ABSTRACT
Any noetherian scheme can be reconstructed uniquely up to isomorphism from the category of quasi-coherent sheaves on this scheme. This chapter studies some ‘noncommutative spaces’ of interest which are introduced as abelian categories thought as categories of quasi-coherent sheaves on ‘would-be spaces’. It also studies the geometrical structure of an abelian category in more detail than is strictly necessary to prove the main Gabriel theorem. The chapter contains some basic facts of spectral theory of abelian categories and studies Zariski closed subschemes and Zariski topology on the spectrum. It further studies the reduced and Zariski reduced subschemes, and establishes the stability of subschemes and reduced subschemes with respect to flat localizations. The chapter introduces the prime spectrum and Levitzki spectum of an abelian category which are naturally related (especially the Levitzki spectrum) with the Zariski topology on the spectrum. It defines a ringed space associated to any choice of a topology on the spectrum of an abelian category.
