Direct images. I: Rigid analytic spaces and direct images
Let \(\mathcal{V}\) be a discrete valuation ring with fraction field \(K\) of characteristic 0 and residue field \(k\) of characteristic \(p>0\). Let \(f : X \to S\) be a morphism of \(k\)-varieties. Without going into the details, let us recall that an isocrystal \(E\) over \(X\) is a module with an integrable connection which is defined on some \(K\)-analytic rigid space associated to \(X\) (a strict neighbourhood of its tube in a certain formal scheme). The article under review gives sufficient conditions in order that the direct images \(R^i f_{\mathrm{rig}*}\) of an overconvergent isocrystal on \(X\) remain overconvergent. This will be the case when \(S\) is smooth over \(k\), \(f\) is smooth and projective and either \(X\) lifts to a flat scheme over \(\mathcal{V}\) or \(X\) is a relative complete intersection inside projective spaces over \(S\).NEWLINENEWLINEThe first section of the text is devoted to base change theorems for proper morphisms, which will be a crucial ingredient in the proof. The author first handle the case of formal schemes, then of rigid analytic spaces. The second section gathers a few technical results on strict neighbourhoods, in particular conditions to ensure that a fundamental system of strict neighbourhoods remains so after pull-back. The third section contains a base change theorem for overconvergent modules by a proper morphism and ends with the proofs of the results concerning direct images of overconvergent isocrystals.NEWLINENEWLINETo finish, let us mention that the author announces two articles to follow, in which he will consider isocrystals endowed with a Frobenius structure in the convergent and overconvergent settings.
scientific article; zbMATH DE number 6075024
scientific article