Here we study the notion of local category and a different notion of local Grothendieck category. Local Grothendieck category is not simply Grothendieck category which is local in above sense, but has a different definition. However, both notions have the category of modules over a local commutative unital ring as the main example.

Definitions

(RosenbergSpectraNSp 1.1) A category $C$ is local if the full subcategory generated by all objects which are not initial, has itself an initial object. In particular, every local category has initial objects.

The center of a local category $C$ which is Abelian (that is the ring of endomorphisms of the identity functor of $C$) is a commutative local ring.

(RosenbergSpectraNSp 1.2) The local spectrum of an arbitrary small category $C$ is the full subcategory $\mathfrak{Spec}^1(C)\hookrightarrow C$ whose objects are all $x$ in $Ob C$ such that the undercategory$x\backslash C$ is local.

(Popescu 4.20) A local Grothendieck category is a Grothendieck category (= Ab5-category with a generator) $A$ having a simple object $S$ whose injective envelope$E(S)$ is a cogenerator of $A$.

Let $R$ be a unital ring, and ${}_R Mod$ the category of left $R$-modules. Following (Goldman 1969), define a left prime (left prime system) a Gabriel filter such that the corresponding Gabriel localization is a local Grothendieck category. All left primes form another spectrum, $Spel(R)$. Regarding that Gabriel filters of left ideals correspond to localizing subcategories of the category of left modules, it is obvious how to generalize this. Following Popescu, a localizing subcategory$T$ of any Grothendieck category $A$ is a prime localizing subcategory if the Serre quotient category $A/T$ is local Grothendieck category. Prime localizing subcategories form certain spectrum.

Literature

(RosenbergSpectraNSp) A. L. Rosenberg, Spectra of noncommutative spaces, MPIM2003-110 psdvi (2003)

N. Popescu, Abelian categories with applications to rings and modules, London Math. Soc. Monographs 3, Academic Press 1973. xii+467 pp. MR0340375

N. Popescu, Le spectre a gauche d’un anneau, J. Algebra 18 (1971), 213-228.

(Goldman 1969) O. Goldman, Rings and modules of quotients, J. Algebra 13, 1969 10–47, MR245608, doi