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Theory of tubular neighborhood in étale topology. (English) Zbl 0872.14014
Summary: To begin with the author spends some time proving results in the affine and hence algebraic case. Inverting admissible blowing ups, he makes the category of coherent henselian schemes into the category of rigid-henselian spaces (with an analogue to rigid-analytic spaces for formal schemes). Next, a rigid space is defined as a sheaf on a site determined by some rigid-henselian space. Topological properties of rigid spaces and their associated Zariski-Riemann spaces are developed. The direct limit of a sequence of blowing-ups of a closed subspace \(Y\) of a coherent model \(X\) associated to a rigid space \({\mathcal X}\) is called a tube \({\mathcal C}_{X/Y}\) of \(Y\) in \(X\). On the other hand, the rigid-henselian étale ringed topos of the henselization of \(X\) along \(Y\) is called the deleted tubular neighborhood \(T_{X/Y}\) of \(X\) along \(Y\). Among the main results are:
There is a categorical equivalence between the rigid-henselian étale spaces over a good henselian scheme \(S\) and the rigid-analytic étale spaces over \(\widehat S\).
Comparison theorem: For a torsion étale sheaf \({\mathcal F}\) on the étale site \(U_{et}\), \[ i^*Rj_* {\mathcal F} \simeq R\alpha_{X^*} (i^{unr*} Rj^{unr}_* {\mathcal F}), \] where using the above notation \(U= X-Y\), \(\overline U\) is obtained by patching together \(T_{X/Y}\) and \(U_{et}\), \(i\) and \(j\) are the inclusions, respectively, of \(T_{X/Y}\) and \(U_{et}\) into \(\overline U\) and the other maps arise on passing over to the unramified situation.
Gabber formal base change theorem (I): For a noetherian henselian couple \((A,I)\) and a torsion sheaf \({\mathcal F}\) on \(\text{Spec} A \setminus V(I)\), \[ H^q_{et} (\text{Spec} A \setminus V(I), {\mathcal F}) \simeq H^q_{et} (\text{Spec} \widehat A\setminus V(I\widehat A),\;\widetilde {\mathcal F}) \] where \(\widetilde {\mathcal F}\) denotes the inverse image of \({\mathcal F}\).
Comparison theorem in the nonproper case: Let \(V\) be a valuation ring with separably closed quotient field \(K\). Let \(f: X\to Y\) be a morphism between finite-type schemes over \(K\). Then \[ (R^qf_* {\mathcal F})^{rig} \to R^q f^{rig}_* {\mathcal F})^{rig} \] is an isomorphism for \({\mathcal F}\in D_c^b (X,\Lambda)\), a constructible complex of \(\Lambda\)-modules with \(\Lambda\) a finite torsion ring having order prime to \(p\), the residual characteristic of \(V\).

MSC:
14F20 Étale and other Grothendieck topologies and (co)homologies
14F30 \(p\)-adic cohomology, crystalline cohomology
18F10 Grothendieck topologies and Grothendieck topoi
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