{"id":4828,"date":"2023-06-09T15:03:07","date_gmt":"2023-06-09T15:03:07","guid":{"rendered":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=4828"},"modified":"2025-11-17T14:45:15","modified_gmt":"2025-11-17T14:45:15","slug":"a-jouanolou-device","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=4828","title":{"rendered":"A Jouanolou device"},"content":{"rendered":"

Let X be a scheme or an algebraic space. A Jouanolou device<\/em> is a morphism Y → X such that Y is an affine scheme and such that Y is a torsor for a vector bundle over X.<\/p>\n

A scheme with an ample family of invertible modules has a Jouanolou device (due to Jouanolou and Thomason). This is called the “Jouanolou trick”.<\/p>\n

Let X be quasi-compact with affine diagonal. Consider the following conditions<\/p>\n

    \n
  1. X has the resolution property<\/li>\n
  2. X has a Jouanolou device<\/li>\n
  3. X is the quotient of an affine scheme by a free action of a group scheme<\/li>\n
  4. X is the quotient of a quasi-affine scheme by a free action of GL_n for some n<\/li>\n<\/ol>\n

    We know from Thomason, Totaro, and Grosa that 1, 3, and 4 are equivalent (for some discussion and references see eg this paper<\/a>). It is easy to see that 2 implies 3.<\/p>\n

    I am going to sketch an argument for 1 ⇒ 2. It may be in the literature; if you have a reference, please email me or leave a comment. [Edit:<\/strong> see end of this post.] Thanks!<\/p>\n

    The case of schemes is easier so I will explain that first. So, assume X is a quasi-compact scheme with affine diagonal and with the resolution property. By standard methods we reduce to the case where X is also of finite type over the integers, i.e., we may assume X is Noetherian. Let X = U_1 ∪ … ∪ U_n be a finite affine open covering. Let I_i ⊂ O_X be the ideal sheaf of the complement of U_i. By the resolution property, we may choose a finite locally free O_X-module V and a surjection V → I_1 ⊕ … ⊕ I_n. Thus we have<\/p>\n

    \nV → I_1 ⊕ … ⊕ I_n → O_X\n<\/p><\/blockquote>\n

    Let E be the dual of V and let O_X(m_1 D_1 + … + m_n D_n) be short hand for SheafHom_{O_X}(I_1^{m_1} … I_n^{m_n}, O_X). Warning: this is just notation and we do not think of D_i as an actual divisor. Taking the dual sequence we obtain<\/p>\n

    \nO_X → O_X(D_1) ⊕ … ⊕ O_X(D_n) → E\n<\/p><\/blockquote>\n

    If s : O_X → E is the composition, then we can consider<\/p>\n

    \nf : Y = RelativeSpec_X(Sym^*(E)\/(s – 1)) → X\n<\/p><\/blockquote>\n

    Since the section s is nowhere zero, this is a torsor for a vector bundle. To finish the proof we have to show that Y is affine. To see this it suffices to show that H^1(Y, G) = 0 for every coherent O_Y-module G. Since f_*G is a direct summand of f_*f^*f_*G it suffices to prove that for every coherent O_X-module F the map H^1(X, F) → H^1(X, f_*f^*F) is zero. This will be the case if for every element ξ in H^1(X, F) there exists an m > 0 such that the image of ξ by the map<\/p>\n

    \nF = F ⊗ O_X → F ⊗ Sym^m(E)\n<\/p><\/blockquote>\n

    is zero. Finally, the key point is that the map O_X → Sym^m(E) factors through the map<\/p>\n

    \nO_X → ⨁ O_X(m_1 D_1 + … + m_n D_n)\n<\/p><\/blockquote>\n

    where the sum is over m_1 + … + m_n = m. Thus it suffices to show that ξ dies in H^1(X, F ⊗ O_X(m_1 D_1 + … + m_n D_n)) provided m is large enough. Since m_i > m\/n for at least one i, we see that it suffices to show that ξ dies in H^1(X, F ⊗ O_X(m D_i)) for m large enough. This is true because (U_i → X)_*O_{U_i} is the colimit of the modules O_X(m D_i) and hence we have<\/p>\n

    \ncolim H^1(X, F ⊗ O_X(m D_i)) = H^1(U_i, F|_{U_i}) = 0\n<\/p><\/blockquote>\n

    Here we use that the open immersion U_i → X is affine and that U_i is affine. This proves the schemes case of the assertion.<\/p>\n

    How do we change this argument when X is an algebraic space? We reduce to X of finite type over the integers as before. We replace the affine open covering by a surjective etale morphism U → X where U is an affine scheme. Using Zariski’s main theorem we may assume that U is a dense and schematically dense affine open of an algebraic space Z which comes with a finite morphism π : Z → X. Let I ⊂ O_Z be the ideal sheaf of the complement of U. If π where finite locally free, there would be a trace map tr : π_*O_Z → O_X. The replacement for this is that, after replacing I by a large positive power, there is a map<\/p>\n

    \nτ : π_*I → O_X\n<\/p><\/blockquote>\n

    such that for a point u ∈ U with image x ∈ X the restriction of τ on the summand O^h_{U, u} of (π_*I) ⊗ O^h_{X, x} is given by the trace map for the finite free ring map O^h_{X, x} → O^h_{U, u}. In particular, τ is surjective. Choose a finite locally free O_X-module V and a surjection V → π_*I. Thus we have<\/p>\n

    \nV → π_*I → O_X\n<\/p><\/blockquote>\n

    Let E be the dual of V and let s : O_X → E be the dual section. As before we have to show that Y = RelativeSpec_X(Sym^*(E)\/(s – 1)) is affine.<\/p>\n

    Before we prove this, we can reduce to X integral and even normal. Namely, if X’ → X is finite surjective and if we can show that the base change of Y to X’ is affine, then it follows that Y is affine (see Tag 01ZT<\/a>). The reader can check that the situation after base change has all the same properties as the situation before base change.<\/p>\n

    OK, we continue the proof that Y is affine, but now with X normal irreducible. Arguing as in the case of schemes, we reduce to proving that given F coherent on X and ξ ∈ H^1(X, F) then for m large enough ξ maps to zero in H^1(X, F ⊗ Sym^m(E)). Observe that Sym^m(E) is dual to the sheaf Syt^m(V) of symmetric tensors. Thus we consider the map<\/p>\n

    \nτ^m : Syt^m(π_*I) → O_X\n<\/p><\/blockquote>\n

    Then I claim<\/strong> given m_0 for all m ≫ m_0 the map τ^m is a sum of compositions<\/p>\n

    Syt^m(π_*I) → π_*(I^a) ⊗ Syt^{m – a}(π_*I) → O_X<\/p><\/blockquote>\n

    for a ≥ m_0. Here the first map comes from the multiplication maps and the second uses τ on the first tensor factor and some linear map Syt^{m – a}(π_*I) → O_X on the second tensor factor. It follows that F → F ⊗ Sym^m(E) is a sum of maps which factor through F → SheafHom(π_*(I^a), F) for some a ≥ m_0. Thus it suffices to show that<\/p>\n

    colim SheafHom(π_*(I^a), F) <\/p><\/blockquote>\n

    is the pushforward of a coherent module on U to get the desired vanishing (argument as in the schemes case). This holds because it is the pushforward of the pullback of F as follows from Deligne’s formula and ω_{U\/X} ≅ O_U. This finishes the sketch of the proof.<\/p>\n

    A word about the claim. To prove it we construct global maps over X and then we prove the equality on stalks at the generic point of X (this is why we reduced to the case where X is normal irreducible). Let us first explain what is happening in the generic point. Say L is a finite separable algebra of degree n over a field K. Let τ : L → K be the trace map. Denote Syt^m(L) the symmetric tensors in the mth tensor power of L over K. For m large enough we will write<\/p>\n

    \nτ^m : Syt^m(L) → K\n<\/p><\/blockquote>\n

    as a sum of maps<\/p>\n

    \nSyt^m(L) → L ⊗ Syt^{m – a}(L) → K\n<\/p><\/blockquote>\n

    for a ≥ m_0 where the first arrow uses the product on the first a tensors and the second map is of the form τ \\otimes f_{m – a} for some linear map f_{m – a} : Syt^{m – a}(L) → K. In fact f_{m – a}(alpha) will be a universal polynomial in the coefficients of the characteristic polynomial of alpha over K. Thus to prove this we may reduce to the case where K is algebraically closed. Then L = K x … x K is a product of n copies of K. In this case, the dual statement, in terms of symmetric polynomials, asks the following: we have to write the polynomial<\/p>\n

    (x_1 + … + x_n)^m <\/p><\/blockquote>\n

    as a sum of polynomials<\/p>\n

    (x_1^a + … + x_n^a) f_{m – a} <\/p><\/blockquote>\n

    for a ≥ m_0 where f_{m – a} is some element in the Z<\/strong>-algebra S of symmetric polynomials in x_1, …, x_n. Now this is possible because the element x_1 + … + x_n maps to a nilpotent element of<\/p>\n

    S \/(p_a; a ≥ m_0)<\/p><\/blockquote>\n

    as the reader easily verifies. To finish the proof, all that is required is to observe that the polynomials f_{m – a} do indeed produce linear maps Syt^{m – a}(π_*I) → O_X over X as X is normal (details omitted).<\/p>\n

    Thanks for reading!<\/p>\n

    Edit 6\/25\/2023.<\/strong> Burt Totaro emailed to say that one can deduce the existence of the Jouanolou device more directly from his and Gross’s papers. I agree. For example, we may using their papers write X as W\/GL_n with W quasi-affine and then use the “equivariant Jouanolou trick” in Section 3 of Burt’s paper to get the Jouanolou device for X. Or one can modify the arguments in Burt’s paper directly and get a simple, short, direct proof. I gave the argument above because for a while I’ve wanted to make the thing about powers of the trace map work but didn’t succeed until the writing of this blog post.<\/p>\n","protected":false},"excerpt":{"rendered":"

    Let X be a scheme or an algebraic space. A Jouanolou device is a morphism Y → X such that Y is an affine scheme and such that Y is a torsor for a vector bundle over X. A scheme … 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-4828","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\/4828","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=4828"}],"version-history":[{"count":75,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/4828\/revisions"}],"predecessor-version":[{"id":5008,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/4828\/revisions\/5008"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4828"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4828"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4828"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}