  We consider the category of smooth $W(k)[GL_n(F)]$-modules, where F is a
p-adic field and k is an algebraically closed field of characteristic l
different from p. We describe a factorization of this category into blocks, and
show that the center of each block is a reduced, finite type, l-torsion free
W(k)-algebra. Moreover, the k-points of the center of each block are in
bijection with the possible "supercuspidal supports" of the smooth
$k[GL_n(F)]$-modules that lie in the block. Finally, we describe a large,
explicit subalgebra of the center of each block and give a description of the
action of this subalgebra on the simple objects of the block, in terms of the
description of the classical "characteristic zero" Bernstein center.
