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HEREDITARY TORSION THEORIES OF A LOCALLY NOETHERIAN GROTHENDIECK CATEGORY
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 94 / Issue 3 / December 2016
- Published online by Cambridge University Press:
- 26 September 2016, pp. 421-430
- Print publication:
- December 2016
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- Article
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- You have access
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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.
