Spectrum of an abelian category via premonoform objects

Abstract

Let $\cA$ be an abelian category. In this paper, we study ${\rm (n)PSpec}\cA$, a topological space formed by equivalence classes derived from an equivalence relation on (noetherian) premonoform objects. We classify torsion classes of $\cA$ via closed subclasses of $\nPSpec\cA$. We introduce a new topology on $\PSpec\cA$ and we classify Serre subcategories of $\noeth A$ and localizing subcategories of $\cA$ using this topology. If $A$ is a commutative noetherian ring, we show that $\nPSpec A$ is homeomorphic to $\Spec A$. Moreover, there is a one-to-one correspondence between the closed subsets of $\nPSpec A$ and the open subsets of $\ASpec A$, the atom spectrum of $A$. Finally, we explore the relationships between the new subctegories of $\Mod A$ and subsets of $\nPSpec A$ introduced in this paper, and the known subcategories of $\Mod A$ and subsets of other spectra of $A$.