In this appendix, we prove Theorem 3.1, i.e., we show that N !(N) := H0(A1/k, j0!N) is a fibre functor on the Tannakian category Pgeom of those perverse sheaves on Gm/k satisfying P, under middle convolution. Throughout this appendix, we work entirely over k, explicit mention of which we will omit. Thus we will write !(N) simply as H0(A1, j0!N). And when we wish to emphasize the roles of both 0 and ∞ in its definition, we will write it as !(N) := H0(P1, Rj∞?j0!N) = H0(P1, j0!Rj∞?N).
It will be convenient to define, for any object M ∈ Dbc(Gm, Qℓ),
!(M) := H?(P1, j0!Rj∞?M) := ⊕i∈ZHi(P1, j0!Rj∞?M).