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 Local Systems and Constructible Sheaves February 1, 2007 Convention. Henceforth, all sheaves will be sheaves of complex vector spaces. All topological spaces will be locally compact, Hausdorff, second-countable, locally path-connected, and semilocally simply connected. Unless otherwise specified, they will also be path-connected. Remark 1. If F and G are sheaves of complex vector spaces, objects like Hom(F, G) and F G depend on whether one is working in the category of sheaves of abelian groups, or in the category of sheaves of vector spaces. Indeed, the same phenomenon is already visible with ordinary Hom and : in the category of complex vector spaces, we have C C C, while in the category of real vector spaces, C C R4 . In the category of abelian groups, C C is an uncountable-rank free Z-module for which we cannot give an explicit basis. Henceforth, all Hom-groups, sheaf Hom's, and tensor products are to be computed in the category of sheaves of complex vector spaces. Definition 2. A sheaf F on X is locally constant, or F is a local system, if for all x X, there is a neighborhood U containing x such that F|U is a constant sheaf. Example 3. A constant sheaf is locally constant. Example 4. The square-root sheaf Q on C {0} is locally constant, but not constant. Collections: Mathematics