By Gebhard Bockle and Richard Pink

ISBN-10: 3037190744

ISBN-13: 9783037190746

This booklet develops a brand new cohomological thought for schemes in confident attribute $p$ and it applies this concept to offer a only 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 kinfolk of Drinfeld $A$-modules or, extra quite often, of $A$-motives on a number of finite style over the finite box $\mathbb{F}_p$. by way of 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 through analytic tools a los angeles Dwork. the current textual content introduces $A$-crystals, which might be considered as generalizations of households of $A$-motives, and experiences 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 valuable result's a Lefschetz hint formulation for $L$-functions of $A$-crystals, from which the rationality of those $L$-functions is fast. past its program to Goss's $L$-functions, the idea of $A$-crystals is heavily relating to the paintings of Emerton and Kisin on unit root $F$-crystals, and it truly is crucial in an Eichler - Shimura sort isomorphism for Drinfeld modular types as developed by means of 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 types in optimistic attribute. It assumes a great operating wisdom in algebraic geometry in addition to familiarity with homological algebra and derived different types, as supplied by way of usual textbooks. past that the presentation is basically self contained

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**Extra resources for Cohomological Theory of Crystals over Function Fields**

**Sample text**

A proof under much weaker assumptions can be found in [38], where the case D ¿ was left out but can be handled in the same fashion. 4 x ! 8 to give a short proof, as we can use the splitting s W A follows. First we observe that the fact that ÃBqi is essentially locally small follows once we have proved that the functor induces a bijection on morphisms. A/, we may already use that this localization exists in the weaker sense where 1 morphisms form only classes instead of sets. A/ in one step by localizing at the multiplicative system of morphisms whose cones have cohomology in B.

N / are r-acyclic; hence E2ij D 0 whenever i 6D 0. N // for every j . N /. N / ,! , a quasi-isomorphism, as desired. N // ! A/; hence i Rr Š id, as desired. Chapter 3 Fundamental concepts In this chapter we introduce the basic objects that are being studied in this book. The main concept is that of an A-crystal, which is a coherent sheaf together with a semilinear endomorphism, viewed modulo a certain equivalence relation. This equivalence relation annihilates precisely those objects whose semi-linear endomorphism is nilpotent.

Thus the direct limit exists by the definition of Grothendieck category. y . By the exactness of filtered direct Next consider the natural morphism M ! M limits its kernel and cokernel are the direct limits of the kernels and cokernels of the morphisms M ) M 0 . By assumption these lie in B, and B is closed under filtered y lie in B. Thus this morphism direct limits; hence the kernel and cokernel of M ! M is an element of Ã. In particular, this morphism is itself part of the system of all M ) M 0 ; so it alone forms a cofinal subsystem.