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On higher direct images of convergent isocrystals. (English) Zbl 1430.14050

For a scheme over a field of perfect field \(k\) of characteristic \(p>0\), A. Ogus [Prog. Math. 88, 133–162 (1990; Zbl 0728.14020)] defined a “convergent site” \(\mathrm{Conv}(X/V)\), \(V\) being a discrete valuation ring of characteristic \(0\) with residue field \(k\). On this site one can consider the category of convergent crystals. Let \(\mathfrak{S}\) be an admissible formal scheme over \(V\), and \(X\) be a scheme over \((\mathfrak{S}\otimes k)^{\text{red}}\), one can also define a relative convergent site \(\mathrm{Conv}(X/\mathfrak{S})\), and the relative convergent topos. A morphism \(g:X \to Y\) of varieties over \(k\) induces a morphism of topoi \[ g_{\text{conv}}: (X/V)_{\text{conv}} \to (Y/V)_{\text{conv}} \]
In the article under review, assuming that \(V = W(k)\), the author proves two theorems.
Theorem 1 (Relative Frobenius descent theorem). Let \(F_{X/S_0}:X' = X \times_{S_0,\mathrm{Frob}_{S_0}} S_0 \to X\) be the relative Frobenius. Then \(F_{X/S_0}\) induces an equivalence between the fppf-convergent topos of \(X'/ \mathfrak{S}\) and that of \(X/\mathfrak{S}\). Moreover, it induces an equivalence between the category of isocrystals on \(X\) and on \(X'\).
Theorem 2 (“Berthelot’s conjecture” for convergent isocrystals). Let \(g:X \to Y\) be a smooth and proper morphism of \(k\)-varieties. Let \(\mathcal{E}\) be an isocrystal on \(X\). Then \(\mathcal{F}=R^{i}g_{\text{conv},\ast}\mathcal{E}\) is a convergent isocrystal on \(Y\).
The proof of theorem 2 for a smooth \(Y\) is based on the fact that \(\mathcal{F}\) is a “\(p\)-adically convergent isocrystal” in the sense of A. Ogus [Duke Math. J. 51, 765–850 (1984; Zbl 0584.14008)] (proved in §4). By Theorem 1, \(\mathcal{E}\) is a Frobenius descendant, hence \(\mathcal{F}\) is a Frobenius descendant too. Iterating this process, Dwork’s trick then implies \(\mathcal{F}\) is convergent (§5).
In order to treat singular \(Y\), the author defines a variant of the convergent site which allows him to apply Ogus’s “proper descent theorem”, and reduces the problem back to the smooth case (§8).

MSC:

14F30 \(p\)-adic cohomology, crystalline cohomology
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
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References:

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