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

By Gebhard Bockle and Richard Pink

This publication develops a brand new cohomological thought for schemes in optimistic attribute $p$ and it applies this conception to provide a merely algebraic evidence of a conjecture of Goss at the rationality of convinced $L$-functions bobbing up within the mathematics of functionality fields. those $L$-functions are strength sequence over a undeniable ring $A$, linked to any family members of Drinfeld $A$-modules or, extra ordinarily, of $A$-motives on quite a few finite sort over the finite box $\mathbb{F}_p$. by way of analogy to the Weil conjecture, Goss conjectured that those $L$-functions are in reality rational capabilities. In 1996 Taguchi and Wan gave a primary evidence of Goss's conjecture by way of analytic equipment 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 when it comes to coherent sheaves including a Frobenius map, in lots of methods they really behave like constructible etale sheaves. A primary 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 on the topic of the paintings of Emerton and Kisin on unit root $F$-crystals, and it really is crucial in an Eichler - Shimura style isomorphism for Drinfeld modular types as developed by way of 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 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 through usual textbooks. past that the presentation is essentially self contained

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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 . 1 is equivalent to its dual version obtained by inverting all arrows and asserting the existence of a right derived functor of a left exact functor. 2. B/ exist. Let F W A ! B be a right exact functor. A0 / and a quasi-isomorphism M0 ) M . A0 / to quasi-isomorphisms. A/ ! A0 / coincides with F . 1. For simplicity we drop the upper index .

F 00 ! 0 where F 00 is coherent and F 0 is ind-coherent. Then there exists a coherent -sheaf G F which maps epimorphically to F 00 . Proof. Assume first that X is affine, and choose global sections m1 ; : : : ; mr of F whose images generate F 00 as a coherent exist global sections ˛i;j P sheaf. mi / j ˛i;j mj lie in F . By assumption there exists a coherent -subsheaf G 0 F 0 containing all these sections. By construction the coherent subsheaf G F generated by G 0 together with all sections mi underlies a coherent -subsheaf G F .

4, and consider a morphism f W M ! N in A. (a) x is zero if and only if M 2 B. f / is zero if and only if im f 2 B. f / is a monomorphism if and only if ker f 2 B. f / is an epimorphism if and only if coker f 2 B. f / is an isomorphism if and only if both ker f , coker f 2 B. f / is an isomorphism, then f can be factored as f D gh where h is an epimorphism with kernel in B and h is a monomorphism with cokernel in B. x is isomorphic to the image of a short exact (c) Every short exact sequence in A sequence in A.

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