By Gebhard Bockle and Richard Pink

ISBN-10: 3037190744

ISBN-13: 9783037190746

This publication develops a brand new cohomological concept for schemes in confident attribute $p$ and it applies this conception to provide a in basic terms algebraic facts of a conjecture of Goss at the rationality of yes $L$-functions coming 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 normally, of $A$-motives on quite a few finite variety over the finite box $\mathbb{F}_p$. through analogy to the Weil conjecture, Goss conjectured that those $L$-functions are in truth rational capabilities. In 1996 Taguchi and Wan gave a primary facts of Goss's conjecture by means of analytic tools a los angeles Dwork. the current textual content introduces $A$-crystals, which might be seen 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 crucial 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 speculation of $A$-crystals is heavily with regards to the paintings of Emerton and Kisin on unit root $F$-crystals, and it really is crucial in an Eichler - Shimura sort isomorphism for Drinfeld modular types as built via the 1st writer. The publication is meant for researchers and complex graduate scholars attracted to the mathematics of functionality fields and/or cohomology theories for kinds in optimistic attribute. It assumes an excellent 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 basically self contained

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RF Ä. Similarly for every exact x ! A/ natural transformations Ä Fx ! GÄ and Ä Fxq ! GÄq D GqÄ, and composition with Ä and q yields bijections between natural transformations RF ! G and RF Ä ! GÄ and RF qÄ ! GqÄ. Using this one easily sees that RF satisfies the universal property for the right derived functor of Fx, as desired. 1. B/ exist. Let F W A ! B be a right exact functor. A/ ! A0 / to itself. A0 / and a quasi-isomorphism M 0 ) CM . v/ W F ! A/ ! B/. A0 / to quasi-isomorphisms. A/ ! A0 / coincides with F .

GÄ there exists a unique natural transformation Á W RF ! G such that D Á . If a right derived functor exists, it is unique up to unique isomorphism. A/ ! B/ as follows. First, for every object M 2 A there exists an infinite exact sequence 0 ! M ! I 0 ! I 1 ! with all I i injective, called an injective resolution of M . A/, one can construct a quasi-isomorphism M ! A/ with all I i injective. A/, turning M 7! A/ ! I / defines a right derived functor of F . Identify an object M 2 A with the complex having M in degree 0 and the zero object in all other degrees.

In the following a multiplicative system in a triangulated category will always be assumed compatible with the triangulation. 2. For any triangulated category C, there is a bijection between the class of saturated multiplicative systems Ã and the class of strictly full triangulated subcategories N of C which are closed under taking direct summands. Explicitly, given N one defines Ã as the set of those morphisms f W L ! M which are part of f a distinguished triangle L ! M ! N ! L/ such that N lies in N.

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

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