Verdier duality

In mathematics, Verdier duality is a duality in sheaf theory that generalizes Poincaré duality for manifolds. Verdier duality was introduced by Verdier (1967, 1995) as an analog for locally compact spaces of the coherent duality for schemes due to Grothendieck. It is commonly encountered when studying constructible or perverse sheaves. In mathematics, Verdier duality is a duality in sheaf theory that generalizes Poincaré duality for manifolds. Verdier duality was introduced by Verdier (1967, 1995) as an analog for locally compact spaces of the coherent duality for schemes due to Grothendieck. It is commonly encountered when studying constructible or perverse sheaves. 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 f : X → Y {displaystyle f:X o Y} , the derived functor of the direct image with proper supports Rf! has a right adjoint f! in the derived category of sheaves, in other words, for a sheaf F {displaystyle {mathcal {F}}} on X and G {displaystyle {mathcal {G}}} on Y we have The exclamation mark is often pronounced 'shriek' (slang for exclamation mark), and the maps called 'f shriek' or 'f lower shriek' and 'f upper shriek' – see also shriek 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 (complexes of) 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 DX on X is defined to be where p is the map from X to a point. Part of what makes Verdier duality interesting in the singular setting is that when X is not a manifold (a graph or singular algebraic variety for example) 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 X is a finite-dimensional locally compact space, and Db(X) the bounded derived category of sheaves of abelian groups over X, then the Verdier dual is a contravariant functor