Atom Spectra of Grothendieck Categories

Specialeforsvar ved Beatrix Miranda Ginn Nielsen

Titel: Atom Spectra of Grothendieck Categories

Abstract: The thesis defines the atom spectrum, ASpec(A) of an abelian category A, as a collection of equivalence classes of monoform objects. The atom spectrum is given a topology, which in the case of a locally Noetherian Grothendieck category corresponds to the Ziegler spectrum, and in the case of RMod for a commutative Noetherian ring, R, corresponds to the Hochster dual of the Zariski topology on SpecR. Atoms as H. H. Storrer defines them and strongly uniform modules are also introduced, and it is shown that a module is monoform if and only if it is strongly uniform. It is also shown that H. H. Storrer’s atoms correspond to Ryo Kanda’s atoms when working with RMod, R a unital ring. Serre subcategories are defined in the thesis and it is shown that if A is a locally Noetherian Grothendieck category, then there is a one-to-one correspondence between the localising subcategories of A, the Serre subcategories of the full subcategory of Noetherian objects of A and the open subsets of ASpec(A). It is also shown that in the case of a locally Noetherian Grothendieck category A, there is a one-to-one correspondence between isomorphism classes of indecomposable injective objects and the atoms of A. When R is a commutative ring, it is shown that ASpec(RMod) is in one-to-one correspondence with {R/P, where P is a prime ideal of R}. In the case of an Artinian ring, R, it is shown that ASpec(RMod) is the equivalence classes of R/m_1, ..., R/m_n where m_1, ..., m_n are maximal ideals and {R/m_1, ..., R/m_n} is a maximal set of simple modules which are pairwise non-isomorphic

 Vejleder: Henrik G. Holm
Censor:    Henning Haahr Andersen, Aarhus Universitet