373 research outputs found
Constructible isocrystals
We introduce a new category of coefficients for p-adic cohomology called
constructible isocrystals. Conjecturally, the category of constructible
isocrystals endowed with a Frobenius structure is equivalent to the category of
perverse holonomic arithmetic D-modules. We prove here that a constructible
isocrystal is completely determined by any of its geometric realizations.Comment: Pr\'epublication de l'IRMAR 2016-0
The Overconvergent Site I. Coefficients
We define and study the overconvergent site of an algebraic variety, the
sheaf of overconvergent functions on this site and show that the modules of
finite presentations correspond to Berthelot's overconvergent isocrystals. We
work with Berkovich theory instead of rigid analytic geometry and do not use
any of Berthelot's results. This gives a complete alternative approach to rigid
cohomology.Comment: 53 page
The Overconvergent Site II. Cohomology
We prove that rigid cohomology can be computed as the cohomology of a site
analogous to the crystalline site. Berthelot designed rigid cohomology as a
common generalization of crystalline and Monsky-Washnitzer cohomology.
Unfortunately, unlike the former, the functoriality of the theory is not
built-in. We defined somewhere else the "overconvergent site" which is
functorially attached to an algebraic variety and proved that the category of
modules of finite presentation on this ringed site is equivalent to the
category of over- convergent isocrystals on the variety. We show here that
their cohomology also coincides.Comment: 27 page
On quantum state of numbers
We introduce the notions of quantum characteristic and quantum flatness for
arbitrary rings. More generally, we develop the theory of quantum integers in a
ring and show that the hypothesis of quantum flatness together with positive
quantum characteristic generalizes the usual notion of prime positive
characteristic. We also explain how one can define quantum rational numbers in
a ring and introduce the notion of twisted powers. These results play an
important role in many different areas of mathematics and will also be quite
useful in a subsequent work of the authors.Comment: 2013 - 8
Constructible nabla-modules on curves
39 pagesInternational audienceLet be a discrete valuation ring of mixed characteristic with perfect residue field. Let be a geometrically connected smooth proper curve over . We introduce the notion of constructible convergent -module on the analytification of the generic fibre of . A constructible module is an -module which is not necessarily coherent, but becomes coherent on a stratification by locally closed subsets of the special fiber of . The notions of connection, of (over-) convergence and of Frobenius structure carry over to this situation. We describe a specialization functor from the category of constructible convergent -modules to the category of -modules. We show that if is endowed with a lifting of the absolute Frobenius of , then specialization induces an equivalence between constructible --modules and perverse holonomic --modules
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