Applications of Homological Algebra Introduction to Perverse Sheaves Spring 2007 P. Achar Summary: Applications of Homological Algebra Introduction to Perverse Sheaves Spring 2007 P. Achar Operations on Sheaves; Adjointness Theorems February 1, 2007 Definition 1. Let f : X Y be a continuous map of topological spaces. If F is a sheaf on X, the push-forward by f of F, also called its direct image, and denoted fF, is the sheaf on Y defined by (fF)(U) = F(f-1 (U)). Remark 2. Note that specifying a sheaf on a one-point topological space is the same as specifying a single abelian group. One-point spaces occur in several examples below, and we will frequently treat abelian groups as sheaves on these spaces without further comment. Example 3. Let f : X {x} be the constant map to a one-point space. Then fF (X, F) for any sheaf F on X. Example 4. Let x0 X, and let i : {x0} X be the inclusion of the point. Let A be an abelian group, thought of as a sheaf on {x0}. Then iA is sheaf on X whose stalks are all 0 except at x0, where its stalk is isomorphic to A. This kind of sheaf is called a skyscraper sheaf. Definition 5. Let F and G be sheaves on X. Their direct sum is the sheaf F G defined by (F G)(U) = F(U) G(U) for all open sets U X. Example 6. Let X = C {0}, and let f : X X be the map f(z) = z2 . Then fC C Q, where Q is Collections: Mathematics