Grothendieck topology, coherent sheaves and Serre's theorem for schematic algebras
type: Item
instance of: scholarly article
title: Grothendieck topology, coherent sheaves and Serre's theorem for schematic algebras
zbMATH Open document ID: 0854.16027
DOI: 10.1016/0022-4049(94)00118-3
author: Luc Willaert
author: Fred Van Oystaeyen
published in: Journal of Pure and Applied Algebra
publication date: 1996-06-26T00:00:00Z
full work available at URL: https://hdl.handle.net/10067/124190151162165141
review text: Associated to a commutative graded ring, in this note, the authors aim to associate to schematic algebras (several of which occur in the framework of quantum groups) a ``geometric'' object. Schematic algebras are characterized by the fact that they possess ``many'' Ore sets, the latter permitting the authors to introduce a suitable notion of ``noncommutative Grothendieck topology'' (based upon the free monoid over the Ore sets), as well as constructing an associated notion of ``sheaf'' over these. In general, as pointed out in the text, the structure ``sheaf'' thus associated to a schematic algebra \(R\) is not a ``sheaf of rings'', nor does the construction specialize to that of \(\text{Proj} (R)\) in the commutative case (as only covers of open sets induced by global ones are considered). However, the authors introduce a suitable notion of quasicoherent sheaf over these geometric objects and prove that the category of these is equivalent to Artin's \(\text{Proj} (R)\), thus providing an interesting step towards a full geometric understanding of the latter.
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 16S60
Mathematics Subject Classification ID: 16U20
Mathematics Subject Classification ID: 14A22
Mathematics Subject Classification ID: 16S90
zbMATH DE Number: 825329
zbMATH Keywords: noncommutative Grothendieck topology
zbMATH Keywords: schematic algebras
zbMATH Keywords: quantum groups
zbMATH Keywords: Ore sets
zbMATH Keywords: quasicoherent sheaf
zbMATH Keywords: geometric objects
Wikidata QID: Q116689256
MaRDI profile type: Publication
OpenAlex ID: W2080327789
cites work: Q4011045
cites work: Q4143433
cites work: Faisceaux algébriques cohérents
cites work: Q4049223
cites work: Examples and quantum sections of schematic algebras
description: scientific article; zbMATH DE number 825329