IRMA, UMR 7501 
7 rue RenéDescartes 
67084 Strasbourg Cedex 
Tél. 33 (0)3 68 85 01 29 
Fax. 33 (0)3 68 85 03 28 
Accueil du site > Agenda > Colloques et rencontres > Archives > Agenda 2008 > Recent Progress in Arithmetic Dmodules theory
IRMA, Friday 3rd October 2008
Recent Progress in Arithmetic Dmodules theory
on Friday 3rd October 2008, at Institut de Recherche Mathematique Avancée, Strasbourg.
The aim of this conference is the exposition of recent progress on arithmetic Dmodules theory. We will focus in particular on the work of Caro and Tsuzuki, which, combined with the semistable theorem of Kedlaya, proves that the category of overholonomic arithmetic Dmodules with Frobenius, contructed by Caro, is stable by the 6 Grothendieck cohomological operations.
Organizers : C.NootHuyghe and A.Marmora


08h00 
A confirmerA confirmerCoffee and reception of participants 


08h30 
Pierre Berthelot  Univ Rennes 1An introduction to finiteness conditions in arithmetic Dmodule theoryIn this introductory talk, we will review the main initeness conditions introduced in the theory of arithmetic Dmodules, and we will discuss their stability properties under the basic cohomological operations. 


09h05 
Kiran Kedlaya  MITSemistable reduction for overconvergent FisocrystalsLet k be a field of characteristic p > 0, and let X be a kvariety. The alterations theorem of de Jong states that we can find a proper generically finite morphism from some smooth variety Y to X, such that Y admits a smooth compactification overline{Y} with boundary divisor having strict normal crossings. Now let E be an overconvergent Fisocrystal on X. In its simplest form, the semistable reduction problem (conjecture of Shiho) is to prove that we can choose Y and overline{Y} as above so that the pullback of E to Y admits a logarithmic extension to overline{Y} with nilpotent residues on the boundary. One can also state a refined conjecture applying to partially overconvergent Fisocrystals. We will give a detailed account of our proof of the conjecture of shiho. We start by recallying the ZariskiNagata purity theorem for overconvergent Fisocrystals and logarithmic extensions. We then formulate the problem of local semistable reduction, which allows us to work in a neighborhood of a single valuation on the function field of an irreducible X. We reduce this problem to the case where the valuation has height 1 and residual transcendence degree 0, then set up an induction on a certain numerical invariant (the corank). The base case is that of an Abhyankar valuation; using the strong form of local uniformization available for such varieties, we can proceed much as the onedimensional case (i.e., the conjecture of Crew). For the induction, we fiber X in curves over a variety X_0 of dimension one lower, and view our given valuation as a point in a Berkovich affine line (over the completion of the function field of X_0). We then make an intricate calculation (using our quantitative refinement of ChristolMebkhout decomposition theory) to how that we can reduce from our given valuation to another point in this Berkovich line corresponding to a valuation of lower corank. 


10h20 
Nobuo Tsuzuki  Tohoku Univ.On the overholonomicity of overconvergent Fisocrystals on smooth varieties I. Comparison between logrigid and rigid cohomologiesThis is the first part of the joint work with D.Caro. The goal is to prove that an overconvergent Fisocrystal on a smooth variety is verholonomic arithmetic Dmodules. In this talk we discuss a relative version of comparison between logrigid cohomology and rigid cohomology with oefficients in logisocrystals. This is one of the keys to prove our main result which will be explained in the talk of D. Caro. This type of comparison between logrigid cohomology and rigid cohomology was first studied by BaldassarriChiarellotto and by many others. We assume that an O_X^daggercoherent module with connection is locally free only in the rigid analytic side. Hence it might be locally projective in the formal scheme side in general. By this reason we can not apply Christol's transfer theorem, generalized by BaldassarriChiarellotto, directly. We will explain the technical part of our proof. 


11h20 
Coffee break. 


11h45 
Daniel Caro  Univ. CaenOn the overholonomicity of overconvergent Fisocrystals on smooth varieties IIThis is the second part of the joint work with N. Tsuzuki. The goal is to prove that an overconvergent Fisocrystal on a smooth variety is an overholonomic arithmetic Dmodules. In this talk we will give some explanations on this result by using the relative version of comparison between logrigid cohomology and rigid cohomology with coefficients in logisocrystals explained by N. Tsuzuki in the previous talk. We will give some applications of this result in theory of arithmetic Dmodules of Berthelot. 


12h45 
Lunch. 


14h30 
Richard Crew  Florida Univ.Rings of padic differential operators on tubes 


15h45 
Takeshi Tsuji  Tokyo Univ.Nearby cycles and Dmodules of log schemes in characteristic p>0For a semistable scheme over a complete discrete valuation ring of mixed characteristic (0,p), I define nearby cycles as a Dmodule and study its properties. P. Berthelot recently proved that the cohomology of the nearby cycles coincides with HyodoKato cohomology (log crystalline cohomology). Combining his result, we obtain a new construction of a weight spectral sequence, which was constructed by A. Mokrane using de RhamWitt complex. 
Dernière mise à jour le 5082009