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
$begingroup$
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.
ag.algebraic-geometry schemes regular-rings
edited May 31 at 13:59
Wojowu
8,03613462
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add a comment |
$begingroup$
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.
ag.algebraic-geometry schemes regular-rings
edited May 31 at 13:59
Wojowu
8,03613462
$endgroup$
6
$begingroup$
Probably related: meta.mathoverflow.net/questions/4200/flood-of-new-users
$endgroup$
– YCor
May 31 at 13:57
add a comment |
$begingroup$
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.
ag.algebraic-geometry schemes regular-rings
edited May 31 at 13:59
Wojowu
8,03613462
$endgroup$
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
ag.algebraic-geometry schemes regular-rings
edited May 31 at 13:59
Wojowu
8,03613462
edited May 31 at 13:59
Wojowu
8,03613462
edited May 31 at 13:59
Wojowu
8,03613462
edited May 31 at 13:59
Wojowu
8,03613462
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
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– YCor
May 31 at 13:57
add a comment |
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
add a comment |
1 Answer
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oldest
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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.
answered May 31 at 15:54
SashaSasha
21.9k22860
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1 Answer
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$begingroup$
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.
answered May 31 at 15:54
SashaSasha
21.9k22860
$endgroup$
add a comment |
$begingroup$
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.
answered May 31 at 15:54
SashaSasha
21.9k22860
$endgroup$
add a comment |
$begingroup$
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.
answered May 31 at 15:54
SashaSasha
21.9k22860
$endgroup$
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.
answered May 31 at 15:54
SashaSasha
21.9k22860
answered May 31 at 15:54
SashaSasha
21.9k22860
answered May 31 at 15:54
SashaSasha
21.9k22860
answered May 31 at 15:54
SashaSasha
21.9k22860
21.9k22860
add a comment |
add a comment |
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Probably related: meta.mathoverflow.net/questions/4200/flood-of-new-users
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– YCor
May 31 at 13:57