{"id":2040,"date":"2023-12-08T20:28:41","date_gmt":"2023-12-08T20:28:41","guid":{"rendered":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/?page_id=2040"},"modified":"2023-12-11T13:41:06","modified_gmt":"2023-12-11T13:41:06","slug":"trivial-ring-extension-a-fresh-geometric-point-of-view","status":"publish","type":"page","link":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/trivial-ring-extension-a-fresh-geometric-point-of-view\/","title":{"rendered":"Trivial ring extension: A fresh geometric point of view"},"content":{"rendered":"\n
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Moroccan Journal of Algebra and Geometry with Applications<\/strong><\/p>\n\n\n\n

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Trivial ring extension: A fresh geometric point of view<\/strong><\/p>\n<\/div>\n<\/div>\n\n\n\n

Mohamed Aqalmoun<\/strong> <\/i>
Department of Mathematics, Higher Normal School, Sidi Mohamed Ben Abdellah University, Fez, Morocco.<\/p>\n\n\n\n

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Pages 315\u2013324 | <\/font> Received 12 May 2023, Accepted<\/span><\/span> 15 September 2023<\/span><\/span>, Published 11 December 2023 <\/span> <\/span><\/p>\n\n\n\n

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Abstract<\/span><\/strong><\/p>\n\n\n\n

Let X<\/span> be a scheme and \\mathcal{M}<\/span> be a quasi-coherent \\mathcal{O}_X<\/span>-module. In this paper we introduce the concept of trivial scheme extension denoted X\\propto \\mathcal{M}<\/span> which behave as the trivial ring extension in the affine case, that is X\\propto \\mathcal{M}<\/span> is the affine scheme associated to the commutative ring R\\propto M <\/span> when X=\\mathrm{Spec}(R)<\/span> and \\mathcal{M}<\/span> is the quasi-coherent module associated to the R<\/span>-module M<\/span>. 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<\/span>-algebra is considered. Finally we discuss the quasi-affineness of the trivial scheme extension X\\propto \\mathcal{M}<\/span>.<\/p>\n\n\n\n

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Keywords<\/strong>:<\/span>  trivial ring extension, scheme, quasi-coherent module, affine morphism, trivial scheme extension. <\/p>\n\n\n\n

MSC numbers<\/strong>:<\/span> 14A15, 14F06.<\/p>\n\n\n\n

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Moroccan Journal of Algebra and Geometry with Applications Latest articles 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\u2013324 | Received 12 May 2023, Accepted 15 September 2023, Published 11 December 2023 Abstract Let be a scheme and Read More<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"template-page-builder.php","meta":{"footnotes":""},"class_list":["post-2040","page","type-page","status-publish","hentry","entry"],"_links":{"self":[{"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/pages\/2040","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/comments?post=2040"}],"version-history":[{"count":5,"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/pages\/2040\/revisions"}],"predecessor-version":[{"id":2090,"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/pages\/2040\/revisions\/2090"}],"wp:attachment":[{"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/media?parent=2040"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}