abstract duality: opposite category,
concrete duality: dual object, dualizable object, fully dualizable object, dualizing object
between higher geometry/higher algebra
Langlands duality, geometric Langlands duality, quantum geometric Langlands duality
Given a homomorphism $f$ of schemes, one says that it satisfies Grothendieck duality if the (derived) direct image functor $f_\ast$ on quasicoherent sheaves has a (derived) right adjoint $f^!$. This is Verdier duality in a “Grothendieck context” of six operations.
Suppose $f\colon X \to Y$ is a quasi-compact and quasi-separated morphism of schemes; then the triangulated functor $\mathbf{R}f_*\colon D_{qc}(X)\to D(Y)$ has a bounded below right adjoint. In other words, $\mathbf{R}Hom_X(\mathcal{F}, f^\times \mathcal{G})\stackrel{\sim}{\to} \mathbf{R}Hom_Y(\mathbf{R}f_*\mathcal{F}, \mathcal{G})$ is a natural isomorphism.
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