{"id":2259,"date":"2012-03-28T02:11:11","date_gmt":"2012-03-28T02:11:11","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=2259"},"modified":"2012-03-28T02:11:11","modified_gmt":"2012-03-28T02:11:11","slug":"generically-a-quotient","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=2259","title":{"rendered":"Generically a quotient"},"content":{"rendered":"

In this post I want to outline an argument that proves “most” algebraic stacks are generically “global” quotient stacks. I don’t have the time to add this to the stacks project now, but I hope to return to it in the not too distant future.<\/p>\n

To fix ideas suppose that X is a Noetherian, reduced, irreducible algebraic stack whose geometric generic stabilizer is affine. Then I would like to show there exists a dense open substack U ⊂ X such that U ≅ [W\/GL_n] for some Noetherian scheme W endowed with action of GL_n. The proof consists in repeatedly replacing X by dense open substacks each of which has some additional property:<\/p>\n

    \n
  1. We may assume that X is a gerbe, i.e., that there exists an algebraic space Y and a morphism X —> Y such that X is a gerbe over Y. This follows from Proposition Tag 06RC<\/a>.<\/li>\n
  2. We may assume Y is an affine Noetherian integral scheme. This holds because X —> Y is surjective, flat, and locally of finite presentation, so Y is reduced, irreducible, and locally Noetherian by descent. Thus we get what we want by replacing Y be a nonempty affine open.<\/li>\n
  3. We may assume there exists a surjective finite locally free morphism Z —> Y such that there exists a morphism s : Z —> X over Y. Namely, pick a finite type point of the generic fibre of X —> Y and do a limit argument.<\/li>\n
  4. We may assume the projections R = Z ×_X Z —> Z are affine. Namely, the geometric generic fibres of R —> Z ×_Y Z are torsors under the geometric generic stabilizer which we assumed to be affine. A limit argument does the rest (note that we may shrink Z and Z ×_Y Z by shrinking Y).<\/li>\n
  5. We may assume the projections s, t : R —> Z are free, i.e., s_*O_R and t_*O_R are free O_Z-modules. This follows from generic freeness.<\/li>\n
  6. General principle. Suppose that (U, R, s, t, c) is a groupoid scheme with U, R affine and s, t free and of finite presentation. Consider the morphism p : U —> [U\/R]. Then p_*O_U is a filtered colimit of finite free modules V_i on the algebraic stack [U\/R]. This follows from a well known trick with basis elements.<\/li>\n
  7. General principle, continued. For sufficiently large i the stabilizer groups of [U\/R] act faithfully on the fibres of the vector bundle V_i.<\/li>\n
  8. General principle, continued. [U\/R] ≅ [W\/GL_n] for some algebraic space W and integer n. Namely W is the quotient by R of the frame bundle of the vector bundle V_i.<\/li>\n
  9. We conclude that X = [W\/GL_n] for some Noetherian, reduced irreducible algebraic space W.<\/li>\n
  10. The set of points where W is not a scheme is GL_n-invariant and not dense, hence we may assume W is a scheme by shrinking. (I think this works — there should be something easier you can do here, but I don’t see it right now.)<\/li>\n<\/ol>\n

    Note that we can’t assume that W is affine (a counter example is X = [Spec(k)\/B] where B is the Borel subgroup of SL_{2, k} and k is a field). But with a bit more work it should be possible to get W quasi-affine as in the paper by Totaro (which talks about the harder question of when the entire stack X is of the form [W\/GL_n] and relates it to the resolution property).<\/p>\n","protected":false},"excerpt":{"rendered":"

    In this post I want to outline an argument that proves “most” algebraic stacks are generically “global” quotient stacks. I don’t have the time to add this to the stacks project now, but I hope to return to it in … 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-2259","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\/2259","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=2259"}],"version-history":[{"count":6,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/2259\/revisions"}],"predecessor-version":[{"id":2265,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/2259\/revisions\/2265"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2259"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2259"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2259"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}