Trivial ring extension: A fresh geometric point of view
Mohamed Aqalmoun
Department of Mathematics, Higher Normal School, Sidi Mohamed Ben Abdellah University, Fez, Morocco.
Pages 315–324 | Received 12 May 2023, Accepted 15 September 2023, Published 11 December 2023
Abstract
Let X be a scheme and \mathcal{M} be a quasi-coherent \mathcal{O}_X-module. In this paper we introduce the concept of trivial scheme extension denoted X\propto \mathcal{M} which behave as the trivial ring extension in the affine case, that is X\propto \mathcal{M} is the affine scheme associated to the commutative ring R\propto M when X=\mathrm{Spec}(R) and \mathcal{M} is the quasi-coherent module associated to the R-module M. Two constructions are presented, one of which is based on the gluing of the local data, another construction in terms of quasi-coherent \mathcal{O}_X-algebra is considered. Finally we discuss the quasi-affineness of the trivial scheme extension X\propto \mathcal{M}.
Keywords: trivial ring extension, scheme, quasi-coherent module, affine morphism, trivial scheme extension.
MSC numbers: 14A15, 14F06.
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