Grothendieck duality

Grothendieck duality


Given a homomorphism ff of schemes, one says that it satisfies Grothendieck duality if the (derived) direct image functor f *f_\ast on quasicoherent sheaves has a (derived) right adjoint f !f^!. This is Verdier duality in a “Grothendieck context” of six operations.


Suppose f:XYf\colon X \to Y is a quasi-compact and quasi-separated morphism of schemes; then the triangulated functor Rf *:D qc(X)D(Y)\mathbf{R}f_*\colon D_{qc}(X)\to D(Y) has a bounded below right adjoint. In other words, RHom X(,f ×𝒢)RHom Y(Rf *,𝒢)\mathbf{R}Hom_X(\mathcal{F}, f^\times \mathcal{G})\stackrel{\sim}{\to} \mathbf{R}Hom_Y(\mathbf{R}f_*\mathcal{F}, \mathcal{G}) is a natural isomorphism.


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Revised on July 6, 2014 07:56:13 by Adeel Khan (