{"id":3951,"date":"2014-10-09T17:21:34","date_gmt":"2014-10-09T17:21:34","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=3951"},"modified":"2014-10-09T17:21:34","modified_gmt":"2014-10-09T17:21:34","slug":"dilatations-2","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=3951","title":{"rendered":"Dilatations"},"content":{"rendered":"

We clarify the discussion in this post<\/a> resulting in a generalization of a result of Mike Artin.<\/p>\n

Let X be a Noetherian algebraic space. Let T ⊂ X be a closed subspace. Let us denote X\/T<\/sub> the formal completion of X along T (Tag 0AIX<\/a>). Let W —> X\/T<\/sub> be a rig-etale morphism<\/i> of formal algebraic spaces, which means that<\/p>\n

    \n
  1. W —> X\/T<\/sub> is representable by algebraic spaces, i.e., it is an adic morphism of formal algebraic spaces Tag 0AQ2<\/a>),<\/li>\n
  2. W —> X\/T<\/sub> is locally of finite type, i.e., it is etale locally on affine formal algebraic pieces given by a continuous ring map A —> B which is topologically of finite type Tag 0ALL<\/a>),<\/li>\n
  3. these ring maps A —> B have a completed cotangent complex whose cohomology groups are annihilated by an ideal of definition of A, for more details see Tag 0ALP<\/a> and Tag 0AQK<\/a>.<\/li>\n<\/ol>\n

    These three conditions correspond to condition (i) of Definition 1.7 of Artin’s paper “Formal Moduli: II”. The first result is that<\/p>\n

    Given W —> X\/T<\/sub> rig-etale there exists a morphism of algebraic spaces Y —> X which is an isomorphism over X – T and whose completion Y\/T<\/sub> —> X\/T<\/sub> is isomorphic to W —> X\/T<\/sub>.<\/p><\/blockquote>\n

    In fact, Theorem 0ARB<\/a> tells us we obtain an equivalence of categories.<\/p>\n

    This theorem does not often produce separated morphisms Y —> X if we start with a random W —> X\/T<\/sub>. A typical thing that happens can be seen by starting with X equal to the affine line, T = {0} and W two copies of the completion of X at 0. Then the resulting Y is the affine line with 0 doubled.<\/p>\n

    Thus to get Artin’s theorem on dilatations we need to impose conditions on W —> X\/T<\/sub> guaranteeing that Y —> X is separated or even proper. To do this we will use the notion of a rig-surjective morphism<\/i> W’ —> W of locally Noetherian formal algebraic spaces W, W’ defined by requiring adic morphisms Spf(R) —> W with R a cdvr to lift to Spf(R’) —> W’ for some extension of cdvr R ⊂ R’ (Tag 0AQP<\/a>). Let’s say an adic morphism W —> W’ of locally Noetherian formal algebraic spaces is a rig-monomorphism<\/i> if the diagonal morphism is rig-surjective. In this language the conditions (ii) and (iii) from Definition 1.7 of Artin’s paper have the following interpretations:<\/p>\n

      \n
    1. If W —> X\/T<\/sub> as above is separated and a rig-monomorphism, then Y —> X is separated (Tag 0ARW<\/a>),<\/li>\n
    2. If W —> X\/T<\/sub> as above is proper, a rig-monomorphism, and rig-surjective, then Y —> X is proper (Tag 0ARX<\/a>).<\/li>\n<\/ol>\n

      The second statement recovers exactly Artin’s theorem on dilatations.<\/p>\n

      One typically applies the result to construct modifications<\/i> Y —> X (Tag 0AD7<\/a>) by taking the complete local ring A of X at a closed point x and setting W equal to the formal completion of the blow up of A at an ideal I ⊂ A which is locally principal on the puctured spectrum of A. Here the funny situation occurs that we can first read Tag 0ARX<\/a> backwards over Spec(A) to conclude that W —> X\/x<\/sub> has the required properties and then forwards to conclude that Y —> X is proper. In other situations it may not be that easy to verify the assumptions needed for the application of the theorem and it would behoove us to prove a few lemmas that help with this task.<\/p>\n

      Your help with this and other <\/a> tasks<\/a> is always welcome!<\/p>\n","protected":false},"excerpt":{"rendered":"

      We clarify the discussion in this post resulting in a generalization of a result of Mike Artin. Let X be a Noetherian algebraic space. Let T ⊂ X be a closed subspace. Let us denote X\/T the formal completion of … 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-3951","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\/3951","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=3951"}],"version-history":[{"count":16,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/3951\/revisions"}],"predecessor-version":[{"id":3967,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/3951\/revisions\/3967"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3951"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3951"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3951"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}