Moroccan Journal of Algebra and Geometry with Applications

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Trivial ring extension: A fresh geometric point of view

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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