# Torsion classes of finite type and spectra

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

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.