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Notes on the local p-adic Simpson correspondence

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Notes on the local p-adic Simpson correspondence
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The local p-adic Simpson correspondence by Faltings asserts that there is a natural equivalence of categories between small generalized representations and small Higgs modules for an affine semi-stable scheme over a complete discrete valuation ring of mixed characteristic with algebraically closed residue field. However, in the case of rational coefficients, the construction of the functor from the former to the latter, reducing to the theory for integral coefficients, does not seem to work as it is written, as pointed out by Ahmed Abbes. In this talk, I give an alternative argument based on a generalized Sen's theory for the semi-stable scheme and complete the local theory for rational coefficients.
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