{"id":802,"date":"2010-09-05T18:50:27","date_gmt":"2010-09-05T18:50:27","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=802"},"modified":"2010-09-05T18:50:27","modified_gmt":"2010-09-05T18:50:27","slug":"a-fun-lemma","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=802","title":{"rendered":"A fun lemma"},"content":{"rendered":"<p>Lemma <a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/locate.php?tag=055J\">Tag 055J<\/a>: Let R be a dvr. Let X be flat over Spec(R), with reduced special fibre, and connected total space. Then the generic fibre of the structure morphism f : X &#8212;&gt; Spec(R) is connected.<\/p>\n<p>You can find a version of this lemma as EGA IV, Lemma 15.5.6 where the hypotheses are that f is locally of finite type and open instead of flat. But in the proof of the lemma in EGA it is remarked that the hypotheses &#8220;loc. fin. type + open over dvr&#8221; imply &#8220;flat&#8221;, hence the lemma above implies the lemma in EGA. I urge you to try to prove the lemma above before looking it up, because it is fun when you find it!<\/p>\n<p>Why is flatness necessary? If X is not flat over R, then a counter example is X = Spec(R[x]\/(px(x-1))) where p \u2208 R is a uniformizer. In words: X is a union of two copies of Spec(R) glued at 0, 1 of the affine line over the residue field of R.<\/p>\n<p>Why is a reduced special fibre necessary? If not then a counter example is X = Spec(R[x]\/(x(x &#8211; p))) where p is a uniformizer in R. In words: X is a union of two copies of Spec(R) glued at their special points.<\/p>\n<p>Wie het kleine niet eert is het grote niet weerd!<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Lemma Tag 055J: Let R be a dvr. Let X be flat over Spec(R), with reduced special fibre, and connected total space. Then the generic fibre of the structure morphism f : X &#8212;&gt; Spec(R) is connected. You can find &hellip; <a href=\"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=802\">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-802","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\/802","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=802"}],"version-history":[{"count":3,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/802\/revisions"}],"predecessor-version":[{"id":805,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/802\/revisions\/805"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=802"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=802"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=802"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}