of E on X such that every point in X has an open neighborhood on which S is isomorphic to E, with E acting on itself by translation. For example, on a ringed space (X,OX), it follows that the Picard group of invertible sheaves on X is isomorphic to the sheaf cohomology group H1(X,OX*), where OX* is the sheaf of units in OX. Relative cohomology For a subset Y of a topological space X and a sheaf E of abelian groups on X, one can define relative cohomology groups: for integers j. Other names are the cohomology of X with support in Y, or (when Y is closed in X) local cohomology. A long exact sequence relates relative cohomology to sheaf cohomology in the usual sense: When Y is closed in X, cohomology with support in Y can be defined as the derived functors of the functor the group of sections of E that are supported on Y. There are several isomorphisms known as excision. For example, if X is a topological space with subspaces Y and U such that the closure of Y is contained in the interior of U, and E is a sheaf on X, then the restriction is an isomorphism. (So cohomology with support in a closed subset Y only depends on the behavior of the space X and the sheaf E near Y.) Also, if X is a paracompact Hausdorff space that is the union of closed subsets A and B, and E is a sheaf on X, then the restriction is an isomorphism. Cohomology with compact support Let X be a locally compact topological space. (In this article, a locally compact space is understood to be Hausdorff.) For a sheaf E of abelian groups on X, one can define cohomology with compact support Hcj(X,E). These groups are defined as the derived functors of the functor of compactly supported sections: There is a natural homomorphism Hcj(X,E) → Hj(X,E), which is an isomorphism for X compact. For a sheaf E on a locally compact space X, the compactly supported cohomology of X × R with coefficients in the pullback of E is a shift of the compactly supported cohomology of X: It follows, for example, that Hcj(Rn,Z) is isomorphic to Z if j = n and is zero otherwise. Compactly supported cohomology is not functorial with respect to arbitrary continuous maps. For a proper map f: