We let Z denote a certain chain of projective lines over a field k of characteristic p, and we let A denote the category of smooth (but not necessarily admissible) GL2(Qp)-representations on k-vector spaces (with some fixed central character). Our goal in this talk is to explain how the category A may be \localized" over the scheme Z, and to explain various applications of this localization. This is work-in-progress with Andrea Dotto and Toby Gee.June 4th. Aula 1 AD 100, Dipartimento di Matematica "Tullio-Levi Civita".