Translation of Dualité Dans La Cohomologie Des Espaces Localement Compacts

We propose to prove a theorem analogues to the duality theorem on the étale cohomology of schemes proved by Grothendieck ([SGA72a, SGA72b, SGA73] and [Ver67]). This theorem generalize the classical theorem of Poincaré duality on topological spaces and more generally homological and cohomological varieties ([Bor57] and [BM60]). We will utilize hypercohomology as well as derived categories of abelian categories [Ver96]. Let A be an abelian category, C(A) the category of complexes in A (differentials have degree +1, and morphisms have degree 0), and K(A) the category whose objects are the objects of C(A) and morphisms are homotopy classes of morphisms of C(A). A morphism in C(A) (resp. K(A)) is called a quasi-isomorphism if it induces isomorphisms on cohomology objects. A quasi-isomorphism is often called a resolution of its source or target depending on the context. The category D(A) (the derived category of A) is then obtained by formally inverting the quasiisomorphisms in K(A). Performing the same construction on the category C+(A) of complexes bounded below (the complex in which all objects of sufficiently small degrees vanish), we obtain the category D+(A), which is a full subcategory of D(A). An object in D(A) is isomorphic to an object in D+(A) if and only if all of its objects vanishes for degree sufficiently small. Denote by I+(A) the full subcategory K+(A) consisting of the complexes whose objects are injective in each degree. If each object in A is a sub-object of an injective object, the natural functor K+(A) → D+(A) induced an equivalence of I+(A) and D+(A). Dually we have categories K−(A), D−(A), and P −(A) (consisting of complexes of projective objects). Finally we denote Db(A) the full subcategory of D(A) consisting of complexes whose cohomology objects vanish for all but finitely many degrees. Let F : A → B be an additive functor between abelian categories. The total right derived functor of F: RF :D+(A) −→D+(B)