Torsion classes of finite type and spectra

Garkusha, Grigory and Prest, Mike (2006) Torsion classes of finite type and spectra. [MIMS Preprint]

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Abstract

Given a commutative ring $R$ (respectively a positively graded commutative ring $A=\ps_{j\geq 0}A_j$ which is finitely generated as an $A_0$-algebra), a bijection between the torsion classes of finite type in $\Rfp$ (respectively tensor torsion classes of finite type in $\QGr A$) and the set of all subsets $Y\subseteq\spec R$ (respectively $Y\subseteq\Proj A$) of the form $Y=\bigcup_{i\in\Omega}Y_i$, with $\spec R\setminus Y_i$ (respectively $\Proj A\setminus Y_i$) quasi-compact and open for all $i\in\Omega$, is established. Using these bijections, there are constructed isomorphisms of ringed spaces $$(\spec R,\cc O_{R})\lra{\sim}(\spec(\Rfp),\cc O_{\Rfp})$$ and $$(\Proj A,\cc O_{\Proj A})\lra{\sim}(\spec(\QGr A),\cc O_{\QGr A}),$$ where $(\spec(\Rfp),\cc O_{\Rfp})$ and $(\spec(\QGr A),\cc O_{\QGr A})$ are ringed spaces associated to the lattices $L_{\serre}(\Rfp)$ and $L_{\serre}(\QGr A)$ of torsion classes of finite type. Also, a bijective correspondence between the thick subcategories of perfect complexes $\perf(R)$ and the torsion classes of finite type in $\Rfp$ is established.

Item Type: MIMS Preprint
Uncontrolled Keywords: affine and projective schemes, torsion classes of finite type, thick subcategories
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 03 Mathematical logic and foundations
MSC 2010, the AMS's Mathematics Subject Classification > 16 Associative rings and algebras
MSC 2010, the AMS's Mathematics Subject Classification > 18 Category theory; homological algebra
Depositing User: Professor Mike Prest
Date Deposited: 18 Dec 2006
Last Modified: 08 Nov 2017 18:18
URI: http://eprints.maths.manchester.ac.uk/id/eprint/668

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