In mathematics, Verdier duality is a cohomological duality in algebraic topology that generalizes Poincaré duality for manifolds. Verdier duality was introduced in 1965 by Jean-Louis Verdier (1965) as an analog for locally compact topological spaces of Alexander Grothendieck's theory of Poincaré duality in étale cohomology for schemes in algebraic geometry. It is thus (together with the said étale theory and for example Grothendieck's coherent duality) one instance of Grothendieck's six operations formalism.

Verdier duality generalises the classical Poincaré duality of manifolds in two directions: it applies to continuous maps from one space to another (reducing to the classical case for the unique map from a manifold to a one-point space), and it applies to spaces that fail to be manifolds due to the presence of singularities. It is commonly encountered when studying constructible or perverse sheaves.

Verdier duality states that (subject to suitable finiteness conditions discussed below) certain derived image functors for sheaves are actually adjoint functors. There are two versions.

Global Verdier duality states that for a continuous map ${\displaystyle f\colon X\to Y}$ of locally compact Hausdorff spaces, the derived functor of the direct image with compact (or proper) supports ${\displaystyle Rf_{!}}$ has a right adjoint ${\displaystyle f^{!}}$ in the derived category of sheaves, in other words, for (complexes of) sheaves (of abelian groups) ${\displaystyle {\mathcal {F}}}$ on ${\displaystyle X}$ and ${\displaystyle {\mathcal {G}}}$ on ${\displaystyle Y}$ we have

Local Verdier duality states that

in the derived category of sheaves on Y. It is important to note that the distinction between the global and local versions is that the former relates morphisms between complexes of sheaves in the derived categories, whereas the latter relates internal Hom-complexes and so can be evaluated locally. Taking global sections of both sides in the local statement gives the global Verdier duality.

These results hold subject to the compactly supported direct image functor ${\displaystyle f_{!}}$ having finite cohomological dimension. This is the case if there is a bound ${\displaystyle d\in \mathbf {N} }$ such that the compactly supported cohomology ${\displaystyle H_{c}^{r}(X_{y},\mathbf {Z} )}$ vanishes for all fibres ${\displaystyle X_{y}=f^{-1}(y)}$ (where ${\displaystyle y\in Y}$) and ${\displaystyle r>d}$. This holds if all the fibres ${\displaystyle X_{y}}$ are at most ${\displaystyle d}$-dimensional manifolds or more generally at most ${\displaystyle d}$-dimensional CW-complexes.

The discussion above is about derived categories of sheaves of abelian groups. It is instead possible to consider a ring ${\displaystyle A}$ and (derived categories of) sheaves of ${\displaystyle A}$-modules; the case above corresponds to ${\displaystyle A=\mathbf {Z} }$.