Let
G
G
be the group of rational points of a split connected reductive group over a nonarchimedean local field of residue characteristic
p
p
. Let
I
I
be a pro-
p
p
Iwahori subgroup of
G
G
and let
R
R
be a commutative quasi-Frobenius ring. If
H
=
R
[
I
∖
G
/
I
]
H=R[I\backslash G/I]
denotes the pro-
p
p
Iwahori-Hecke algebra of
G
G
over
R
R
we clarify the relation between the category of
H
H
-modules and the category of
G
G
-equivariant coefficient systems on the semisimple Bruhat-Tits building of
G
G
. If
R
R
is a field of characteristic zero this yields alternative proofs of the exactness of the Schneider-Stuhler resolution and of the Zelevinski conjecture for smooth
G
G
-representations generated by their
I
I
-invariants. In general, it gives a description of the derived category of
H
H
-modules in terms of smooth
G
G
-representations and yields a functor to generalized
(
φ
,
Γ
)
(\varphi ,\Gamma )
-modules extending the constructions of Colmez, Schneider and Vignéras.