3.1 Flat, unramified and etale morphisms 


3.2 Nilpotent deformations of almost algebras and modules 


3.3 Nilpotent deformations of torsors 


3.4 Descent 


3.5 Behaviour of etale morphisms under Frobenius

Almost Ring Theory

/pdf/points-and-topologies-in-rigid-geometry-1qjziyamj6.pdf

Points and topologies in rigid geometry

We investigate the p -adic properties of higher coherent cohomology of automorphic vector bundles of singular weights on Siegel threefolds.

Higher coherent cohomology and $p$-adic modular forms of singular weights

Given a generically etale morphism $f\colon Y\to X$ of quasi-smooth Berkovich curves, we define a different function $\delta_f\colon Y\to[0,1]$ that measures the wildness of the topological ramification locus of $f$. This provides a new invariant for studying $f$, which cannot be obtained by the usual reduction techniques. We prove that $\delta_f$ is a piecewise monomial function satisfying a balancing condition at type 2 points analogous to the classical Riemann-Hurwitz formula, and show that $\delta_f$ can be used to explicitly construct the simultaneous skeletons of $X$ and $Y$. As an application, we use our results to completely describe the topological ramification locus of $f$ when its degree equals to the residue characteristic $p$.

Morphisms of Berkovich curves and the different function

© Publications mathématiques de l’I.H.É.S., 1981, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

/pdf/equivalence-of-differentiable-mappings-and-analytic-mappings-16342qeydi.pdf

Equivalence of differentiable mappings and analytic mappings

Cohomology on affinoid spaces

A problem on coefficient fields and equations over local rings

/pdf/some-properties-of-the-ring-of-germs-of-c-functions-1gb8zlu4iv.pdf

Some properties of the ring of germs of C∞-functions

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Cohomology on affinoid spaces

A problem on coefficient fields and equations over local rings

Some properties of the ring of germs of C∞-functions

Cohomology on affinoid spaces