# Local cohomology in Grothendieck categories

## TL;DR

This paper develops a local cohomology theory within Grothendieck categories by defining section functors relative to open subsets of the spectrum, and explores their properties in the derived category setting.

## Contribution

It introduces a new framework for local cohomology in Grothendieck categories using section functors and extends the theory to the derived category context.

## Key findings

- Section functor defined for open subsets of ASpec\mathcal{A}
- Local cohomology theory formulated in the category\mathcal{A}
- Abstract local cohomology functor studied on the derived category

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