Resolutions of tempered representations of reductive p-adic groups

Let G be a reductive group over a non-archimedean local field and let S(G) be its Schwartz algebra. We compare Ext-groups of tempered G-representations in several module categories: smooth G-representations, algebraic S(G)-modules, bornological S(G)-modules and an exact category of S(G)-modules on LF-spaces, which contains all admissible S(G)-modules. We simplify the proofs of known comparison theorems for these Ext-groups, due to Meyer and Schneider–Zink. Our method is based on the Bruhat–Tits building of G and on analytic properties of the Schneider–Stuhler resolutions.