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Register a new userOdd-dimensional cohomology with finite coefficients and roots of unity
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Authors
Yuri G. Zarhin
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We prove that the triviality of the Galois action on the suitably twisted odd-dimensional etale cohomogy group of a smooth projective varietiy with finite coefficients implies the existence of certain primitive roots of unity in the field of definition of the variety. This text was inspired by an exercise in Serres Lectures on the Mordell--Weil theorem.
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We prove various finiteness and representability results for flat cohomology of finite flat abelian group schemes. In particular, we show that if $f:Xrightarrow mathrm{Spec} (k)$ is a projective scheme over a field $k$ and $G$ is a finite flat abelian group scheme over $X$ then $R^if_*G$ is an algebraic space for all $i$. More generally, we study the derived pushforwards $R^if_*G$ for $f:Xrightarrow S$ a projective morphism and $G$ a finite flat abelian group scheme over $X$. We also develop a theory of compactly supported cohomology for finite flat abelian group schemes, describe cohomology in terms of the cotangent complex for group schemes of height $1$, and relate the Dieudonne modules of the group schemes $R^if_*mu _p$ to cohomology generalizing work of Illusie. A higher categorical version of our main representability results is also presented.
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At roots of unity the $N$-state integrable chiral Potts model and the six-vertex model descend from each other with the $tau_2$ model as the intermediate. We shall discuss how different gauge choices in the six-vertex model lead to two different quantum group constructions with different $q$-Pochhammer symbols, one construction only working well for $N$ odd, the other equally well for all $N$. We also address the generalization based on the sl$(m,n)$ vertex model.
The goal of this paper is to calculate the trace of the composition of a Hecke correspondence and a (high enough) power of the Frobenius at a good place on the intersection cohomology of the Satake-Baily-Borel compactification of certain Shimura varieties, to stabilize the result for Shimura varieties associated to unitary groups over $mathbb{Q}$ and to give applications of this calculations using base change from these unitary groups to $GL_n$. ----- Le but de ce texte est de calculer la trace dune correspondance de Hecke composee avec une puissance (assez grande) du Frobenius en une bonne place sur la cohomologie dintersection de la compactification de Satake-Baily-Borel de certaines varietes de Shimura, de stabiliser le resultat obtenu pour les varietes de Shimura associees aux groupes unitaires sur $mathbb{Q}$, et de donner des applications de ces calculs en utilisant le changement de base de ces groupes unitaires a $GL_n$.
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