{"id":702,"date":"2010-08-01T14:07:17","date_gmt":"2010-08-01T14:07:17","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=702"},"modified":"2010-08-01T14:07:17","modified_gmt":"2010-08-01T14:07:17","slug":"one-point-compactification","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=702","title":{"rendered":"One point compactification"},"content":{"rendered":"<p>Let f : X &#8212;&gt; S be a separated morphism of finite presentation. Consider the functor F : (Sch)^{opp} \u2014&gt; (Sets) which to a scheme T associates all pairs (a, Z) where a : T &#8212;> S and Z is a closed sub scheme of the base change X_T such that the projection Z &#8212;> T is an open immersion. In other words, this is the functor of flat families of closed sub schemes of degree <= 1 on X\/S, as we discussed briefly in <a href=\"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=694\">this post<\/a>. As we saw there it is not true in general that F is an algebraic space. If X = A^1_S then F is (probably) a directed colimit of schemes. But if X has higher dimension I&#8217;m not sure how to &#8220;compute&#8221; F.<\/p>\n<p>Here are some general properties of this construction. There is a canonical morphism<\/p>\n<p>j : X &#8212;> F<\/p>\n<p>which is an open immersion by construction. Moreover, there is a canonical morphism<\/p>\n<p>&infin; : S &#8212;> F<\/p>\n<p>which associates to a : T &#8212;> S the pair (a, &empty;). And of course on points we have F = j(X) &cup; &infin;(S). The structure morphism p : F &#8212;> S is locally of finite presentation and satisfies the valuative criterion (both existence and uniqueness). These properties tell us p is &#8220;proper&#8221;. Thus F is morally speaking the <em>one point compactification of X\/S<\/em>.<\/p>\n<p>When I was discussing this with Bhargav Bhatt he suggested we think about the etale cohomology of F. Now that I have had some time to think about his suggestion, I think this is a splendid idea. Namely, it seems to me that we could <em>try<\/em> to use Rp_*Rj_! to define Rf_! for the morphism f : X &#8212;> S&#8230;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Let f : X &#8212;&gt; S be a separated morphism of finite presentation. Consider the functor F : (Sch)^{opp} \u2014&gt; (Sets) which to a scheme T associates all pairs (a, Z) where a : T &#8212;> S and Z is &hellip; <a href=\"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=702\">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-702","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\/702","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=702"}],"version-history":[{"count":10,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/702\/revisions"}],"predecessor-version":[{"id":712,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/702\/revisions\/712"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=702"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=702"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=702"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}