Derived functor

introduction

• basic tool to define cohomology theory
• extend a left invariant functor to get a derived functor
• then we get a cohomology theory
• e.g. sheaf cohomology of a topological space X with coefficients in a sheaf $\mathcal F$ = the right derived functor of the global section functor

left invariant functors

global section functor

• a functor from sheaves on $X$ to abelian groups defined by

$$ \mathcal F \mapsto H^{0}(X, \mathcal F) $$

invariants

• $G$ : group
• from modules of $G$ to abelian groups

$$ M\mapsto M^{G} $$