{"id":3800,"date":"2014-07-22T23:33:27","date_gmt":"2014-07-22T23:33:27","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=3800"},"modified":"2014-10-20T15:26:01","modified_gmt":"2014-10-20T15:26:01","slug":"dilatations","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=3800","title":{"rendered":"Dilatations"},"content":{"rendered":"

This is a follow up of Example wanted<\/a>. There I ask for two examples.<\/p>\n

Firstly, I ask for a Noetherian local domain A such that its completion A* has an isolated singularity and such that Spec(A) does not have a resolution of singularities.<\/p>\n

I now think such an example cannot exist. Namely, conjecturally resolution for Spec(A*) would proceed by blowing up nonsingular centers each lying about the closed point, which would transfer over to Spec(A) thereby giving a resolution for Spec(A).<\/p>\n

Secondly, I ask for a Noetherian local ring A and a proper morphism Y —> Spec(A*) of algebraic spaces which is an iso above the puctured spectrum U* which is NOT the base change of a similar morphism X —> Spec(A).<\/p>\n

As Jason pointed out in the comments on the aforementioned blog post, to get such an example we have to assume that A is nonexcellent since otherwise Artin’s result on dilatations kicks in to show that X does exist. In fact we have the following:<\/p>\n

    \n
  1. We may assume A is henselian, see Lemma Tag 0AE4<\/a><\/li>\n
  2. It holds when A is a G-ring, see Lemma Tag 0AE5<\/a><\/li>\n
  3. There exists a blow up Y’ —> Spec(A*) with center supported on the closed point which dominates Y and which is the base change of some X’ —> Spec(A) as above, see Lemma Tag 0AE6<\/a> and Lemma Tag 0AFK<\/a>.\n<\/ol>\n

    I’ve tried to make a counter example for non-G-rings, but failed. So now I am beginning to wonder: maybe there isn’t one? [Edit 20 October 2014:<\/strong> there is none as can be seen by visiting the upgraded Lemma Tag 0AE5<\/a>.]<\/p>\n

    If so, then perhaps Artin’s result on dilatations (in formal moduli II<\/a>) holds for Noetherian algebraic spaces without any supplementary conditions. Yes, this is a ridiculous step to take (Artin’s result is about formal algebraic spaces and a lot stronger than the question asked above), and I say this, not because I have a good reason to think this is true, but just to make it easier for you and me to make a counter example. I don’t have one, do you? [Edit 20 October 2014:<\/strong> there is none as can be seen by visiting this blog post<\/a>.]<\/p>\n

    You should probably stop reading here, because now things become really vague. Looking at affine schemes \\’etale over Y leads to the following type of question. Suppose that f : V* —> Spec(A*) is a finite type morphism with V* affine and f^{-1}(U*) —> U* \\’etale. Then we can ask whether V* is the base change of a similar type of morphism V —> Spec(A). The answer is a resounding NO because for example the morphism f could be an open immersion whose complement is a closed subscheme of Spec(A*) which is not the base change of a closed subscheme of Spec(A). But suppose we only ask for a V —> Spec(A) such that the m_A-adic formal completion of V is isomorphic to the m_{A*}-adic formal completion of V*? Namely, if this question has a positive answer, then we might be able to use this to construct an X as above whose base change is Y by glueing affine pieces. I also would dearly love a counter example to this question (again it holds if A is a G-ring so a counter example would have to involve some kind of bad ring). [Edit 20 October 2014:<\/strong> The existence of these algebras follows from the paper by Elkik on solutions of equations over henselian rings, see this blog post<\/a>.]<\/p>\n

    Anyway, any suggestions, ideas, references, etc are very welcome. Thanks!<\/p>\n","protected":false},"excerpt":{"rendered":"

    This is a follow up of Example wanted. There I ask for two examples. Firstly, I ask for a Noetherian local domain A such that its completion A* has an isolated singularity and such that Spec(A) does not have a … 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-3800","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\/3800","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=3800"}],"version-history":[{"count":13,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/3800\/revisions"}],"predecessor-version":[{"id":3970,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/3800\/revisions\/3970"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3800"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3800"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3800"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}