Algebraic versus rigid cohomology with logarithmic coefficients. (English) Zbl 0833.14010

Cristante, Valentino (ed.) et al., Barsotti symposium in algebraic geometry. Memorial meeting in honor of Iacopo Barsotti, in Abano Terme, Italy, June 24-27, 1991. San Diego, CA: Academic Press. Perspect. Math. 15, 11-50 (1994).
Let \((K, |\cdot |)\) be a complete valued field extension of \((\mathbb{Q}_p, |\cdot |)\) for some prime integer \(p\). Denote by \(A\) the ring of integers of \(K\), by \(m\) the maximal ideal of \(A\), and by \(k\) the residue field of \(A\). Let \(K'\) be a discretely valued complete subfield of \(K\), of valuation ring \(A'\) and residue field \(k'\). Let \(Y\) be a proper smooth \(A'\)-scheme, and consider a divisor \(Z\) in \(Y\) with normal crossings relative to \(A'\). Set \(X = Y - Z\). Let \(E\) be a locally free \({\mathcal O}_Y\)-module of finite rank endowed with an integrable connection \(\nabla\) with logarithmic singularities along \(Z_{K'}\). The main aim of this paper is to relate (under certain conditions) the algebraic cohomology \(H^\bullet (X_K/K, (E, \nabla))\) with the rigid cohomology \(H^\bullet_{\text{rig}} (X_k/K, (E^+, \nabla^+))\) naturally associated to this situation. The results obtained are analogous to some GAGA-type results previously proved by the first named author.
For the entire collection see [Zbl 0802.00020].


14F30 \(p\)-adic cohomology, crystalline cohomology
14G20 Local ground fields in algebraic geometry
14F25 Classical real and complex (co)homology in algebraic geometry