{"id":1671,"date":"2011-07-03T19:42:04","date_gmt":"2011-07-03T19:42:04","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=1671"},"modified":"2011-07-03T19:42:04","modified_gmt":"2011-07-03T19:42:04","slug":"clarification","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=1671","title":{"rendered":"Clarification"},"content":{"rendered":"

Let me just clarify what I was trying to say in the previous post<\/a>.<\/p>\n

Setup. Let A be a Noetherian local ring. Set S = Spec(A). Denote U the punctured spectrum of A. Let A \u2282 C \u2282 A^* be a finite type A-algebra contained in the completion of A. Set V = Spec(C) \u2210 U. Consider the functor F : (Sch\/S)^{opp} —> Sets<\/em> which to a scheme T\/S assigns<\/p>\n

    \n
  1. F(T) = {*} = a singleton if there exists an fppf covering {T_i —> T} such that each T_i —> S factors through V.<\/li>\n
  2. F(T) = \u2205 else.<\/li>\n<\/ol>\n

    I claim that all of Artin’s criteria<\/a> are satisfied for this fppf sheaf (details omitted). Moreover, note that both F(Spec(A^*)) = {*} and F(U) = {*} are nonempty.<\/p>\n

    If Artin’s criteria imply that F is an algebraic space, then, choosing a surjective etale morphism X —> F where X is an affine scheme, we conclude that X is surjective and etale over A (this takes a little argument). Using the definition of F we find a faithfully flat, finite type A-algebra B and an A-algebra map C —> B.<\/p>\n

    Conversely, if there exists an A-algebra map C —> B with B a faithfully flat, finite type A-algebra, then Spec(B) —> F is a flat, surjective, finitely presented morphism and F is an algebraic space (by a result of Artin we blogged about<\/a> recently).<\/p>\n

    This analysis singles out the following condition on a Noetherian local ring A: Every finite type A-algebra C contained in the completion of A should have an A-algebra map to a faithfully flat, finite type A-algebra B. But it isn’t necessary for B to also map into A^*! I missed this earlier when I was thinking about this issue. For example any dvr has this property (but there exist non-excellent dvrs).<\/p>\n

    Finally, if Artin’s criteria characterize algebraic spaces over Spec(R) for some Noetherian ring R then this property holds for any local ring of any finite type R-algebra. Likely this isn’t a sufficient condition.<\/p>\n","protected":false},"excerpt":{"rendered":"

    Let me just clarify what I was trying to say in the previous post. Setup. Let A be a Noetherian local ring. Set S = Spec(A). Denote U the punctured spectrum of A. Let A \u2282 C \u2282 A^* be … 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-1671","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\/1671","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=1671"}],"version-history":[{"count":18,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1671\/revisions"}],"predecessor-version":[{"id":1689,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1671\/revisions\/1689"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1671"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1671"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1671"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}