (arithmetic) diﬀerential operators of level m ≥ 0. We study the Azumaya nature of this algebra and show how to construct, using an additional data, a splitting of a completion (along some ideal contained in its center) of it.
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Michel Gros, Bernard Le Stum, Adolfo Quirós, A Simpson Correspondence in Positive Characteristic. Publ. Res. Inst. Math. Sci. 46 (2010), no. 1, pp. 1–35DOI 10.2977/PRIMS/1