{"id":1711,"date":"2023-06-14T12:32:29","date_gmt":"2023-06-14T12:32:29","guid":{"rendered":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/?page_id=1711"},"modified":"2023-10-12T22:03:14","modified_gmt":"2023-10-12T22:03:14","slug":"when-two-definitions-of-an-additive-functor-of-commutative-algebras-agree","status":"publish","type":"page","link":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/when-two-definitions-of-an-additive-functor-of-commutative-algebras-agree\/","title":{"rendered":"When Two Definitions of an Additive Functor of Commutative Algebras Agree"},"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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When Two Definitions of an Additive Functor of Commutative Algebras<\/em><\/strong>
Agree<\/em><\/strong><\/p>\n<\/div>\n<\/div>\n\n\n\n

David E. Dobbs <\/strong><\/i>
Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1320, USA<\/p>\n\n\n\n

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Pages 38-69 | Received 20 October 2022, Accepted 29 December 2022, Published 16 June 2023 <\/span><\/p>\n\n\n\n

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

Let R<\/span> be a commutative ring and \\underline{C}_R<\/span> the category of commutative unital R<\/span>-algebras. We show that \\underline{C}_R<\/span> is a pre-additive category if and only if R<\/span> is a zero ring. When these conditions hold, a functor F<\/span> from \\underline{C}_R<\/span> to a pre-additive category \\underline{D}<\/span> with finite products is an additive functor (in the classical sense) if and only if F<\/span> is additive in the sense due to Chase-Harrison-Rosenberg (the latter sense of “additive functor” meaning that F<\/span> commutes with finite products), if and only if F(R)<\/span> is a terminal object of \\underline{D}<\/span>. More generally, if \\underline{C}<\/span> and \\underline{D}<\/span> are additive categories (that is, pre-additive categories with finite products) and F:\\underline{C} \\to \\underline{D}<\/span> is a functor, then F<\/span> is additive if and only if F<\/span> commutes with finite products. For such categories \\underline{C}<\/span> and \\underline{D}<\/span>, we also give four other new characterizations of the additive functors F:\\underline{C} \\to \\underline{D}<\/span>. <\/p>\n\n\n\n

Keywords<\/strong>:<\/span>  Commutative ring, unital algebra, pre-additive category, additive functor, CHR-additive functor, zero ring, sheaf. <\/p>\n\n\n\n

MSC numbers<\/strong>:<\/span> Primary 13A99; Secondary 18E05.<\/p>\n\n\n\n

Downloads:<\/strong> Full-text PDF<\/a> <\/p>\n<\/div>\n\n\n\n

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Moroccan Journal of Algebra and Geometry with Applications Latest articles When Two Definitions of an Additive Functor of Commutative Algebras Agree David E. Dobbs  Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1320, USA Pages 38-69 | Received 20 October 2022, Accepted 29 December 2022, Published 16 June 2023 Abstract Let be a commutative ring 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-1711","page","type-page","status-publish","hentry","entry"],"_links":{"self":[{"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/pages\/1711","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=1711"}],"version-history":[{"count":9,"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/pages\/1711\/revisions"}],"predecessor-version":[{"id":1936,"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/pages\/1711\/revisions\/1936"}],"wp:attachment":[{"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/media?parent=1711"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}