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# Frobenius Tori and Weights; Getting Elements of Garith

In this chapter, we work on Gm/k. We consider an arithmetically semisimple object N ∈ Parith which is pure of weight zero. We assume it is of the form G[1], with G a middle extension sheaf. Thus for some open set j : UGm, we have G = j? F, for F := j?G a lisse sheaf on U which is pure of weight –1 and arithmetically semisimple, and having no geometric constituent isomorphic to (the restriction to Uk of) a Kummer sheaf. Recall that Deligne’s fibre functor is (for j0 : GmA1 the inclusion)

M !(N) := H0(A1k k, j0!N).

The action of Frobk on the restriction to <N>arith of this ...

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