“…For any perfect complex of Λ-sheafs F • on the étale site of X we have defined in [16] an L-function L(F • , T ) attached to F • . This is an element in the first K-group K 1 (Λ[ Aside from the result of Emerton and Kisin, a central ingredient for the proof is the recent work of Chinburg, Pappas, and Taylor [2,3] on the first K-group of p-adic group rings. In fact, the main strategy of the proof is to reduce the assertion first to the case Λ = Z p [G] for a finite group G and then use the results of Chinburg, Pappas, and Taylor to reduce it further to the case already treated by Emerton and Kisin.…”