global sections of locally free sheaf on projective space












4














Let $mathcal{E}$ be a locally free sheaf on $mathbb{P}^n_A=mathbb{P}^ntimes_{Spec k} Spec A$, where $A$ is a finitely generated algebra over a field $k$. By a well known theorem (see e.g. Hartshorne's Algebraic Geometry, Thm 5.19) $H^0(mathbb{P}^n_A, mathcal{E})$ is a finitely generated $A$- module.




  1. Is it true that $H^0(mathbb{P}^n_A, mathcal{E})$ is a projective module?


Let $B$ be a finitely generated $k$-algebra, $f: Spec B to Spec A$ a morphism and $f^*mathcal{E}$ the pullback of $mathcal{E}$ to $Spec B$.




  1. Is it true that $H^0(mathbb{P}^n_A, mathcal{E})otimes _A B= H^0(mathbb{P}^n_B, f^*mathcal{E})$ ?


1 and 2 above are true when $mathcal {E}$ is a direct sum of line bundles of the form $mathcal O(n)$. I was wondering if they are true for general $mathcal{E}$.






ag.algebraic-geometry vector-bundles







share|cite|improve this question




















  • 1




    Have a look at Base change for quasi-coherent sheaves.
    – abx
    Nov 27 '18 at 5:27
















4














Let $mathcal{E}$ be a locally free sheaf on $mathbb{P}^n_A=mathbb{P}^ntimes_{Spec k} Spec A$, where $A$ is a finitely generated algebra over a field $k$. By a well known theorem (see e.g. Hartshorne's Algebraic Geometry, Thm 5.19) $H^0(mathbb{P}^n_A, mathcal{E})$ is a finitely generated $A$- module.




  1. Is it true that $H^0(mathbb{P}^n_A, mathcal{E})$ is a projective module?


Let $B$ be a finitely generated $k$-algebra, $f: Spec B to Spec A$ a morphism and $f^*mathcal{E}$ the pullback of $mathcal{E}$ to $Spec B$.




  1. Is it true that $H^0(mathbb{P}^n_A, mathcal{E})otimes _A B= H^0(mathbb{P}^n_B, f^*mathcal{E})$ ?


1 and 2 above are true when $mathcal {E}$ is a direct sum of line bundles of the form $mathcal O(n)$. I was wondering if they are true for general $mathcal{E}$.






ag.algebraic-geometry vector-bundles







share|cite|improve this question




















  • 1




    Have a look at Base change for quasi-coherent sheaves.
    – abx
    Nov 27 '18 at 5:27














4












4








4


1





Let $mathcal{E}$ be a locally free sheaf on $mathbb{P}^n_A=mathbb{P}^ntimes_{Spec k} Spec A$, where $A$ is a finitely generated algebra over a field $k$. By a well known theorem (see e.g. Hartshorne's Algebraic Geometry, Thm 5.19) $H^0(mathbb{P}^n_A, mathcal{E})$ is a finitely generated $A$- module.




  1. Is it true that $H^0(mathbb{P}^n_A, mathcal{E})$ is a projective module?


Let $B$ be a finitely generated $k$-algebra, $f: Spec B to Spec A$ a morphism and $f^*mathcal{E}$ the pullback of $mathcal{E}$ to $Spec B$.




  1. Is it true that $H^0(mathbb{P}^n_A, mathcal{E})otimes _A B= H^0(mathbb{P}^n_B, f^*mathcal{E})$ ?


1 and 2 above are true when $mathcal {E}$ is a direct sum of line bundles of the form $mathcal O(n)$. I was wondering if they are true for general $mathcal{E}$.






ag.algebraic-geometry vector-bundles







share|cite|improve this question















Let $mathcal{E}$ be a locally free sheaf on $mathbb{P}^n_A=mathbb{P}^ntimes_{Spec k} Spec A$, where $A$ is a finitely generated algebra over a field $k$. By a well known theorem (see e.g. Hartshorne's Algebraic Geometry, Thm 5.19) $H^0(mathbb{P}^n_A, mathcal{E})$ is a finitely generated $A$- module.




  1. Is it true that $H^0(mathbb{P}^n_A, mathcal{E})$ is a projective module?


Let $B$ be a finitely generated $k$-algebra, $f: Spec B to Spec A$ a morphism and $f^*mathcal{E}$ the pullback of $mathcal{E}$ to $Spec B$.




  1. Is it true that $H^0(mathbb{P}^n_A, mathcal{E})otimes _A B= H^0(mathbb{P}^n_B, f^*mathcal{E})$ ?


1 and 2 above are true when $mathcal {E}$ is a direct sum of line bundles of the form $mathcal O(n)$. I was wondering if they are true for general $mathcal{E}$.






ag.algebraic-geometry vector-bundles




ag.algebraic-geometry vector-bundles






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Nov 26 '18 at 22:30

























asked Nov 26 '18 at 22:23









mike

263




263








  • 1




    Have a look at Base change for quasi-coherent sheaves.
    – abx
    Nov 27 '18 at 5:27














  • 1




    Have a look at Base change for quasi-coherent sheaves.
    – abx
    Nov 27 '18 at 5:27








1




1




Have a look at Base change for quasi-coherent sheaves.
– abx
Nov 27 '18 at 5:27




Have a look at Base change for quasi-coherent sheaves.
– abx
Nov 27 '18 at 5:27










1 Answer
1






active

oldest

votes


















8














The answer to both questions is negative, see counterexamples below.



1) Let $A = k[x,y,z]$, $n = 1$. Note that
$$
H^1(mathbb{P}^n_A,O(-2)) cong A.
$$
Consider the extension
$$
0 to O(-2) to E to O oplus O oplus O to 0
$$
whose extension class is $(x,y,z)$. Then the cohomolopgy exact sequence
$$
0 to H^0(mathbb{P}^n_A,E) to A oplus A oplus A stackrel{(x,y,z)}to A
$$
shows that $H^0(mathbb{P}^n_A,E)$ is reflexive but not locally free (hence not projective). In fact, this is the simplest example of a reflexive non-locally free sheaf.



2) Take $A = k[x,y,z,w]$ and define $E$ as the extension
$$
0 to O(-2) to E to O oplus O oplus O oplus O to 0
$$
whose extension class is $(x,y,z,w)$. Let $B = k$ with the morphism $A to B$ defined by $x,y,z,w mapsto 0$. Then $f^*E cong O(-2) oplus O oplus O oplus O oplus O$, hence
$$
H^0(mathbb{P}^n_B,f^*E) = B oplus B oplus B oplus B.
$$
On the other hand, tensoring
$$
0 to H^0(mathbb{P}^n_A,E) to A oplus A oplus A oplus A stackrel{(x,y,z,w)}to A to B to 0
$$
by $B$ (over $A$), we deduce
$$
H^0(mathbb{P}^n_A,E) otimes_A B cong Tor_2^A(B,B) cong B^{oplus 6}.
$$






share|cite|improve this answer





















    Your Answer





    StackExchange.ifUsing("editor", function () {
    return StackExchange.using("mathjaxEditing", function () {
    StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
    StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
    });
    });
    }, "mathjax-editing");

    StackExchange.ready(function() {
    var channelOptions = {
    tags: "".split(" "),
    id: "504"
    };
    initTagRenderer("".split(" "), "".split(" "), channelOptions);

    StackExchange.using("externalEditor", function() {
    // Have to fire editor after snippets, if snippets enabled
    if (StackExchange.settings.snippets.snippetsEnabled) {
    StackExchange.using("snippets", function() {
    createEditor();
    });
    }
    else {
    createEditor();
    }
    });

    function createEditor() {
    StackExchange.prepareEditor({
    heartbeatType: 'answer',
    autoActivateHeartbeat: false,
    convertImagesToLinks: true,
    noModals: true,
    showLowRepImageUploadWarning: true,
    reputationToPostImages: 10,
    bindNavPrevention: true,
    postfix: "",
    imageUploader: {
    brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
    contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
    allowUrls: true
    },
    noCode: true, onDemand: true,
    discardSelector: ".discard-answer"
    ,immediatelyShowMarkdownHelp:true
    });


    }
    });














    draft saved

    draft discarded


















    StackExchange.ready(
    function () {
    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f316285%2fglobal-sections-of-locally-free-sheaf-on-projective-space%23new-answer', 'question_page');
    }
    );

    Post as a guest















    Required, but never shown

























    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    8














    The answer to both questions is negative, see counterexamples below.



    1) Let $A = k[x,y,z]$, $n = 1$. Note that
    $$
    H^1(mathbb{P}^n_A,O(-2)) cong A.
    $$
    Consider the extension
    $$
    0 to O(-2) to E to O oplus O oplus O to 0
    $$
    whose extension class is $(x,y,z)$. Then the cohomolopgy exact sequence
    $$
    0 to H^0(mathbb{P}^n_A,E) to A oplus A oplus A stackrel{(x,y,z)}to A
    $$
    shows that $H^0(mathbb{P}^n_A,E)$ is reflexive but not locally free (hence not projective). In fact, this is the simplest example of a reflexive non-locally free sheaf.



    2) Take $A = k[x,y,z,w]$ and define $E$ as the extension
    $$
    0 to O(-2) to E to O oplus O oplus O oplus O to 0
    $$
    whose extension class is $(x,y,z,w)$. Let $B = k$ with the morphism $A to B$ defined by $x,y,z,w mapsto 0$. Then $f^*E cong O(-2) oplus O oplus O oplus O oplus O$, hence
    $$
    H^0(mathbb{P}^n_B,f^*E) = B oplus B oplus B oplus B.
    $$
    On the other hand, tensoring
    $$
    0 to H^0(mathbb{P}^n_A,E) to A oplus A oplus A oplus A stackrel{(x,y,z,w)}to A to B to 0
    $$
    by $B$ (over $A$), we deduce
    $$
    H^0(mathbb{P}^n_A,E) otimes_A B cong Tor_2^A(B,B) cong B^{oplus 6}.
    $$






    share|cite|improve this answer


























      8














      The answer to both questions is negative, see counterexamples below.



      1) Let $A = k[x,y,z]$, $n = 1$. Note that
      $$
      H^1(mathbb{P}^n_A,O(-2)) cong A.
      $$
      Consider the extension
      $$
      0 to O(-2) to E to O oplus O oplus O to 0
      $$
      whose extension class is $(x,y,z)$. Then the cohomolopgy exact sequence
      $$
      0 to H^0(mathbb{P}^n_A,E) to A oplus A oplus A stackrel{(x,y,z)}to A
      $$
      shows that $H^0(mathbb{P}^n_A,E)$ is reflexive but not locally free (hence not projective). In fact, this is the simplest example of a reflexive non-locally free sheaf.



      2) Take $A = k[x,y,z,w]$ and define $E$ as the extension
      $$
      0 to O(-2) to E to O oplus O oplus O oplus O to 0
      $$
      whose extension class is $(x,y,z,w)$. Let $B = k$ with the morphism $A to B$ defined by $x,y,z,w mapsto 0$. Then $f^*E cong O(-2) oplus O oplus O oplus O oplus O$, hence
      $$
      H^0(mathbb{P}^n_B,f^*E) = B oplus B oplus B oplus B.
      $$
      On the other hand, tensoring
      $$
      0 to H^0(mathbb{P}^n_A,E) to A oplus A oplus A oplus A stackrel{(x,y,z,w)}to A to B to 0
      $$
      by $B$ (over $A$), we deduce
      $$
      H^0(mathbb{P}^n_A,E) otimes_A B cong Tor_2^A(B,B) cong B^{oplus 6}.
      $$






      share|cite|improve this answer
























        8












        8








        8






        The answer to both questions is negative, see counterexamples below.



        1) Let $A = k[x,y,z]$, $n = 1$. Note that
        $$
        H^1(mathbb{P}^n_A,O(-2)) cong A.
        $$
        Consider the extension
        $$
        0 to O(-2) to E to O oplus O oplus O to 0
        $$
        whose extension class is $(x,y,z)$. Then the cohomolopgy exact sequence
        $$
        0 to H^0(mathbb{P}^n_A,E) to A oplus A oplus A stackrel{(x,y,z)}to A
        $$
        shows that $H^0(mathbb{P}^n_A,E)$ is reflexive but not locally free (hence not projective). In fact, this is the simplest example of a reflexive non-locally free sheaf.



        2) Take $A = k[x,y,z,w]$ and define $E$ as the extension
        $$
        0 to O(-2) to E to O oplus O oplus O oplus O to 0
        $$
        whose extension class is $(x,y,z,w)$. Let $B = k$ with the morphism $A to B$ defined by $x,y,z,w mapsto 0$. Then $f^*E cong O(-2) oplus O oplus O oplus O oplus O$, hence
        $$
        H^0(mathbb{P}^n_B,f^*E) = B oplus B oplus B oplus B.
        $$
        On the other hand, tensoring
        $$
        0 to H^0(mathbb{P}^n_A,E) to A oplus A oplus A oplus A stackrel{(x,y,z,w)}to A to B to 0
        $$
        by $B$ (over $A$), we deduce
        $$
        H^0(mathbb{P}^n_A,E) otimes_A B cong Tor_2^A(B,B) cong B^{oplus 6}.
        $$






        share|cite|improve this answer












        The answer to both questions is negative, see counterexamples below.



        1) Let $A = k[x,y,z]$, $n = 1$. Note that
        $$
        H^1(mathbb{P}^n_A,O(-2)) cong A.
        $$
        Consider the extension
        $$
        0 to O(-2) to E to O oplus O oplus O to 0
        $$
        whose extension class is $(x,y,z)$. Then the cohomolopgy exact sequence
        $$
        0 to H^0(mathbb{P}^n_A,E) to A oplus A oplus A stackrel{(x,y,z)}to A
        $$
        shows that $H^0(mathbb{P}^n_A,E)$ is reflexive but not locally free (hence not projective). In fact, this is the simplest example of a reflexive non-locally free sheaf.



        2) Take $A = k[x,y,z,w]$ and define $E$ as the extension
        $$
        0 to O(-2) to E to O oplus O oplus O oplus O to 0
        $$
        whose extension class is $(x,y,z,w)$. Let $B = k$ with the morphism $A to B$ defined by $x,y,z,w mapsto 0$. Then $f^*E cong O(-2) oplus O oplus O oplus O oplus O$, hence
        $$
        H^0(mathbb{P}^n_B,f^*E) = B oplus B oplus B oplus B.
        $$
        On the other hand, tensoring
        $$
        0 to H^0(mathbb{P}^n_A,E) to A oplus A oplus A oplus A stackrel{(x,y,z,w)}to A to B to 0
        $$
        by $B$ (over $A$), we deduce
        $$
        H^0(mathbb{P}^n_A,E) otimes_A B cong Tor_2^A(B,B) cong B^{oplus 6}.
        $$







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Nov 27 '18 at 0:01









        Sasha

        20.2k22654




        20.2k22654






























            draft saved

            draft discarded




















































            Thanks for contributing an answer to MathOverflow!


            • Please be sure to answer the question. Provide details and share your research!

            But avoid …



            • Asking for help, clarification, or responding to other answers.

            • Making statements based on opinion; back them up with references or personal experience.


            Use MathJax to format equations. MathJax reference.


            To learn more, see our tips on writing great answers.





            Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


            Please pay close attention to the following guidance:


            • Please be sure to answer the question. Provide details and share your research!

            But avoid …



            • Asking for help, clarification, or responding to other answers.

            • Making statements based on opinion; back them up with references or personal experience.


            To learn more, see our tips on writing great answers.




            draft saved


            draft discarded














            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f316285%2fglobal-sections-of-locally-free-sheaf-on-projective-space%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown





















































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown

































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown







            -

            Popular posts from this blog

            Plaza Victoria

            Image
            Plaza Victoria Plaza Victoria en 2017 Otros nombres Plaza de La Victoria Localización El Almendral, Valparaíso, Chile Vías adyacentes Chacabuco, Molina, Plaza Victoria y Edwards Creación 1841 Metro Bellavista Ubicación 33°02′46″S 71°37′11″O  /  -33.04624167, -71.61980833 Coordenadas: 33°02′46″S 71°37′11″O  /  -33.04624167, -71.61980833 [editar datos en Wikidata] La Plaza Victoria , también llamada Plaza de la Victoria , [ 1 ] ​ es una plaza ubicada en el sector El Almendral del plan de la ciudad de Valparaíso, Chile, de importancia patrimonial, que incluye valiosas estatuas y especies vegetales. [ 2 ] ​ La plaza se creó como tal en la primera mitad del siglo XIX, tomando previamente los nombres de Plaza Nueva y Plaza de Orrego . Si bien desde sus inicios tuvo relevancia histórica, no fue sino a partir de fines de los años 1860 cuando comenzó a cobrar mayor importancia y se convirtió en uno de los principales centros sociales de l...
            Read more

            Puebla de Zaragoza

            Image
            Para otros usos de este término, véase Puebla (desambiguación). Heroica Puebla de Zaragoza Localidad De arriba a abajo de izquierda a derecha: Estrella de Puebla, Volcán Popocatépetl, Catedral de Puebla, Monumento a Ignacio Zaragoza, Estadio Cuauhtémoc, Angelópolis y Vista panorámica de la ciudad. Escudo Otros nombres: El relicario de América/ La ciudad de los Ángeles/ La Angelópolis Lema: Muy noble y muy leal ciudad de Puebla de los Ángeles Heroica Puebla de Zaragoza Localización de Heroica Puebla de Zaragoza en México Heroica Puebla de Zaragoza Localización de Heroica Puebla de Zaragoza en Puebla Mapa interactivo Coordenadas 19°03′05″N 98°13′04″O  /  19.051388888889, -98.217777777778 Coordenadas: 19°03′05″N 98°13′04″O  /  19.051388888889, -98.217777777778 Idioma oficial Español Entidad Localidad  • País   México  • Estado Puebla  • Municipio Puebla Presidente municipal Claudia Rivera Vivanc...
            Read more

            Musa

            Image
            Para el género de plantas, véase Musa (planta). Apolo y la Nueve Musas En la mitología griega, las musas (en griego antiguo μοῦσαι « mousai» ) son, según los escritores más antiguos, las divinidades inspiradoras de las Artes: cada una de ellas está relacionada con ramas artísticas y del conocimiento. Son hijas de Zeus y de Mnemósine, compañeras del séquito de Apolo, dios olímpico de la música y patrón de las bellas artes, quien tuvo romances con cada una de ellas, dejando descendientes. Bajaban a la tierra a susurrar ideas e inspirar a aquellos mortales que las invocaran. En la época más arcaica eran las ninfas inspiradoras de las fuentes, en las cuales eran adoradas. Finalmente, alrededor de los siglos VIII-VII a. C. [ 1 ] ​ prevaleció en todo el territorio de la Hélade la adoración de las nueve Musas, que son Calíope, Clío, Erato, Euterpe, Melpómene, Polimnia, Talía, Terpsícore y Urania. El culto a las musas era originalmente de Tracia y Beocia, y fueron de vital impo...
            Read more