{"id":4702,"date":"2021-07-19T14:45:12","date_gmt":"2021-07-19T14:45:12","guid":{"rendered":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=4702"},"modified":"2024-05-07T14:26:22","modified_gmt":"2024-05-07T14:26:22","slug":"silly-question","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=4702","title":{"rendered":"Silly question"},"content":{"rendered":"<p>So, recently I was looking at <a href=\"https:\/\/stacks.math.columbia.edu\/tag\/03L7\">Lemma 03L7<\/a> because of a question asked a comment on <a href=\"https:\/\/stacks.math.columbia.edu\/tag\/022B\">Definition 022B<\/a>. The condition that a single flat morphism f : X &#8212;> Y determines an fpqc covering {f : X &#8212;> Y} is the following topological condition on f: given any affine open V of Y there should exist a quasi-compact open U of X with f(U) = V.<\/p>\n<p>My question is this: is it sufficient to ask for a quasi-compact open U with V &subset; f(U)?<\/p>\n<p>The lemma says &#8220;yes&#8221; if Y is quasi-separated (so in practice always).<\/p>\n<p>Does anybody have a counterexample to the general case? Or is it sufficient?<\/p>\n<p>I remember successfully avoiding thinking about this when I first wrote the material on fpqc coverings. But I guess no harm is done thinking about it a little bit during these hot summer days&#8230;<\/p>\n<p><strong>Update 5\/7\/2024:<\/strong> The answer is no by an <a href=\"https:\/\/stacks.math.columbia.edu\/tag\/03L7#comment-8641\">example of Tony Scholl.<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>So, recently I was looking at Lemma 03L7 because of a question asked a comment on Definition 022B. The condition that a single flat morphism f : X &#8212;> Y determines an fpqc covering {f : X &#8212;> Y} is &hellip; <a href=\"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=4702\">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-4702","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\/4702","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=4702"}],"version-history":[{"count":12,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/4702\/revisions"}],"predecessor-version":[{"id":4939,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/4702\/revisions\/4939"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4702"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4702"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4702"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}