 
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, secondcountable, locally pathconnected, and semilocally simply connected.
Unless otherwise specified, they will also be pathconnected.
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 uncountablerank free Zmodule for which we cannot give an
explicit basis.
Henceforth, all Homgroups, 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 FU is a constant sheaf.
Example 3. A constant sheaf is locally constant.
Example 4. The squareroot sheaf Q on C {0} is locally constant, but not constant.
