nLab
local category

Local categories

Idea and warning

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 CC 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 CC which is Abelian (that is the ring of endomorphisms of the identity functor of CC) is a commutative local ring.

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

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

Let RR be a unital ring, and RMod{}_R Mod the category of left RR-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)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 TT of any Grothendieck category AA is a prime localizing subcategory if the Serre quotient category A/TA/T is local Grothendieck category. Prime localizing subcategories form certain spectrum.

Literature

  • (RosenbergSpectraNSp) A. L. Rosenberg, Spectra of noncommutative spaces, MPIM2003-110 ps dvi (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
category: algebra

Last revised on March 9, 2014 at 10:31:05. See the history of this page for a list of all contributions to it.