Noncommutative supports, local cohomology and spectral sequences

TL;DR

This paper develops a framework for local cohomology in noncommutative algebraic geometry, introducing supports, primes, and spectral sequences to analyze derived functors in this setting.

Contribution

It introduces a new theory of supports and associated primes in noncommutative spaces, and constructs spectral sequences for local cohomology and related functors.

Findings

Established a theory of supports and primes in noncommutative categories

Derived spectral sequences for local cohomology and generalized functors

Applied the framework to noncommutative algebraic geometry settings

Abstract

The purpose of this paper is to study local cohomology in the noncommutative algebraic geometry framework of Artin and Zhang. The noncommutative spaces are obtained by base change of a Grothendieck category that is locally noetherian or strongly locally noetherian. Using what we call elementary objects and their injective hulls, we develop a theory of supports and associated primes in these categories. We apply our theory to study a general functorial setup that requires certain conditions on the injective hulls of elementary objects and gives us spectral sequences for derived functors associated to local cohomology objects, as well as generalized local cohomology and also generalized Nagata ideal transforms.