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 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), (f-1 , 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. Collections: Mathematics