Reconstructing projective schemes from Serre subcategories

Grigory Garkusha; Mike Prest

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Reconstructing projective schemes from Serre subcategories

Grigory Garkusha·Mike Prest

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Abstract

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\inΩ}Y_i with quasi-compact open complement Proj A\Y_i for all i\inΩis 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.

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BibTeX RIS

Grigory Garkusha, Mike Prest. 2007-04-12. Reconstructing projective schemes from Serre subcategories. https://arxiv.org/abs/math/0608574

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