Coherence of certain overconvergent isocrystals without Frobenius structures on curves. (English) Zbl 1264.12005

This paper proves coherence as \(\mathcal D^\dagger\)-modules of certain overconvergent isocrystals on curves, even at the absence of Frobenius structures.
Let \(R\) be a complete discrete valuation ring of mixed characteristic, with perfect residue field \(k\). Let \(K\) denote its field of fractions. Let \(\mathcal X\) be a smooth formal curve over \(R\), \(\mathcal U\) a dense open subscheme, and \(S: = \mathcal X - \mathcal U\) (which we may assume to be a finite set of \(k\)-points). Let \(M\) be a convergent isocrystal on \(\mathcal U\) overconvergent along \(S\). The author makes the following assumption: at each point \(s \in S\), the \(0\)-slope part (tame part) of the differential modules at \(s\) is trivial; here the slope refers to applying the deep theorem of Christol-Mebkhout to the base change of \(M\) to the formal completion of \(\mathcal X\) at \(s\), so that we obtain a decomposition of the differential module into a direct sum of ones with pure slopes (which is a measure of irregularity in the \(p\)-adic setting). If we use \(\mathrm{sp}: \mathcal X_K \to \mathcal X\) to denote the specialization map, then the main theorem of this paper states that \(\mathrm{sp}_*(M)\) is a coherent \(\mathcal D^\dagger_{\mathcal X, \mathbb Q}\)-module. Note that there is no assumption on the existence of Frobenius.
The proof of the theorem is to construct a finite presentation of \(\mathrm{sp}_*(M)\) as a \(\mathcal D^\dagger_{\mathcal X, \mathbb Q}\)-module directly. For this, one may work locally. If \(e_1, \dots, e_\mu\) form a basis of \(M\), the idea is to construct differential operators \(P^{(i)}_{k,j}\) such that \(\sum_j P^{(i)}_{k,j}e_j = x^{-k} e_i\), where \(x\) is a local parameter at a point \(s \in S\). The technical core lies in, when \(M\) is a differential module pure of some positive slope near \(s\), one can write down such differential operators \(P^{(i)}_{k,j}\) satisfying certain nice bounds on their norms.
The paper is very technical. Although the author gives clear and detailed presentation, the gist of the proof still seems to be buried in the delicate computation and complicated notation.


12H25 \(p\)-adic differential equations
14F30 \(p\)-adic cohomology, crystalline cohomology
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