  In this paper we study homological properties of modules over an affine Hecke
algebra H. In particular we prove a comparison result for higher extensions of
tempered modules when passing to the Schwartz algebra S, a certain topological
completion of the affine Hecke algebra. The proof is self-contained and based
on a direct construction of a bounded contraction of certain standard
resolutions of H-modules.
  This construction applies for all positive parameters of the affine Hecke
algebra. This is an important feature since it is an ingredient to analyse how
the irreducible discrete series representations of H arise in generic families
over the parameter space of H. For irreducible non-simply laced affine Hecke
algebras this will enable us to give a complete classification of the discrete
series characters for all positive parameters (we will report on this
application in a separate article).
