Rigid gauges and F-zips, and the fundamental sheaf of gauges G_1tnn [Elektronische Ressource] / vorgelegt von Felix Schnellinger

Rigid gauges and F-zips, and thefundamental sheaf of gauges GnDissertation zur Erlangung des Doktorgrades der Naturwissenschaften(Dr. rer. nat.) der mathematischen Fakultat¨ der Universit¨at Regensburgvorgelegt vonFelix Schnellinger aus Regensburg2009Promotionsgesuch eingereicht am: 30.01.2009Die Arbeit wurde angeleitet von Prof. Dr. Uwe JannsenPrufungsaussc¨ huss:Prof. Dr. Felix Finster (Vorsitzender)Prof. Dr. Uwe Jannsen (1. Gutachter)Prof. Dr. Torsten Wedhorn, Universit¨at Paderborn (2. Gutachter)Prof. Dr. Klaus Kunne¨ mannProf. Dr. Guido Kings (Ersatzprufer)¨IntroductionIn this paper we study D−ϕ-gauges introduced by J.M. Fontaine and U. Jannsen, theauthor’s advisor. These are objects of (Frobenius)-linear algebra over a perfect field kof positive characteristic. Fontaine and Jannsen define invariants for smooth projectivevarieties over k, which take values in gauges, by means of e.g. syntomic cohomology of a· crissheaf G built essentially of the sheavesO .n nIn the first section we study D −ϕ-gauges over a perfect field k. A D −ϕ−gauge1 1over k is a graded module M of finite type over the graded ring k[f,v]/(fv) (with fin degree 1 and v in degree−1) together with an Frobenius-semi-linear isomorphism∼∞ −∞ϕ :M →M . Fontaine defined the subcategory of rigid gauges to be those D −ϕ-1gauges with imv = kerf, imf = kerv and ker(f,v) = 0.WestudythestructureandmorphismsofrigidD−ϕ-gauges.