  Let $\mathcal{A}$ be a locally noetherian Grothendieck category. In this
paper we define and study the section functor on $\mathcal{A}$ with respect to
an open subset of ASpec$\mathcal{A}$. Next we define and study local cohomology
theory in $\mathcal{A}$ in terms of the section functors. Finally we study
abstract local cohomology functor on the derived category
$\mathcal{D}^+(\mathcal{A})$.
