Torsion classes of finite type and spectra
Torsion classes of finite type and spectra
复制标题
有限类型和谱的扭转类别
DOI：
10.4171/060-1/12
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发表时间：
2006
期刊：
影响因子：
--
通讯作者：
M. Prest
中科院分区：
文献类型：
--
作者：
G. Garkusha;M. Prest
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.
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.