{"id":80,"date":"2010-02-20T15:47:01","date_gmt":"2010-02-20T15:47:01","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=80"},"modified":"2010-02-20T15:47:01","modified_gmt":"2010-02-20T15:47:01","slug":"schemes-spaces-and-points","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=80","title":{"rendered":"Schemes, Spaces, and Points"},"content":{"rendered":"

Suppose you have two morphisms a, b : X —> Y and you want to know whether a = b. If<\/p>\n

    \n
  1. X, Y are schemes,<\/li>\n
  2. X is reduced,<\/li>\n
  3. a(x) = b(x) for all x in X, and<\/li>\n
  4. the induced maps on residue fields are the same too,<\/li>\n<\/ol>\n

    then a = b. If<\/p>\n

      \n
    1. X, Y are algebraic spaces,<\/li>\n
    2. Y is locally separated,<\/li>\n
    3. X is reduced, and<\/li>\n
    4. a(x) = b(x) in Y(K) for every x in X(K) and any field K,<\/li>\n<\/ol>\n

      then a = b. But the last statement does not<\/em> hold if we replace condition 2 by the condition that Y is quasi-separated. Recall that quasi-separated algebraic spaces are the “usual” algebraic spaces, i.e., the ones in Knutson’s book, not some bizarre ultra general class of algebraic spaces.<\/p>\n

      This comes up when you consider quotient maps for groupoids in algebraic spaces, and it is just the first small sign that things get a little more confusing when dealing with algebraic spaces. Namely, the above means that if we have a groupoid in algebraic spaces (U, R, s, t, c) and a morphism U —> X then even if all of U,R,X are reduced to see whether U —> X is R-invariant (i.e. a quotient map), it is not enough to check that this holds on field valued points.<\/p>\n","protected":false},"excerpt":{"rendered":"

      Suppose you have two morphisms a, b : X —> Y and you want to know whether a = b. If X, Y are schemes, X is reduced, a(x) = b(x) for all x in X, and the induced maps … Continue reading →<\/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-80","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\/80","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=80"}],"version-history":[{"count":3,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/80\/revisions"}],"predecessor-version":[{"id":83,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/80\/revisions\/83"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=80"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=80"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=80"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}