Mixing
Intuition and technique for (strict) Henselization of nodal cubic at node
$begingroup$
Consider the union of the axes $frac{Bbbk[x,y]}{(xy)}$. There are two irreducible components passing through the origin which correspond to the two minimal primes of the local ring at the origin.
Now consider the nodal cubic $frac{Bbbk[x,y]}{(y^2-x^2(x+1))}$. There are two "branches" passing through the node, but they are not seen by the local ring at the origin, which is a domain (as the localization of a domain).
In this comment, Roland explains this issue is resolved by passing to the (strict) Henselization of the local ring at the origin, which does have two minimal primes.
How to calculate the (strict) Henselization of the local ring of the nodal cubic at the node? (How to calculate Henselizations in general?)
Thinking of (strict) Henselizations as universal covers, why intuitively does the (strict) Henselization of the node acquire the "missing" minimal prime?
algebraic-geometry commutative-algebra affine-schemes hensels-lemma
asked Jan 25 at 16:58
ArrowArrow
5,20131546
$endgroup$
add a comment |
$begingroup$
Consider the union of the axes $frac{Bbbk[x,y]}{(xy)}$. There are two irreducible components passing through the origin which correspond to the two minimal primes of the local ring at the origin.
Now consider the nodal cubic $frac{Bbbk[x,y]}{(y^2-x^2(x+1))}$. There are two "branches" passing through the node, but they are not seen by the local ring at the origin, which is a domain (as the localization of a domain).
In this comment, Roland explains this issue is resolved by passing to the (strict) Henselization of the local ring at the origin, which does have two minimal primes.
How to calculate the (strict) Henselization of the local ring of the nodal cubic at the node? (How to calculate Henselizations in general?)
Thinking of (strict) Henselizations as universal covers, why intuitively does the (strict) Henselization of the node acquire the "missing" minimal prime?
algebraic-geometry commutative-algebra affine-schemes hensels-lemma
asked Jan 25 at 16:58
ArrowArrow
5,20131546
$endgroup$
$begingroup$
Henselization is for local ring, so I don't understand what you mean by the Henselization of the entire nodal cubic.
$endgroup$
– Roland
Jan 25 at 17:15
$begingroup$
I mean separable closure in the sense of Magid's Galois theory of commutative rings. Perhaps this is not really a standard term, so I will edit the question.
$endgroup$
– Arrow
Jan 25 at 17:17
$begingroup$
Ok I will have a look at the book. Or maybe you can explain a bit its content, I think it will be helpful for your second question. Indeed, I believe that in 2. you want to speak about this "global henselization" (or separable closure).
$endgroup$
– Roland
Jan 25 at 17:40
$begingroup$
Dear @Roland, forgive me for bothering you, but do you by any chance have some time to answer the local aspect of this question? Thank you!
$endgroup$
– Arrow
Feb 19 at 10:05
$begingroup$
Sorry, I forgot this question. Also, I don't have a good answer to propose. For 1., I guess I have the intuition of what it should be and then I show that this works (but I did not have to do it very often so...). For 2. I don't know because I don't think of Henselization as universal cover. I guess I could say something along the line : in classical topology, a space is locally contractible. In Zariski topology, this is not true, but "taking a universal cover of a local neighborhood" may resolve the issue (we kill the $pi_1$ of the nodal cubic so there is only the two branches intersecting).
$endgroup$
– Roland
Feb 22 at 8:23
add a comment |
$begingroup$
Consider the union of the axes $frac{Bbbk[x,y]}{(xy)}$. There are two irreducible components passing through the origin which correspond to the two minimal primes of the local ring at the origin.
Now consider the nodal cubic $frac{Bbbk[x,y]}{(y^2-x^2(x+1))}$. There are two "branches" passing through the node, but they are not seen by the local ring at the origin, which is a domain (as the localization of a domain).
In this comment, Roland explains this issue is resolved by passing to the (strict) Henselization of the local ring at the origin, which does have two minimal primes.
How to calculate the (strict) Henselization of the local ring of the nodal cubic at the node? (How to calculate Henselizations in general?)
Thinking of (strict) Henselizations as universal covers, why intuitively does the (strict) Henselization of the node acquire the "missing" minimal prime?
algebraic-geometry commutative-algebra affine-schemes hensels-lemma
asked Jan 25 at 16:58
ArrowArrow
5,20131546
$endgroup$
Consider the union of the axes $frac{Bbbk[x,y]}{(xy)}$. There are two irreducible components passing through the origin which correspond to the two minimal primes of the local ring at the origin.
Now consider the nodal cubic $frac{Bbbk[x,y]}{(y^2-x^2(x+1))}$. There are two "branches" passing through the node, but they are not seen by the local ring at the origin, which is a domain (as the localization of a domain).
In this comment, Roland explains this issue is resolved by passing to the (strict) Henselization of the local ring at the origin, which does have two minimal primes.
How to calculate the (strict) Henselization of the local ring of the nodal cubic at the node? (How to calculate Henselizations in general?)
Thinking of (strict) Henselizations as universal covers, why intuitively does the (strict) Henselization of the node acquire the "missing" minimal prime?
algebraic-geometry commutative-algebra affine-schemes hensels-lemma
algebraic-geometry commutative-algebra affine-schemes hensels-lemma
asked Jan 25 at 16:58
ArrowArrow
5,20131546
asked Jan 25 at 16:58
ArrowArrow
5,20131546
edited Jan 25 at 17:18
Arrow
asked Jan 25 at 16:58
ArrowArrow
5,20131546
asked Jan 25 at 16:58
ArrowArrow
5,20131546
asked Jan 25 at 16:58
ArrowArrow
5,20131546
5,20131546
$begingroup$
Henselization is for local ring, so I don't understand what you mean by the Henselization of the entire nodal cubic.
$endgroup$
– Roland
Jan 25 at 17:15
$begingroup$
I mean separable closure in the sense of Magid's Galois theory of commutative rings. Perhaps this is not really a standard term, so I will edit the question.
$endgroup$
– Arrow
Jan 25 at 17:17
$begingroup$
Ok I will have a look at the book. Or maybe you can explain a bit its content, I think it will be helpful for your second question. Indeed, I believe that in 2. you want to speak about this "global henselization" (or separable closure).
$endgroup$
– Roland
Jan 25 at 17:40
$begingroup$
Dear @Roland, forgive me for bothering you, but do you by any chance have some time to answer the local aspect of this question? Thank you!
$endgroup$
– Arrow
Feb 19 at 10:05
$begingroup$
Sorry, I forgot this question. Also, I don't have a good answer to propose. For 1., I guess I have the intuition of what it should be and then I show that this works (but I did not have to do it very often so...). For 2. I don't know because I don't think of Henselization as universal cover. I guess I could say something along the line : in classical topology, a space is locally contractible. In Zariski topology, this is not true, but "taking a universal cover of a local neighborhood" may resolve the issue (we kill the $pi_1$ of the nodal cubic so there is only the two branches intersecting).
$endgroup$
– Roland
Feb 22 at 8:23
add a comment |
$begingroup$
Henselization is for local ring, so I don't understand what you mean by the Henselization of the entire nodal cubic.
$endgroup$
– Roland
Jan 25 at 17:15
$begingroup$
I mean separable closure in the sense of Magid's Galois theory of commutative rings. Perhaps this is not really a standard term, so I will edit the question.
$endgroup$
– Arrow
Jan 25 at 17:17
$begingroup$
Ok I will have a look at the book. Or maybe you can explain a bit its content, I think it will be helpful for your second question. Indeed, I believe that in 2. you want to speak about this "global henselization" (or separable closure).
$endgroup$
– Roland
Jan 25 at 17:40
$begingroup$
Dear @Roland, forgive me for bothering you, but do you by any chance have some time to answer the local aspect of this question? Thank you!
$endgroup$
– Arrow
Feb 19 at 10:05
$begingroup$
Sorry, I forgot this question. Also, I don't have a good answer to propose. For 1., I guess I have the intuition of what it should be and then I show that this works (but I did not have to do it very often so...). For 2. I don't know because I don't think of Henselization as universal cover. I guess I could say something along the line : in classical topology, a space is locally contractible. In Zariski topology, this is not true, but "taking a universal cover of a local neighborhood" may resolve the issue (we kill the $pi_1$ of the nodal cubic so there is only the two branches intersecting).
$endgroup$
– Roland
Feb 22 at 8:23
$begingroup$
Henselization is for local ring, so I don't understand what you mean by the Henselization of the entire nodal cubic.
$endgroup$
– Roland
Jan 25 at 17:15
$begingroup$
Henselization is for local ring, so I don't understand what you mean by the Henselization of the entire nodal cubic.
$endgroup$
– Roland
Jan 25 at 17:15
$begingroup$
I mean separable closure in the sense of Magid's Galois theory of commutative rings. Perhaps this is not really a standard term, so I will edit the question.
$endgroup$
– Arrow
Jan 25 at 17:17
$begingroup$
I mean separable closure in the sense of Magid's Galois theory of commutative rings. Perhaps this is not really a standard term, so I will edit the question.
$endgroup$
– Arrow
Jan 25 at 17:17
$begingroup$
Ok I will have a look at the book. Or maybe you can explain a bit its content, I think it will be helpful for your second question. Indeed, I believe that in 2. you want to speak about this "global henselization" (or separable closure).
$endgroup$
– Roland
Jan 25 at 17:40
$begingroup$
Ok I will have a look at the book. Or maybe you can explain a bit its content, I think it will be helpful for your second question. Indeed, I believe that in 2. you want to speak about this "global henselization" (or separable closure).
$endgroup$
– Roland
Jan 25 at 17:40
$begingroup$
Dear @Roland, forgive me for bothering you, but do you by any chance have some time to answer the local aspect of this question? Thank you!
$endgroup$
– Arrow
Feb 19 at 10:05
$begingroup$
Dear @Roland, forgive me for bothering you, but do you by any chance have some time to answer the local aspect of this question? Thank you!
$endgroup$
– Arrow
Feb 19 at 10:05
$begingroup$
Sorry, I forgot this question. Also, I don't have a good answer to propose. For 1., I guess I have the intuition of what it should be and then I show that this works (but I did not have to do it very often so...). For 2. I don't know because I don't think of Henselization as universal cover. I guess I could say something along the line : in classical topology, a space is locally contractible. In Zariski topology, this is not true, but "taking a universal cover of a local neighborhood" may resolve the issue (we kill the $pi_1$ of the nodal cubic so there is only the two branches intersecting).
$endgroup$
– Roland
Feb 22 at 8:23
$begingroup$
Sorry, I forgot this question. Also, I don't have a good answer to propose. For 1., I guess I have the intuition of what it should be and then I show that this works (but I did not have to do it very often so...). For 2. I don't know because I don't think of Henselization as universal cover. I guess I could say something along the line : in classical topology, a space is locally contractible. In Zariski topology, this is not true, but "taking a universal cover of a local neighborhood" may resolve the issue (we kill the $pi_1$ of the nodal cubic so there is only the two branches intersecting).
$endgroup$
– Roland
Feb 22 at 8:23
add a comment |
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$begingroup$
Henselization is for local ring, so I don't understand what you mean by the Henselization of the entire nodal cubic.
$endgroup$
– Roland
Jan 25 at 17:15
$begingroup$
I mean separable closure in the sense of Magid's Galois theory of commutative rings. Perhaps this is not really a standard term, so I will edit the question.
$endgroup$
– Arrow
Jan 25 at 17:17
$begingroup$
Ok I will have a look at the book. Or maybe you can explain a bit its content, I think it will be helpful for your second question. Indeed, I believe that in 2. you want to speak about this "global henselization" (or separable closure).
$endgroup$
– Roland
Jan 25 at 17:40
$begingroup$
Dear @Roland, forgive me for bothering you, but do you by any chance have some time to answer the local aspect of this question? Thank you!
$endgroup$
– Arrow
Feb 19 at 10:05
$begingroup$
Sorry, I forgot this question. Also, I don't have a good answer to propose. For 1., I guess I have the intuition of what it should be and then I show that this works (but I did not have to do it very often so...). For 2. I don't know because I don't think of Henselization as universal cover. I guess I could say something along the line : in classical topology, a space is locally contractible. In Zariski topology, this is not true, but "taking a universal cover of a local neighborhood" may resolve the issue (we kill the $pi_1$ of the nodal cubic so there is only the two branches intersecting).
$endgroup$
– Roland
Feb 22 at 8:23