HEREDITARY TORSION THEORIES OF A LOCALLY NOETHERIAN GROTHENDIECK CATEGORY

Abstract

Let${\mathcal{A}}$be a locally noetherian Grothendieck category. We construct closure operators on the lattice of subcategories of${\mathcal{A}}$and the lattice of subsets of$\text{ASpec}\,{\mathcal{A}}$in terms of associated atoms. This establishes a one-to-one correspondence between hereditary torsion theories of${\mathcal{A}}$and closed subsets of$\text{ASpec}\,{\mathcal{A}}$. If${\mathcal{A}}$is locally stable, then the hereditary torsion theories can be studied locally. In this case, we show that the topological space$\text{ASpec}\,{\mathcal{A}}$is Alexandroff.