Y → X of locally compact spaces and a sheaf E on X, however, there is a pullback homomorphism on compactly supported cohomology. Also, for an open subset U of a locally compact space X and a sheaf E on X, there is a pushforward homomorphism known as extension by zero: Both homomorphisms occur in the long exact localization sequence for compactly supported cohomology, for a locally compact space X and a closed subset Y: Cup product For any sheaves A and B of abelian groups on a topological space X, there is a bilinear map, the cup product for all i and j. Here A⊗B denotes the tensor product over Z, but if A and B are sheaves of modules over some sheaf OX of commutative rings, then one can map further from Hi+j(X,A⊗ZB) to Hi+j(X,A⊗OXB). In particular, for a sheaf OX of commutative rings, the cup product makes the direct sum into a graded-commutative ring, meaning that for all u in Hi and v in Hj. Complexes of sheaves The definition of sheaf cohomology as a derived functor extends to define cohomology of a topological space X with coefficients in any complex E of sheaves: In particular, if the complex E is bounded below (the sheaf Ej is zero for j sufficiently negative), then E has an injective resolution I just as a single sheaf does. (By definition, I is a bounded below complex of injective sheaves with a chain map E → I that is a quasi-isomorphism.) Then the cohomology groups Hj(X,E) are defined as the cohomology of the complex of abelian groups The cohomology of a space with coefficients in a complex of sheaves was earlier called hypercohomology, but usually now just "cohomology". More generally, for any complex of sheaves E (not necessarily bounded below) on a space X, the cohomology group Hj(X,E) is defined as a group of morphisms in the derived category of sheaves on X: where ZX is the constant sheaf associated to the integers, and E[j] means the complex E shifted j steps to the left. Poincaré duality and generalizations A central result in topology is the Poincaré duality theorem: for a closed oriented connected topological manifold X of dimension n and a field k, the group Hn(X,k) is isomorphic to k, and the cup product is a perfect pairing for all integers j. That is, the resulting map from Hj(X,k) to