 
Summary: Applications of Homological Algebra Introduction to Perverse Sheaves
Spring 2007 P. Achar
Derived Functors in Categories of Sheaves
March 8, 2007
In the philosophy expounded by Grothendieck, there are six important operations on sheaves, occuring in
three adjoint pairs:
(
L
, R Hom), (f1
, Rf), (Rf!, f!
).
All other operations can be built out of these. (For example, R is the same as Ra, where a : X {pt} is
the constant map to a point, and R Hom = R R Hom.) In this set of notes, we will review the definitions
of all of these except f!
, list adapted classes, and collect various composition and adjointness theorems.
Convention. All statements below are correct for sheaves of abelian groups. However, some of them contain
conditions that become superfluous or trivial in the category of sheaves of vector spaces. Specifically, all
sheaves of vector spaces are flat, so is already an exact functor. Nevertheless, we will use the usual notation
L
for the functor induced by on the derived category.
