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The aim of this paper is—for any ideal a of a commutative noetherian ring A—to introduce and study a functor Γ= a (X, Y) of two complexes X and Y of A-modules. If X= A (the ringA considered as a complex concentrated in degree zero), then Γ= a (X, Y)= Γ= a (Y), the usual section functor with support in V (a) introduced by Grothendieck. When X= M and Y= N are A-modules then it turns out that the cohomology module H i (Γ= a (X, Y)), is naturally isomorphic to the generalized local cohomology module H i a (M, N), which was introduced by J. Herzog. Furthermore, it turns out that the functor Γ= a (−, A) is isomorphic to the duality functor D a (−) introduced by Hartshorne when A is a regular ring.