235 research outputs found
Torsion classes of finite type and spectra
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 Mod R (respectively
tensor torsion classes of finite type in QGr A) and the set of all subsets
Y\subset Spec R (respectively Y\subset Proj A) of the form
Y=\cup_{i\in\Omega}Y_i, with Spec R\Y_i (respectively Proj A\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,O_R)-->(Spec(Mod R),O_{Mod R}) and
(Proj A,O_{Proj A})-->(Spec(QGr A),O_{QGr A}), where (Spec(Mod R),O_{Mod R})
and (Spec(QGr A),O_{QGr A}) are ringed spaces associated to the lattices
L_{tor}(Mod R) and L_{tor}(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 Mod R is established
Reconstructing projective schemes from Serre subcategories
Given a positively graded commutative coherent ring A which is finitely
generated as an A_0-algebra, a bijection between the tensor Serre subcategories
of qgr A and the set of all subsets Y\subseteq Proj A of the form
Y=\bigcup_{i\in\Omega}Y_i with quasi-compact open complement Proj A\Y_i for all
i\in\Omega is established. To construct this correspondence, properties of the
Ziegler and Zariski topologies on the set of isomorphism classes of
indecomposable injective graded modules are used in an essential way. Also,
there is constructed an isomorphism of ringed spaces (Proj A,O_{Proj A}) -->
(Spec(qgr A),O_{qgr A}), where (Spec(qgr A),O_{qgr A}) is a ringed space
associated to the lattice L_{serre}(qgr A) of tensor Serre subcategories of qgr
A.Comment: some minor corrections mad
Hochster duality in derived categories and point-free reconstruction of schemes
For a commutative ring , we exploit localization techniques and point-free
topology to give an explicit realization of both the Zariski frame of (the
frame of radical ideals in ) and its Hochster dual frame, as lattices in the
poset of localizing subcategories of the unbounded derived category .
This yields new conceptual proofs of the classical theorems of Hopkins-Neeman
and Thomason. Next we revisit and simplify Balmer's theory of spectra and
supports for tensor triangulated categories from the viewpoint of frames and
Hochster duality. Finally we exploit our results to show how a coherent scheme
can be reconstructed from the tensor triangulated structure
of its derived category of perfect complexes.Comment: v5:Minoir typos corrected the proof of tensor nilpotence is made
totally point-free and self-contained; some simplifications and expository
improvements; section on preliminaries shortened; 50pp. To appear in Trans.
AM
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