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HEREDITARY TORSION THEORIES OF A LOCALLY NOETHERIAN GROTHENDIECK CATEGORY

Part of: Abelian categories

Published online by Cambridge University Press:  26 September 2016

KAIVAN AHMADI  and
REZA SAZEEDEH
KAIVAN AHMADI
Affiliation:
Department of Mathematics, Urmia University, PO Box 165, Urmia, Iran email K.ahmadi@urmia.ac.ir
REZA SAZEEDEH*
Affiliation:
Department of Mathematics, Urmia University, PO Box 165, Urmia, Iran email rsazeedeh@ipm.ir
*
✉rsazeedeh@ipm.ir
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Abstract

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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.

Keywords

atom spectrumGrothendieck categorylocalising subcategory

MSC classification

Primary: 18E15: Grothendieck categories
Secondary: 18E40: Torsion theories, radicals
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 94 , Issue 3 , December 2016 , pp. 421 - 430
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

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