{"id":4149,"date":"2016-10-25T02:05:37","date_gmt":"2016-10-25T02:05:37","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=4149"},"modified":"2016-10-26T20:41:12","modified_gmt":"2016-10-26T20:41:12","slug":"challenge-accepted","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=4149","title":{"rendered":"Challenge Accepted"},"content":{"rendered":"

Remember this challenge<\/a>? Probably not. But wait, don’t click! Namely, I will do something more general in this post.<\/p>\n

Suppose we\u00a0have a ring A and a contravariant functor F on (Sch\/A) with the following properties:<\/p>\n

    \n
  1. F satisfies the sheaf property for fpqc coverings<\/li>\n
  2. the value of F on a scheme is either a singleton or empty<\/li>\n
  3. for every quasi-compact scheme T\/A such that F(T) is nonempty, there is an ideal I of A such that F(Spec(A\/I)) is nonempty and such that T —> Spec(A) factors through Spec(A\/I).<\/li>\n<\/ol>\n

    Example: A = k[x, y] for a field k and F(T) is nonempty if and only if the generic point of Spec(A) is not in the image of T —> Spec(A). Here F is not a representable functor.<\/p>\n

    I’d like to add some conditions that guarantee that F is representable by a closed subscheme of Spec(A). Here is what I just came up with; I think it is obviously correct and the right thing to do. If A is Noetherian we add the following two conditions<\/p>\n

      \n
    1. If s_1, s_2, s_3, … is an infinite sequence of points of Spec(A) such that F(s_i) is nonempty and s is a limit point of the sequence, then F(s) is nonempty.<\/li>\n
    2. If A —> …. —> A_n —> A_{n – 1} —> … —> A_1 are surjections such that the kernels A_n —> A_{n – 1} are locally nilpotent ideals and F(Spec(A_i)) is nonempty, then F(Spec(A_∞)) is nonempty where A_∞ = lim A_n.<\/li>\n<\/ol>\n

      I leave it as an exercise to show that 1 — 5 imply F is representable in the desired manner. If A is not Noetherian, then somehow these should still be enough although maybe you need to replace the natural numbers by a bigger directed set.<\/p>\n

      Why am I excited by this observation? It is because I want to apply this to the situation of the other blog post mentioned above: X is an algebraic space of finite presentation over A, u : H —> G is a map between quasi-coherent O_X-modules. We assume G is flat over A, of finite presentation, and universally pure relative to A (this is a technical condition which is satisfied if the support of G is proper over A). The functor F is defined by F(T) is nonempty if and only if the base change u_T of u is zero.<\/p>\n

      Properties 1, 2, 3 hold for F and are easy to prove. The proof of property 4 still doesn’t use purity of G relative to A (I think because we already have 3 it follows from an argument using generic freeness, but I also have an argument using \\’etale localization). The key is to prove property 5.<\/p>\n

      To see 5 is true, I argue as follows. Suppose that the base change u_∞ to A_∞ is nonzero. Choose a weakly associated point ξ of the image of u_∞. This is also a weakly associated point of G_∞. The image t’ of ξ in Spec(A_∞) specializes to a point t in V(I_1) = Spec(A_1) because I_1 is contained in the radical of A_∞. Because G is universally pure relative to A, there is a specialization θ of ξ which lies over t (indeed this is the definition of being pure relative to the base). Then since u_∞ is zero at θ (in a suitable \\’etale neighbourhood Edit: Argh… I just discovered this doesn’t work!<\/b>) it is zero at ξ, a contradiction.<\/p>\n

      Enjoy!<\/p>\n

      PS: A finitely presented module G on X flat and pure over A is universally pure relative to A. However, this is harder to prove than the above and it is easy to see that support proper over A implies universal purity.<\/p>\n","protected":false},"excerpt":{"rendered":"

      Remember this challenge? Probably not. But wait, don’t click! Namely, I will do something more general in this post. Suppose we\u00a0have a ring A and a contravariant functor F on (Sch\/A) with the following properties: F satisfies the sheaf property … 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-4149","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\/4149","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=4149"}],"version-history":[{"count":16,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/4149\/revisions"}],"predecessor-version":[{"id":4165,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/4149\/revisions\/4165"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4149"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4149"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4149"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}