nLab Gabriel spectrum

Given a Noetherian Grothendieck category $A$, Pierre Peter Gabriel considered the set of isomorphism classes of indecomposable injective objects, together with an appropriate topology on its set. This topological space is called the spectrum of indecomposable injectives or Gabriel spectrum (or Gabriel’s spectrum).

• Pierre Gabriel, Des Catégories Abéliennes, Bulletin de la Société Mathématique de France 90 (1962) 323-448, numdam

For the category of quasicoherent sheaves over quasicompact Noetherian scheme, Gabriel’s spectrum with appropriate construction of structure sheaf reconstructs the scheme.

A refinement is the spectrum of indecomposable pure injectives introduced by Ziegler and studied by Herzog, Krause, Prest and others, often in relation to model theory, called Ziegler spectrum. Herzog has generalized its construction to locally coherent Grothendieck categories.

Gabriel’s spectrum of the category of Abelian presheaves on the category of finitely presentable $R$-modules is related to Ziegler’s spectrum of the category of fp $R$-modules, see

• Ivo Herzog, Contravariant functors on the category of finitely presented modules, Israel J. Math. 167, 347–410 (2008) doi