{"id":1222,"date":"2011-02-16T20:21:36","date_gmt":"2011-02-16T20:21:36","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=1222"},"modified":"2011-02-16T20:21:36","modified_gmt":"2011-02-16T20:21:36","slug":"purity","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=1222","title":{"rendered":"Purity"},"content":{"rendered":"<p>Let f : X &#8212;> S be a morphism of finite type. The <em>relative assassin Ass(X\/S) of X\/S<\/em> is the set of points x of X which are embedded points of their fibres. So if f has reduced fibres or if f has fibres which are S_1, then these are just the generic points of the fibres, but in general there may be more. If T &#8212;> S is a morphism of schemes then it isn&#8217;t quite true that Ass(X_T\/T) is the inverse image of Ass(X\/S), but it is almost true, see <a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/locate.php?tag=05KL\">Remark Tag 05KL<\/a>.<\/p>\n<p>Definition: We say X is <em>S-pure<\/em> if for any x &isin; Ass(X\/S) the image of the closure <span style=\"text-decoration: overline;\">{x}<\/span> is closed in S, and if the same thing remains true after any etale base change.<\/p>\n<p>Clearly if f is proper then X is pure over S. If f is quasi-finite and separated then X is S-pure if and only if X is finite over S (see <a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/locate.php?tag=05K4\">Lemma Tag 05K4<\/a>). It turns out that if S is Noetherian, then purity is preserved under arbitrary base change (see <a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/locate.php?tag=05J8\">Lemma Tag 05J8<\/a>), but in general this is not true (see <a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/locate.php?tag=05JK\">Lemma Tag 05JK<\/a>). If f is flat with geometrically irreducible (nonempty) fibres, then X is S-pure (see <a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/locate.php?tag=05K5\">Lemma Tag 05K5<\/a>).<\/p>\n<p>A key algebraic result is the following statement: Let A &#8212;> B be a flat ring map of finite presentation. Then B is projective as an A-module if and only if Spec(B) is pure over Spec(A), see <a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/locate.php?tag=05MD\">Proposition Tag 05MD<\/a>. The current proof involves several bootstraps and starts with proving the result in case A &#8212;> B is a smooth ring map with geometrically irreducible fibres.<\/p>\n<p>I challenge any commutative algebraist to prove this statement without using the language of schemes. You will find another challenge in the next post.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Let f : X &#8212;> S be a morphism of finite type. The relative assassin Ass(X\/S) of X\/S is the set of points x of X which are embedded points of their fibres. So if f has reduced fibres or &hellip; <a href=\"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=1222\">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-1222","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\/1222","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=1222"}],"version-history":[{"count":6,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1222\/revisions"}],"predecessor-version":[{"id":1228,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1222\/revisions\/1228"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1222"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1222"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1222"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}