{"id":202,"date":"2010-03-30T21:24:36","date_gmt":"2010-03-30T21:24:36","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=202"},"modified":"2010-03-30T21:24:36","modified_gmt":"2010-03-30T21:24:36","slug":"quasi-coherent-modules","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=202","title":{"rendered":"Quasi-coherent modules"},"content":{"rendered":"<p>Let X be a scheme. Let F be a quasi-coherent O_X-module. Let G \u2282 F be an O_X-submodule, not necessarily quasi-coherent. Then there exists a quasi-coherent submodule G&#8217; \u2282 G which is universal for maps of quasi-coherent modules into G. This is <a href=\"http:\/\/math\/~dejong\/algebraic_geometry\/stacks-git\/locate.php?tag=01QZ\">Lemma Tag 01QZ<\/a> in the stacks project.<\/p>\n<p>The condition that G is a submodule of a quasi-coherent module is necessary in order to construct G&#8217; (I think; explicit counter examples welcome). This result has two funny looking applications<\/p>\n<ol>\n<li>Any morphism f : X &#8212;&gt; Y of schemes has a scheme theoretic image (<a href=\"http:\/\/math\/~dejong\/algebraic_geometry\/stacks-git\/locate.php?tag=01R6\">Lemma Tag 01R6<\/a>), and<\/li>\n<li>i_* : QCoh(Z) &#8212;&gt; QCoh(Y) has a right adjoint when i : Z &#8212;&gt; X is a closed immersion of schemes (<a href=\"http:\/\/math\/~dejong\/algebraic_geometry\/stacks-git\/locate.php?tag=01R0\">Lemma Tag 01R0<\/a>).<\/li>\n<\/ol>\n<p>At first sight it may seem that 1 is too strong. But I think it isn&#8217;t simply because there is no way you can deduce anything from the existence of a smallest closed subscheme of Y through which f factors. It is only when f is quasi-compact and quasi-separated that the scheme theoretic image commutes with restriction to open subschemes for example.<\/p>\n<p>It may be that the existence in 2 of a right adjoint i^! of i_* : QCoh(Z) &#8212;&gt; QCoh(Y) follows from general facts, but it is cute that one can explicitly write it down as in the proof of the lemma referenced above. Again, the construction of i^! is not local on X, except in case the sheaf of ideals defining the closed immersion i is of finite type (to see this use <a href=\"http:\/\/math\/~dejong\/algebraic_geometry\/stacks-git\/locate.php?tag=01PO\">Lemma Tag 01PO<\/a>).<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Let X be a scheme. Let F be a quasi-coherent O_X-module. Let G \u2282 F be an O_X-submodule, not necessarily quasi-coherent. Then there exists a quasi-coherent submodule G&#8217; \u2282 G which is universal for maps of quasi-coherent modules into G. &hellip; <a href=\"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=202\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-202","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/202","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=202"}],"version-history":[{"count":13,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/202\/revisions"}],"predecessor-version":[{"id":215,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/202\/revisions\/215"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=202"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=202"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=202"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}