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# The Situation over a Finite Field

Let us now turn our attention to the case of a finite field k, and a groupscheme G/k which is a form of Gm. Concretely, it is either Gm/k itself, or it is the nonsplit form, defined in terms of the unique quadratic extension k2/k inside the chosen k as follows: for any k-algebra A,

G(A) := {x ∈ A ⊗k k2| NormA⊗kk2/A(x) = 1}.

We denote by Parith the full subcategory of the category Pervarith of all perverse sheaves on G/k consisting of those perverse sheaves on G/k which, pulled back to G/k, lie in P. And we denote by Negarith the full subcategory of Pervarith consisting of those objects which, pulled back to G/k, lie in Neg. Then once again we have

ParithPervarith/Negarith,

which endows Parith with ...

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