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By Nai-Phuan Ong, Ravin Bhatt
During this textual content the writer makes use of stack-theoretic strategies to review the crystalline constitution at the de Rham cohomology of a formal soft scheme over a p-adic box and purposes to p-adic Hodge conception. He develops a common thought of crystalline cohomology and de Rham-Witt complexes for algebraic stacks and applies it to the development and examine of the ([phi], N, G)-structure on de Rham cohomology. utilizing the stack-theoretic perspective rather than log geometry, he develops the elements had to turn out the Cst-conjecture utilizing the tactic of Fontaine, Messing, Hyodo, Kato, and Tsuji, with the exception of the major computation of p-adic vanishing cycles. He additionally generalizes the development of the monodromy operator to schemes with extra common forms of aid than semistable and proves new effects approximately tameness of the motion of Galois on cohomology.
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Example text
2). 14. 2). 15. 2) g g u u X −−−−→ X −−−−→ X S −−−−→ S −−−−→ S, where X , X , and X are Deligne–Mumford stacks such that γ (resp. γ , γ ) extends to X (resp. X , X ). 3) (g ◦ g )cris gcris ◦ gcris . Proof. 6). 16. Just as the ordinary lisse-´etale topos of an algebraic stack is not functorial ([50]), there does not in general exist a morphism of topoi between (Xlis-et /S)cris and (Xlis-et /S). 2) still yields adjoint functors (g −1 , g∗ ), but the functor g −1 need not commute with finite inverse limits.
3. 2) Ψ(ξ1 ) ◦ Ψ(ξ2 ) − Ψ(ξ2 ) ◦ Ψ(ξ1 ) = Ψ([ξ1 , ξ2 ]). ∗ Let :E → be the map obtained from the map pr1∗ pr−1 2 E → pr1∗ pr2 E and the observation that since the projections DX (1) → X induce equivalences between the associated ´etale topoi there is a canonical isomorphism E pr1∗ pr−1 2 E. 3) pr1∗ pr∗2 E pr∗ ρ1 (ξ) 2 E −−−− → pr11∗ pr1∗ 2 E −−−−→ E. 4) ∇ξ1 ◦ ∇ξ2 − ∇ξ2 ◦ ∇ξ1 = ∇[ξ1 ,ξ2 ] . 5) ∇ξ (f · e) = ξ(f ) · e + f ∇ξ (e). In other words, giving the maps ρn is equivalent to giving an integrable connection ∇ on E, and this is in turn equivalent to giving E the structure of a left OXet TXet /S – module.
3) X ×S U ↓ U → 1 PX× S U/U ← X ×S U. Thus if (E, ∇) is a module with connection on X/S, then the pullback to X ×S U has a canonical connection obtained by pulling back the isomorphism pr∗1 E pr∗2 E to 1 1 PX×S U/U U ×S PX/S . 4) ρ∗U : M C(X/S) −→ M C(X ×S U/U ). 4). 5) ρ∗ : M C(X/S) −→ M C(X• /S• ) by sending (E, ∇) to the module with connection on X• /S• whose restriction to Xi is ρ∗S i (E, ∇) and whose transition maps are the natural ones. 4) applied to PX/S . 6) π ∗ E −−−−−→ π ∗ E ∇ξ π −1 E −−−−→ π −1 E commutes.