By Gebhard Bockle and Richard Pink

ISBN-10: 3037190744

ISBN-13: 9783037190746

This e-book develops a brand new cohomological conception for schemes in optimistic attribute $p$ and it applies this concept to offer a in simple terms algebraic facts of a conjecture of Goss at the rationality of definite $L$-functions coming up within the mathematics of functionality fields. those $L$-functions are strength sequence over a definite ring $A$, linked to any relations of Drinfeld $A$-modules or, extra in most cases, of $A$-motives on a number of finite variety over the finite box $\mathbb{F}_p$. by means 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 by way of analytic tools a los angeles Dwork. the current textual content introduces $A$-crystals, which are seen as generalizations of households of $A$-motives, and reviews their cohomology. whereas $A$-crystals are outlined when it comes to coherent sheaves including a Frobenius map, in lots of methods they really behave like constructible etale sheaves. A critical result's a Lefschetz hint formulation for $L$-functions of $A$-crystals, from which the rationality of those $L$-functions is instant. past its software 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 kind isomorphism for Drinfeld modular kinds as developed via the 1st writer. The e-book is meant for researchers and complex graduate scholars drawn to the mathematics of functionality fields and/or cohomology theories for types in confident attribute. It assumes an excellent operating wisdom in algebraic geometry in addition to familiarity with homological algebra and derived different types, as supplied by means of commonplace textbooks. past that the presentation is essentially self contained.

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**Additional info for Cohomological Theory of Crystals over Function Fields (Ems Tracts in Mathematics)**

**Example text**

M 0 / D 0 and M 0 a subobject of I form another cofinal subsystem. 3 the isomorphism classes of these M 0 form a set; hence the whole system possesses a cofinal subsystem which is indexed by a set. 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.

C. Then GF is again left exact. GF / exist. GF / D RG B RF if and only if F maps all injectives in A to G-acyclic objects in B. 3] for the details. FM / for all i . M // for all i. All the above facts have direct analogues for left derived functors and projective resolutions. The following result relates derived functors with localization. 1. 2 consider another abelian category x ! C. Assume that the composite functor F ´ C and a left exact functor Fx W A Fxq W A ! A/ ! C/ which factors x ! C/.

L0 / is called a morphism of triangles. A triangulated category is an additive category endowed with a translation functor and a collection of triangles, called distinguished triangles, satisfying a number of axioms. We will almost never use these axioms directly; thus for their formulation and explanation and for more background on triangulated categories we refer the reader to [20], [27], [46], and [47]. An exact functor between two triangulated categories is an additive functor that commutes with translation and preserves distinguished triangles.

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