Let F be a p-adic field with residue class field k. We investigate the
structure of certain mod p universal modules for GL(3,F) over the corresponding
Hecke algebras. To this end, we first study the structure of some mod p
universal modules for the finite group GL(n,k) as modules over the
corresponding Hecke algebras. We then relate this finite case to the p-adic one
by using homological coefficient systems on the the affine Bruhat-Tits building
of GL(3). Suppose now that k has cardinality p. We prove that the mod p
universal module of GL(3,F) relative to the Iwahori subroup is flat and
projective over the Iwahori-Hecke algebra. When replacing the Iwahori subgroup
of GL(3,F) by its pro-p-radical, we prove that the corresponding module is flat
over the pro-p Iwahori-Hecke algebra if and only if p=2