Garkusha, Grigory and Prest, Mike
(2006)
*Reconstructing projective schemes from Serre subcategories.*
[MIMS Preprint]

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## Abstract

Given a positively graded commutative coherent ring $A=\bigoplus_{j\geq 0}A_j$ which is finitely generated as an $A_0$-algebra, a bijection between the tensor Serre subcategories of ${\rm qgr} A$ and the set of all subsets $Y\subseteq{\rm Proj} A$ of the form $Y=\bigcup_{i\in\Omega}Y_i$ with quasi-compact open complement ${\rm Proj} A\setminus Y_i$ 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 $$({\rm Proj} A,\cc O_{{\rm Proj} A})\simeq ({\rm spec}({\rm qgr} A),{\cal O}_{{\rm qgr} A}),$$ where $({\rm spec}({\rm qgr} A),{\cal O}_{{\rm qgr} A})$ is a ringed space associated to the lattice $L_{\rm serre}({\rm qgr} A)$ of tensor Serre subcategories of ${\rm qgr} A$.

Item Type: | MIMS Preprint |
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Uncontrolled Keywords: | non-commutative projective scheme Serre subcategory coherent graded algebra Ziegler Zariski spectrum |

Subjects: | MSC 2010, the AMS's Mathematics Subject Classification > 14 Algebraic geometry 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: | 24 Aug 2006 |

Last Modified: | 08 Nov 2017 18:18 |

URI: | https://eprints.maths.manchester.ac.uk/id/eprint/583 |

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