  We describe the projectives in the category of functors from a graded poset
to abelian groups. Based on this description we define a related condition,
pseudo-projectivity, and we prove that this condition is enough for the
vanishing of the derived direct limits. We apply this result to deduce a
generalized version of a theorem of Whitehead for the pushout. The dual results
for inverse limits are also considered.
  We present two methods to compute integral cohomology of posets, a local one
a and a global one. The local method is applicable as soon as the sub-posets
under each object possess certain structure. This is the case for simplicial
complexes, simplex-like posets and for Quillen's complex of a finite group. The
global method is related to Morse Theory.
