Verdier duality states that certain image functors for sheaves are actually adjoint functors. There are two versions. Global Verdier duality states that for a continuous map, the derived functor of the direct image with proper supports has a right adjoint in the derived category of sheaves, in other words, for a sheaf on and on we have The exclamation mark is often pronounced "shriek", and the maps called " shriek" or "' lower shriek" and "f upper shriek" – see alsoshriek map. Local Verdier duality states that in the derived category of sheaves of k modules over Y. It is important to note that the distinction between the global and local versions is that the former relates maps between sheaves, whereas the latter relates sheaves directly and so can be evaluated locally. Taking global sections of both sides in the local statement gives global Verdier duality. The dualizing complex on is defined to be where p is the map from to a point. Part of what makes Verdier duality interesting in the singular setting is that when is not a manifold then the dualizing complex is not quasi-isomorphic to a sheaf concentrated in a single degree. From this perspective the derived category is necessary in the study of singular spaces. If is a finite-dimensional locally compact space, and the bounded derived category of sheaves of abelian groups over, then the Verdier dual''' is a contravariant functor defined by It has the following properties:
Poincaré duality
can be derived as a special case of Verdier duality. Here one explicitly calculates cohomology of a space using the machinery of sheaf cohomology. Suppose X is a compact orientable n-dimensional manifold, k is a field and kX 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 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 kX replaced with the same sheaf placed in degree i we get the classical Poincaré duality
