{"id":3755,"date":"2014-07-08T11:14:00","date_gmt":"2014-07-08T11:14:00","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=3755"},"modified":"2014-07-08T11:14:23","modified_gmt":"2014-07-08T11:14:23","slug":"3755","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=3755","title":{"rendered":"Lemma of the day"},"content":{"rendered":"

Let h : X –> Y, g : Y –> B be morphisms of algebraic spaces with composition f : X –> B. Let b \u2208 |B| and let Spec(k) \u2192 B be a morphism in the equivalence class of b. Assume<\/p>\n

    \n
  1. X \u2192 B is a proper morphism,<\/li>\n
  2. Y \u2192 B is separated and locally of finite type,<\/li>\n
  3. one of the following is true:\n
      \n
    1. the image of |X_k| \u2192 |Y_k| is finite,<\/li>\n
    2. the image of |f|^{\u22121}({b}) in |Y| is finite and B is decent.<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n

      Then there is an open subspace B\u2032 \u2282 B containing b such that X_{B\u2032} \u2192 Y_{B\u2032} factors through a closed subspace Z \u2282 Y_{B\u2032} finite over B\u2032. See Lemma Tag 0AEJ<\/a>.<\/p>\n

      Slogan: Collapsing a fibre of a proper family forces nearby ones to collapse too.<\/p>\n","protected":false},"excerpt":{"rendered":"

      Let h : X –> Y, g : Y –> B be morphisms of algebraic spaces with composition f : X –> B. Let b \u2208 |B| and let Spec(k) \u2192 B be a morphism in the equivalence class of … Continue reading →<\/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-3755","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\/3755","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=3755"}],"version-history":[{"count":4,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/3755\/revisions"}],"predecessor-version":[{"id":3759,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/3755\/revisions\/3759"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3755"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3755"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3755"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}