Find maximal ideals in semi local ring with singular
$begingroup$
I am trying to analyze the normalization $N$ of a local ring $A_{mathfrak{m}}$ of a variety with Singular. The normalization (integral closure in its total ring of fractions) is semi-local with its maximal ideals lying over $mathfrak{m}A_{mathfrak{m}}$.
According to
https://www.singular.uni-kl.de/Manual/4-0-3/sing_1270.htm#SEC1345
I can calculate a normalization of a local ring $k[x_1, ldots x_n]_{(0)}/I_{(0)}$ and represent it (if i understood correctly) as a factor ring of a polynomial ring $B[y_1, ldots, y_s]$, where $B = k[x_1, ldots x_n]_{(0)}$.
I am doing it like this
LIB "normal.lib";
ring r=0,(x,y),ds;
ideal I = y^2 - x^2*(x+1);
list nor = normal(I);
def R = nor[1][1];
setring R;
norid;
R;
How would I go about finding the maximal ideals in this factor ring and how can they be represented in Singular. For example I would like to know how many there are (for more general examples than this simple node) and if $mathfrak{n}$ is one of those ideals, how can I get to $N_{mathfrak{n}}$. Is this possible? I am aware that this is not a really specific question, sorry. I have unfortunately just a limited knowledge of working with Singular and local orderings. A small hint or a pointer to some resources would be greatly appreciated.
algebraic-geometry commutative-algebra localization singularity formal-completions
asked Nov 28 '18 at 17:38
pyrogenpyrogen
714
$endgroup$
add a comment |
$begingroup$
I am trying to analyze the normalization $N$ of a local ring $A_{mathfrak{m}}$ of a variety with Singular. The normalization (integral closure in its total ring of fractions) is semi-local with its maximal ideals lying over $mathfrak{m}A_{mathfrak{m}}$.
According to
https://www.singular.uni-kl.de/Manual/4-0-3/sing_1270.htm#SEC1345
I can calculate a normalization of a local ring $k[x_1, ldots x_n]_{(0)}/I_{(0)}$ and represent it (if i understood correctly) as a factor ring of a polynomial ring $B[y_1, ldots, y_s]$, where $B = k[x_1, ldots x_n]_{(0)}$.
I am doing it like this
LIB "normal.lib";
ring r=0,(x,y),ds;
ideal I = y^2 - x^2*(x+1);
list nor = normal(I);
def R = nor[1][1];
setring R;
norid;
R;
How would I go about finding the maximal ideals in this factor ring and how can they be represented in Singular. For example I would like to know how many there are (for more general examples than this simple node) and if $mathfrak{n}$ is one of those ideals, how can I get to $N_{mathfrak{n}}$. Is this possible? I am aware that this is not a really specific question, sorry. I have unfortunately just a limited knowledge of working with Singular and local orderings. A small hint or a pointer to some resources would be greatly appreciated.
algebraic-geometry commutative-algebra localization singularity formal-completions
asked Nov 28 '18 at 17:38
pyrogenpyrogen
714
$endgroup$
$begingroup$
I don't know how to do it in Singular. But if you have a particular maximal ideal in mind, then extend this to the normalization and compute its associated primes in the normalization. Those are the maximal ideals lying over the one you extended.
$endgroup$
– Youngsu
Nov 28 '18 at 21:28
add a comment |
$begingroup$
I am trying to analyze the normalization $N$ of a local ring $A_{mathfrak{m}}$ of a variety with Singular. The normalization (integral closure in its total ring of fractions) is semi-local with its maximal ideals lying over $mathfrak{m}A_{mathfrak{m}}$.
According to
https://www.singular.uni-kl.de/Manual/4-0-3/sing_1270.htm#SEC1345
I can calculate a normalization of a local ring $k[x_1, ldots x_n]_{(0)}/I_{(0)}$ and represent it (if i understood correctly) as a factor ring of a polynomial ring $B[y_1, ldots, y_s]$, where $B = k[x_1, ldots x_n]_{(0)}$.
I am doing it like this
LIB "normal.lib";
ring r=0,(x,y),ds;
ideal I = y^2 - x^2*(x+1);
list nor = normal(I);
def R = nor[1][1];
setring R;
norid;
R;
How would I go about finding the maximal ideals in this factor ring and how can they be represented in Singular. For example I would like to know how many there are (for more general examples than this simple node) and if $mathfrak{n}$ is one of those ideals, how can I get to $N_{mathfrak{n}}$. Is this possible? I am aware that this is not a really specific question, sorry. I have unfortunately just a limited knowledge of working with Singular and local orderings. A small hint or a pointer to some resources would be greatly appreciated.
algebraic-geometry commutative-algebra localization singularity formal-completions
asked Nov 28 '18 at 17:38
pyrogenpyrogen
714
$endgroup$
I am trying to analyze the normalization $N$ of a local ring $A_{mathfrak{m}}$ of a variety with Singular. The normalization (integral closure in its total ring of fractions) is semi-local with its maximal ideals lying over $mathfrak{m}A_{mathfrak{m}}$.
According to
https://www.singular.uni-kl.de/Manual/4-0-3/sing_1270.htm#SEC1345
I can calculate a normalization of a local ring $k[x_1, ldots x_n]_{(0)}/I_{(0)}$ and represent it (if i understood correctly) as a factor ring of a polynomial ring $B[y_1, ldots, y_s]$, where $B = k[x_1, ldots x_n]_{(0)}$.
I am doing it like this
LIB "normal.lib";
ring r=0,(x,y),ds;
ideal I = y^2 - x^2*(x+1);
list nor = normal(I);
def R = nor[1][1];
setring R;
norid;
R;
How would I go about finding the maximal ideals in this factor ring and how can they be represented in Singular. For example I would like to know how many there are (for more general examples than this simple node) and if $mathfrak{n}$ is one of those ideals, how can I get to $N_{mathfrak{n}}$. Is this possible? I am aware that this is not a really specific question, sorry. I have unfortunately just a limited knowledge of working with Singular and local orderings. A small hint or a pointer to some resources would be greatly appreciated.
algebraic-geometry commutative-algebra localization singularity formal-completions
algebraic-geometry commutative-algebra localization singularity formal-completions
asked Nov 28 '18 at 17:38
pyrogenpyrogen
714
asked Nov 28 '18 at 17:38
pyrogenpyrogen
714
asked Nov 28 '18 at 17:38
pyrogenpyrogen
714
asked Nov 28 '18 at 17:38
pyrogenpyrogen
714
asked Nov 28 '18 at 17:38
pyrogenpyrogen
714
714
$begingroup$
I don't know how to do it in Singular. But if you have a particular maximal ideal in mind, then extend this to the normalization and compute its associated primes in the normalization. Those are the maximal ideals lying over the one you extended.
$endgroup$
– Youngsu
Nov 28 '18 at 21:28
add a comment |
$begingroup$
I don't know how to do it in Singular. But if you have a particular maximal ideal in mind, then extend this to the normalization and compute its associated primes in the normalization. Those are the maximal ideals lying over the one you extended.
$endgroup$
– Youngsu
Nov 28 '18 at 21:28
$begingroup$
I don't know how to do it in Singular. But if you have a particular maximal ideal in mind, then extend this to the normalization and compute its associated primes in the normalization. Those are the maximal ideals lying over the one you extended.
$endgroup$
– Youngsu
Nov 28 '18 at 21:28
$begingroup$
I don't know how to do it in Singular. But if you have a particular maximal ideal in mind, then extend this to the normalization and compute its associated primes in the normalization. Those are the maximal ideals lying over the one you extended.
$endgroup$
– Youngsu
Nov 28 '18 at 21:28
add a comment |
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I don't know how to do it in Singular. But if you have a particular maximal ideal in mind, then extend this to the normalization and compute its associated primes in the normalization. Those are the maximal ideals lying over the one you extended.
$endgroup$
– Youngsu
Nov 28 '18 at 21:28