Download Cohomological Theory of Crystals over Function Fields by Gebhard Bockle and Richard Pink PDF

By Gebhard Bockle and Richard Pink

This ebook develops a brand new cohomological concept for schemes in optimistic attribute $p$ and it applies this conception to offer a basically algebraic facts of a conjecture of Goss at the rationality of sure $L$-functions bobbing up within the mathematics of functionality fields. those $L$-functions are energy sequence over a definite ring $A$, linked to any relatives of Drinfeld $A$-modules or, extra more often than not, of $A$-motives on numerous finite style over the finite box $\mathbb{F}_p$. via analogy to the Weil conjecture, Goss conjectured that those $L$-functions are in truth rational features. In 1996 Taguchi and Wan gave a primary evidence of Goss's conjecture via analytic equipment a los angeles Dwork. the current textual content introduces $A$-crystals, that are considered as generalizations of households of $A$-motives, and reviews their cohomology. whereas $A$-crystals are outlined by way of coherent sheaves including a Frobenius map, in lots of methods they really behave like constructible etale sheaves. A imperative result's a Lefschetz hint formulation for $L$-functions of $A$-crystals, from which the rationality of those $L$-functions is rapid. past its program to Goss's $L$-functions, the idea of $A$-crystals is heavily with regards to the paintings of Emerton and Kisin on unit root $F$-crystals, and it's crucial in an Eichler - Shimura kind isomorphism for Drinfeld modular types as developed via the 1st writer. The publication is meant for researchers and complicated graduate scholars drawn to the mathematics of functionality fields and/or cohomology theories for forms in optimistic attribute. It assumes a superb operating wisdom in algebraic geometry in addition to familiarity with homological algebra and derived different types, as supplied via usual textbooks. past that the presentation is essentially self contained

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Example text

All morphisms and fiber products, and all tensor products of modules and algebras, are taken over k, unless otherwise specified. By a (quasi)-coherentsheaf on a scheme X we will always mean a (quasi)-coherent sheaf of OX -modules. Any homomorphism of such sheaves is assumed OX -linear, and any tensor product of such sheaves is taken over OX . The Frobenius morphism on X over k, which acts on functions by x 7! x q , is denoted W X ! X . Throughout most of the monograph we fix a scheme C which is assumed to be a localization of a scheme of finite type over k.

Then F contains no non-zero coherent -subsheaf. 4. X; A/. X; A/. Clearly these are abelian A-linear categories, and all constructions like kernel, cokernel, etc. are the usual ones on the underlying quasi-coherent sheaves, with the respective added by functoriality. X; A/. 5. X; A/. 5. We permit ourselves this forward reference, because the careful reader can readily verify that those results depend neither on the lemma nor on any other intervening theory, and should in fact find it easy to simplify the relevant construction for the case at hand.

A/ and whose morphisms are homotopy classes of morphisms with the evident composition. It is again an additive category, because the homotopy class of a sum of morphisms depends only on the homotopy classes of the summands. It also possesses a natural structure of triangulated category. L Œ1/i D LiC1 and dLi Œ1 D dLiC1 . The cone of a morphism f W L ! f /i ´ M i ˚ LiC1 together with the morphisms M i ˚ LiC1 ! m; `/ 7! `//: The inclusion morphism and minus the projection morphism M i ,! M i ˚ LiC1 LiC1 define morphisms of complexes M !

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