A Note on Arithmetic Cohomologies for Number Fields

Let F be a number field with discriminant ∆F . Denote its (normalized) absolute values by SF , and write SF = Sfin · ∪ S∞, where S∞ denotes the collection of all archimedean valuations. For simplicity, we use v (resp. σ) to denote elements in Sfin (resp. S∞). Denote by A = AF the ring of adeles of F , and Glr(A) the rank r general linear group over A, and write A := Afin ⊕A∞ and Glr(A) := Glr(A)fin ×Glr(A)∞ according to their finite and infinite parts. For any g = (gfin; g∞) = (gv; gσ) ∈ Glr(A), define the injective morphism i(g) := i(g∞) : F r → A∞ by (f) 7→ (gσ · f), F (g) := Im ( i(g) ) and