Maximal Ideal of the Stalk generated by Linear Forms The 2019 Stack Overflow Developer Survey Results Are InHartshorne Lemma I.6.5; Why is $mathfrakm_Rcap Bneq 0$?Function field question from Silverman's AECA question about valuations (in the proof of Hartshorne's proposition II6.2 )Why should this ideal be maximal?Smooth scheme of finite type over a field, some questionsTangent space of a scheme of finite type over a fieldProof of the Theorem of GrothendieckWhat is the relationship between different theorems all called Hilbert's Nullstellensatz?locality of a space with functions $mathcalO_X(U)$Why is the algebraic torus an affine variety?
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Maximal Ideal of the Stalk generated by Linear Forms
The 2019 Stack Overflow Developer Survey Results Are InHartshorne Lemma I.6.5; Why is $mathfrakm_Rcap Bneq 0$?Function field question from Silverman's AECA question about valuations (in the proof of Hartshorne's proposition II6.2 )Why should this ideal be maximal?Smooth scheme of finite type over a field, some questionsTangent space of a scheme of finite type over a fieldProof of the Theorem of GrothendieckWhat is the relationship between different theorems all called Hilbert's Nullstellensatz?locality of a space with functions $mathcalO_X(U)$Why is the algebraic torus an affine variety?
$begingroup$
I have a question about an argument used in Hartshorne's "Algebraic Geometry" in the excerpt below (or look up at page 179: Thm 8.18 Bertini's Thm):
Question: When we take a closed point $x$ an assume that $k$ is an algebraically closed field, why the maximal ideal $m_x$ (of the stalk $mathcalO_X,x$) is generated by linear forms in ther coordinates?
Could anybody explain what Hartshorne does here mean? How this identification between $m_x$ and the linear forms is realized? Why $k$ must be alg closed?
algebraic-geometry
asked Apr 8 at 1:34
KarlPeterKarlPeter
6771416
$endgroup$
add a comment |
$begingroup$
I have a question about an argument used in Hartshorne's "Algebraic Geometry" in the excerpt below (or look up at page 179: Thm 8.18 Bertini's Thm):
Question: When we take a closed point $x$ an assume that $k$ is an algebraically closed field, why the maximal ideal $m_x$ (of the stalk $mathcalO_X,x$) is generated by linear forms in ther coordinates?
Could anybody explain what Hartshorne does here mean? How this identification between $m_x$ and the linear forms is realized? Why $k$ must be alg closed?
algebraic-geometry
asked Apr 8 at 1:34
KarlPeterKarlPeter
6771416
$endgroup$
1
$begingroup$
This is an application of the Nullstellensatz. Pick one of the standard affine charts on $Bbb P^n$ which contains the point, then look at the maximal ideal of $x$ as a point in the standard open, which is generated by $fracx_jx_i-c_j$ for fixed $i$ and varying $j$ (by the Nullstellensatz). Since this ring surjects on to the coordinate ring of $X$ intersected with the standard affine open, the result follows. The field being algebraically closed is necessary to get the result on what the maximal ideals look like.
$endgroup$
– KReiser
Apr 8 at 6:14
add a comment |
$begingroup$
I have a question about an argument used in Hartshorne's "Algebraic Geometry" in the excerpt below (or look up at page 179: Thm 8.18 Bertini's Thm):
Question: When we take a closed point $x$ an assume that $k$ is an algebraically closed field, why the maximal ideal $m_x$ (of the stalk $mathcalO_X,x$) is generated by linear forms in ther coordinates?
Could anybody explain what Hartshorne does here mean? How this identification between $m_x$ and the linear forms is realized? Why $k$ must be alg closed?
algebraic-geometry
asked Apr 8 at 1:34
KarlPeterKarlPeter
6771416
$endgroup$
I have a question about an argument used in Hartshorne's "Algebraic Geometry" in the excerpt below (or look up at page 179: Thm 8.18 Bertini's Thm):
Question: When we take a closed point $x$ an assume that $k$ is an algebraically closed field, why the maximal ideal $m_x$ (of the stalk $mathcalO_X,x$) is generated by linear forms in ther coordinates?
Could anybody explain what Hartshorne does here mean? How this identification between $m_x$ and the linear forms is realized? Why $k$ must be alg closed?
algebraic-geometry
algebraic-geometry
asked Apr 8 at 1:34
KarlPeterKarlPeter
6771416
asked Apr 8 at 1:34
KarlPeterKarlPeter
6771416
asked Apr 8 at 1:34
KarlPeterKarlPeter
6771416
asked Apr 8 at 1:34
KarlPeterKarlPeter
6771416
asked Apr 8 at 1:34
KarlPeterKarlPeter
6771416
6771416
1
$begingroup$
This is an application of the Nullstellensatz. Pick one of the standard affine charts on $Bbb P^n$ which contains the point, then look at the maximal ideal of $x$ as a point in the standard open, which is generated by $fracx_jx_i-c_j$ for fixed $i$ and varying $j$ (by the Nullstellensatz). Since this ring surjects on to the coordinate ring of $X$ intersected with the standard affine open, the result follows. The field being algebraically closed is necessary to get the result on what the maximal ideals look like.
$endgroup$
– KReiser
Apr 8 at 6:14
add a comment |
1
$begingroup$
This is an application of the Nullstellensatz. Pick one of the standard affine charts on $Bbb P^n$ which contains the point, then look at the maximal ideal of $x$ as a point in the standard open, which is generated by $fracx_jx_i-c_j$ for fixed $i$ and varying $j$ (by the Nullstellensatz). Since this ring surjects on to the coordinate ring of $X$ intersected with the standard affine open, the result follows. The field being algebraically closed is necessary to get the result on what the maximal ideals look like.
$endgroup$
– KReiser
Apr 8 at 6:14
1
1
$begingroup$
This is an application of the Nullstellensatz. Pick one of the standard affine charts on $Bbb P^n$ which contains the point, then look at the maximal ideal of $x$ as a point in the standard open, which is generated by $fracx_jx_i-c_j$ for fixed $i$ and varying $j$ (by the Nullstellensatz). Since this ring surjects on to the coordinate ring of $X$ intersected with the standard affine open, the result follows. The field being algebraically closed is necessary to get the result on what the maximal ideals look like.
$endgroup$
– KReiser
Apr 8 at 6:14
$begingroup$
This is an application of the Nullstellensatz. Pick one of the standard affine charts on $Bbb P^n$ which contains the point, then look at the maximal ideal of $x$ as a point in the standard open, which is generated by $fracx_jx_i-c_j$ for fixed $i$ and varying $j$ (by the Nullstellensatz). Since this ring surjects on to the coordinate ring of $X$ intersected with the standard affine open, the result follows. The field being algebraically closed is necessary to get the result on what the maximal ideals look like.
$endgroup$
– KReiser
Apr 8 at 6:14
add a comment |
0
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$begingroup$
This is an application of the Nullstellensatz. Pick one of the standard affine charts on $Bbb P^n$ which contains the point, then look at the maximal ideal of $x$ as a point in the standard open, which is generated by $fracx_jx_i-c_j$ for fixed $i$ and varying $j$ (by the Nullstellensatz). Since this ring surjects on to the coordinate ring of $X$ intersected with the standard affine open, the result follows. The field being algebraically closed is necessary to get the result on what the maximal ideals look like.
$endgroup$
– KReiser
Apr 8 at 6:14