## 2006.359: Reconstructing projective schemes from Serre subcategories

2006.359:
Grigory Garkusha and Mike Prest
(2006)
*Reconstructing projective schemes from Serre subcategories.*

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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 2000 > 14 Algebraic geometry MSC 2000 > 16 Associative rings and algebras MSC 2000 > 18 Category theory; homological algebra |

MIMS number: | 2006.359 |

Deposited By: | Professor Mike Prest |

Deposited On: | 24 August 2006 |

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