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 Basic Facts on Sheaves Definition 1. A sheaf of abelian groups F on a topological space X is the following collection of data: · for each open set U X, an abelian group F(U), with F() = 0 · if U V , a restriction map V U : F(V ) F(U), with UU = id such that (1) (restriction) if U V W, then W U = V U W V (2) (gluing) if {Vi} is an open covering of U, and we have si F(Vi) such that for all i, j, Vi,ViVj (si) = Vj ,ViVj (sj), then there exists a unique s F(U) such that U,Vi (s) = si for all i. If one omits the gluing condition from the above definition, one has a presheaf of abelian groups. Elements of F(U) are called sections of F over U, and elements of F(X) are called global sections. One can also define (pre)sheaves of R-modules, vector spaces, etc. Notation 2. (U, F) := F(U). s|V := V U (s). Lemma 3. Let F be a sheaf. If {Vi} is an open cover of U, and s, t F(U) are sections such that s|Vi = t|Vi for all i, then s = t. In particular, if s|Vi = 0 for all i, then s = 0. Definition 4. Let F be a presheaf on X. The stalk of F at a point x X, denoted Fx, is the group whose elements are equivalence classes of pairs Collections: Mathematics