The \(p\)-adic Simpson correspondence (Q2787944)

The paper under review deals with \(p\)-adic Galois representations. It uses the tool of almost ètale extensions. For Galois groups of local fields this goes back to \textit{J. T. Tate} [in: Proc. Conf. local Fields, NUFFIC Summer School Driebergen 1966, 158--183 (1967; Zbl 0157.27601)], and \textit{S. Sen} [Invent. Math. 94, No. 1, 1--12 (1988; Zbl 0695.12009)] has derived a description of semilinear Galois representations via modules with an endomorphism. The reviewer has extended this theory to schemes of higher dimension. The local theory follows from the almost purity theorem, and globalisation poses interesting new problems. The name Simpson correspondence stems from the well-known complex analytic analogue.NEWLINENEWLINEThe authors give a very detailed introduction to the theory, smoothing out some difficulties by introducing new concepts. They also correct some mistakes of the reviewer.