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.

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 ${\displaystyle f:X\to Y}$, 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 ${\displaystyle {\mathcal {F}}}$ on X and ${\displaystyle {\mathcal {G}}}$ on Y we have

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