{"id":3097,"date":"2013-02-20T14:54:15","date_gmt":"2013-02-20T14:54:15","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=3097"},"modified":"2013-02-20T14:54:15","modified_gmt":"2013-02-20T14:54:15","slug":"localize","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=3097","title":{"rendered":"Localize"},"content":{"rendered":"<p>Let (X, O<sub>X<\/sub>) be a ringed space and let A &#8212;> O<sub>X<\/sub> be a surjection of sheaves of algebras. Let S &sub; A be the subsheaf of local sections which map to invertible functions of O<sub>X<\/sub>. Then S(U) is a multiplicative subset of A(U) for every open U of X and we can factor the map as<\/p>\n<blockquote><p>A &#8212;> S<sup>-1<\/sup>A &#8212;> O<sub>X<\/sub><\/p><\/blockquote>\n<p>If X is a locally ringed space, then the stalks of S<sup>-1<\/sup>A are local rings too.<\/p>\n<p>This construction is occasionally useful. For example, consider a fibre product of schemes W = X x_S Y with projections maps p : W &#8212;> X, q : W &#8212;> Y, and structure morphism h : W &#8212;-> S. Then the map<\/p>\n<blockquote><p>A = p<sup>-1<\/sup>O<sub>X<\/sub> &otimes;<sub>h<sup>-1<\/sup>O<sub>S<\/sub><\/sub> q<sup>-1<\/sup>O<sub>Y<\/sub> &#8212;-> O<sub>W<\/sub><\/p><\/blockquote>\n<p>is not an isomorphism in general, but a localization: with the notation above we have S<sup>-1<\/sup>A = O<sub>W<\/sub>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Let (X, OX) be a ringed space and let A &#8212;> OX be a surjection of sheaves of algebras. Let S &sub; A be the subsheaf of local sections which map to invertible functions of OX. Then S(U) is a &hellip; <a href=\"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=3097\">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-3097","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\/3097","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=3097"}],"version-history":[{"count":6,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/3097\/revisions"}],"predecessor-version":[{"id":3103,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/3097\/revisions\/3103"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3097"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3097"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3097"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}