Suppose "X" is a compact orientable "n"-dimensional manifold, "k" is a field and "k"X is the constant sheaf on "X" with coefficients in "k". Let "f=p" be the constant map. Global Verdier duality then states To understand how Poincaré duality is obtained from this statement, it is perhaps easiest to understand both sides piece by piece. Let  be an injective resolution of the constant sheaf. Then by standard facts on right derived functors  is a complex whose cohomology is the compactly supported cohomology of "X". Since morphisms between complexes of sheaves (or vector spaces) themselves form a complex we find that where the last non-zero term is in degree 0 and the ones to the left are in negative degree. Morphisms in the derived category are obtained from the homotopy category of chain complexes of sheaves by taking the zeroth cohomology of the complex, i.e. For the other side of the Verdier duality statement above, we have to take for granted the fact that when "X" is a compact orientable "n"-dimensional manifold  which is the dualizing complex for a manifold. Now we can re-express the right hand side as We finally have obtained the statement that By repeating this argument with the sheaf "k"X replaced with the same sheaf placed in degree "i" we get the classical Poincaré duality