Local cohomology in Grothendieck categories

Abstract

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 $\mathcal{D}^+(\mathcal{A})$.