  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.
