The ring of global sections of a regular schemeGlobal sections of flat scheme also flat?Embedding of a scheme into a regular schemeIf $X,Y$ are regular and of finite type over $S$, can $Xtimes _S Y$ be embedded into a regular $S$-scheme? What are the local properties of schemes preserved under global sections?Sections of morphisms up to fppf coveringFinite generation of global sections of an invertible sheaf on a quasi-projective schemeSchemes monomorphing into affine scheme of dimension 1The underlying space of a scheme remembers its affineness?Size of the ring of functions on open subschemesGeometric regularity for infinitely generated field extensions

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The ring of global sections of a regular scheme


Global sections of flat scheme also flat?Embedding of a scheme into a regular schemeIf $X,Y$ are regular and of finite type over $S$, can $Xtimes _S Y$ be embedded into a regular $S$-scheme? What are the local properties of schemes preserved under global sections?Sections of morphisms up to fppf coveringFinite generation of global sections of an invertible sheaf on a quasi-projective schemeSchemes monomorphing into affine scheme of dimension 1The underlying space of a scheme remembers its affineness?Size of the ring of functions on open subschemesGeometric regularity for infinitely generated field extensions













3












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Let $X$ be a Noetherian regular scheme. Is $mathcalO_X(X)$ a regular ring? For affine schemes this is true, see 02IU on the Stacks project.










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  • 6




    $begingroup$
    Probably related: meta.mathoverflow.net/questions/4200/flood-of-new-users
    $endgroup$
    – YCor
    May 31 at 13:57















3












$begingroup$


Let $X$ be a Noetherian regular scheme. Is $mathcalO_X(X)$ a regular ring? For affine schemes this is true, see 02IU on the Stacks project.










share|cite|improve this question











$endgroup$







  • 6




    $begingroup$
    Probably related: meta.mathoverflow.net/questions/4200/flood-of-new-users
    $endgroup$
    – YCor
    May 31 at 13:57













3












3








3


1



$begingroup$


Let $X$ be a Noetherian regular scheme. Is $mathcalO_X(X)$ a regular ring? For affine schemes this is true, see 02IU on the Stacks project.










share|cite|improve this question











$endgroup$




Let $X$ be a Noetherian regular scheme. Is $mathcalO_X(X)$ a regular ring? For affine schemes this is true, see 02IU on the Stacks project.







ag.algebraic-geometry schemes regular-rings






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edited May 31 at 13:59









Wojowu

8,03613462




8,03613462










asked May 31 at 13:45







user141316














  • 6




    $begingroup$
    Probably related: meta.mathoverflow.net/questions/4200/flood-of-new-users
    $endgroup$
    – YCor
    May 31 at 13:57












  • 6




    $begingroup$
    Probably related: meta.mathoverflow.net/questions/4200/flood-of-new-users
    $endgroup$
    – YCor
    May 31 at 13:57







6




6




$begingroup$
Probably related: meta.mathoverflow.net/questions/4200/flood-of-new-users
$endgroup$
– YCor
May 31 at 13:57




$begingroup$
Probably related: meta.mathoverflow.net/questions/4200/flood-of-new-users
$endgroup$
– YCor
May 31 at 13:57










1 Answer
1






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oldest

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13












$begingroup$

The answer is no. For instance, take the quadratic cone
$$
Y = mathrmSpec(Bbbk[x,y,z]/(xz-y^2))
$$

and let $X$ be its blowup at the vertex. Then $X$ is regular, but
$$
H^0(X,mathcalO_X) = H^0(Y,mathcalO_Y) = Bbbk[x,y,z]/(xz-y^2)
$$

is not.






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    1 Answer
    1






    active

    oldest

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    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    13












    $begingroup$

    The answer is no. For instance, take the quadratic cone
    $$
    Y = mathrmSpec(Bbbk[x,y,z]/(xz-y^2))
    $$

    and let $X$ be its blowup at the vertex. Then $X$ is regular, but
    $$
    H^0(X,mathcalO_X) = H^0(Y,mathcalO_Y) = Bbbk[x,y,z]/(xz-y^2)
    $$

    is not.






    share|cite|improve this answer









    $endgroup$

















      13












      $begingroup$

      The answer is no. For instance, take the quadratic cone
      $$
      Y = mathrmSpec(Bbbk[x,y,z]/(xz-y^2))
      $$

      and let $X$ be its blowup at the vertex. Then $X$ is regular, but
      $$
      H^0(X,mathcalO_X) = H^0(Y,mathcalO_Y) = Bbbk[x,y,z]/(xz-y^2)
      $$

      is not.






      share|cite|improve this answer









      $endgroup$















        13












        13








        13





        $begingroup$

        The answer is no. For instance, take the quadratic cone
        $$
        Y = mathrmSpec(Bbbk[x,y,z]/(xz-y^2))
        $$

        and let $X$ be its blowup at the vertex. Then $X$ is regular, but
        $$
        H^0(X,mathcalO_X) = H^0(Y,mathcalO_Y) = Bbbk[x,y,z]/(xz-y^2)
        $$

        is not.






        share|cite|improve this answer









        $endgroup$



        The answer is no. For instance, take the quadratic cone
        $$
        Y = mathrmSpec(Bbbk[x,y,z]/(xz-y^2))
        $$

        and let $X$ be its blowup at the vertex. Then $X$ is regular, but
        $$
        H^0(X,mathcalO_X) = H^0(Y,mathcalO_Y) = Bbbk[x,y,z]/(xz-y^2)
        $$

        is not.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered May 31 at 15:54









        SashaSasha

        21.9k22860




        21.9k22860



























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