{"id":1297,"date":"2011-03-03T15:32:37","date_gmt":"2011-03-03T15:32:37","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=1297"},"modified":"2011-03-03T15:32:37","modified_gmt":"2011-03-03T15:32:37","slug":"comparing-topologies","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=1297","title":{"rendered":"Comparing topologies"},"content":{"rendered":"<p>Let S be a scheme. There are many ways to turn the category of schemes over S into a site, but some of the things you can do lead to the same topos, i.e., the category of sheaves are identical. In this case we say these sites define the same <em>topology<\/em>. Here are some examples of comparisons of topologies:<\/p>\n<ol>\n<li>The smooth topology and the etale topology are the same. See <a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/locate.php?tag=055V\">Lemma Tag 055V<\/a>.<\/li>\n<li>The fppf topology is the same as the one you get by considering fppf coverings {T_i &#8212;&gt; T} such that each T_i &#8212;&gt; T is locally quasi-finite. See <a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/locate.php?tag=0572\">Lemma Tag 0572<\/a>.<\/li>\n<li>The topology generated by Zariski coverings and {f : T &#8212;&gt; S} with f surjective finite locally free is finer than the etale topoloy, see <a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/locate.php?tag=02LH\">Lemma Tag 02LH<\/a> and <a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/locate.php?tag=02LI\">Remark Tag 02LI<\/a>.<\/li>\n<li>The fppf topology is the same as the one generated by Zariski coverings and finite surjective locally free morphisms. See <a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/locate.php?tag=05WN\">Lemma Tag 05WN<\/a>. <\/li>\n<\/ol>\n<p>Somehow I hadn&#8217;t realized 4 earlier. What made me think of it today was this <a href=\"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=1291&#038;cpage=1#comment-37905\">comment by David Rydh<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Let S be a scheme. There are many ways to turn the category of schemes over S into a site, but some of the things you can do lead to the same topos, i.e., the category of sheaves are identical. &hellip; <a href=\"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=1297\">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-1297","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\/1297","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=1297"}],"version-history":[{"count":11,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1297\/revisions"}],"predecessor-version":[{"id":1308,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1297\/revisions\/1308"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1297"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1297"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1297"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}