Quotient of monomial ideals












0














I've done a lot of calculations but they all seem to lead nowhere:




Consider the ring of multivariate polynomials with field coefficients $K[X_1,dots,X_n]$, and two monomial ideals, say $alpha=left<X^aright>_{ain A}$ and $beta=left<X^bright>_{bin B}$, I need to prove their quotient $(alpha : beta)$ is also monomial.




Considering that $(alpha : beta)=cap_{bin B}(alpha :X^b)$ and that (I've proved) intersection of monomial ideals is monomial too, I only need to fix a $bin B$ and prove the corresponding $(alpha :X^b)$ is monomial (correct me if I'm wrong). Trying to figure out what can be its generating set I've managed to prove, as long as my calculations are right, that for every polynomial $F$ in such ideal, being $N(F)$ its Newton's diagram we have:



$$b+N(F)subseteq A+mathbb{N}^n$$



but I cannot see if this helps me to conclude. I really appreciate any other idea on how to proceed, but I cannot see a more straightforward road.










share|cite|improve this question
























  • If $I$ is an ideal generated by some monomials $m_i$, and $n$ is a monomial, then $(I:n)$ is generated by $m_i/gcd(m_i,n)$.
    – user26857
    Dec 7 at 20:27










  • en.wikipedia.org/wiki/Ideal_quotient
    – user26857
    Dec 7 at 20:29
















0














I've done a lot of calculations but they all seem to lead nowhere:




Consider the ring of multivariate polynomials with field coefficients $K[X_1,dots,X_n]$, and two monomial ideals, say $alpha=left<X^aright>_{ain A}$ and $beta=left<X^bright>_{bin B}$, I need to prove their quotient $(alpha : beta)$ is also monomial.




Considering that $(alpha : beta)=cap_{bin B}(alpha :X^b)$ and that (I've proved) intersection of monomial ideals is monomial too, I only need to fix a $bin B$ and prove the corresponding $(alpha :X^b)$ is monomial (correct me if I'm wrong). Trying to figure out what can be its generating set I've managed to prove, as long as my calculations are right, that for every polynomial $F$ in such ideal, being $N(F)$ its Newton's diagram we have:



$$b+N(F)subseteq A+mathbb{N}^n$$



but I cannot see if this helps me to conclude. I really appreciate any other idea on how to proceed, but I cannot see a more straightforward road.










share|cite|improve this question
























  • If $I$ is an ideal generated by some monomials $m_i$, and $n$ is a monomial, then $(I:n)$ is generated by $m_i/gcd(m_i,n)$.
    – user26857
    Dec 7 at 20:27










  • en.wikipedia.org/wiki/Ideal_quotient
    – user26857
    Dec 7 at 20:29














0












0








0







I've done a lot of calculations but they all seem to lead nowhere:




Consider the ring of multivariate polynomials with field coefficients $K[X_1,dots,X_n]$, and two monomial ideals, say $alpha=left<X^aright>_{ain A}$ and $beta=left<X^bright>_{bin B}$, I need to prove their quotient $(alpha : beta)$ is also monomial.




Considering that $(alpha : beta)=cap_{bin B}(alpha :X^b)$ and that (I've proved) intersection of monomial ideals is monomial too, I only need to fix a $bin B$ and prove the corresponding $(alpha :X^b)$ is monomial (correct me if I'm wrong). Trying to figure out what can be its generating set I've managed to prove, as long as my calculations are right, that for every polynomial $F$ in such ideal, being $N(F)$ its Newton's diagram we have:



$$b+N(F)subseteq A+mathbb{N}^n$$



but I cannot see if this helps me to conclude. I really appreciate any other idea on how to proceed, but I cannot see a more straightforward road.










share|cite|improve this question















I've done a lot of calculations but they all seem to lead nowhere:




Consider the ring of multivariate polynomials with field coefficients $K[X_1,dots,X_n]$, and two monomial ideals, say $alpha=left<X^aright>_{ain A}$ and $beta=left<X^bright>_{bin B}$, I need to prove their quotient $(alpha : beta)$ is also monomial.




Considering that $(alpha : beta)=cap_{bin B}(alpha :X^b)$ and that (I've proved) intersection of monomial ideals is monomial too, I only need to fix a $bin B$ and prove the corresponding $(alpha :X^b)$ is monomial (correct me if I'm wrong). Trying to figure out what can be its generating set I've managed to prove, as long as my calculations are right, that for every polynomial $F$ in such ideal, being $N(F)$ its Newton's diagram we have:



$$b+N(F)subseteq A+mathbb{N}^n$$



but I cannot see if this helps me to conclude. I really appreciate any other idea on how to proceed, but I cannot see a more straightforward road.







polynomials ring-theory commutative-algebra monomial-ideals






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 7 at 20:32









user26857

39.2k123882




39.2k123882










asked Dec 6 at 17:24









Renato Faraone

2,31411627




2,31411627












  • If $I$ is an ideal generated by some monomials $m_i$, and $n$ is a monomial, then $(I:n)$ is generated by $m_i/gcd(m_i,n)$.
    – user26857
    Dec 7 at 20:27










  • en.wikipedia.org/wiki/Ideal_quotient
    – user26857
    Dec 7 at 20:29


















  • If $I$ is an ideal generated by some monomials $m_i$, and $n$ is a monomial, then $(I:n)$ is generated by $m_i/gcd(m_i,n)$.
    – user26857
    Dec 7 at 20:27










  • en.wikipedia.org/wiki/Ideal_quotient
    – user26857
    Dec 7 at 20:29
















If $I$ is an ideal generated by some monomials $m_i$, and $n$ is a monomial, then $(I:n)$ is generated by $m_i/gcd(m_i,n)$.
– user26857
Dec 7 at 20:27




If $I$ is an ideal generated by some monomials $m_i$, and $n$ is a monomial, then $(I:n)$ is generated by $m_i/gcd(m_i,n)$.
– user26857
Dec 7 at 20:27












en.wikipedia.org/wiki/Ideal_quotient
– user26857
Dec 7 at 20:29




en.wikipedia.org/wiki/Ideal_quotient
– user26857
Dec 7 at 20:29















active

oldest

votes











Your Answer





StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");

StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});

function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});


}
});














draft saved

draft discarded


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3028787%2fquotient-of-monomial-ideals%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown






























active

oldest

votes













active

oldest

votes









active

oldest

votes






active

oldest

votes
















draft saved

draft discarded




















































Thanks for contributing an answer to Mathematics Stack Exchange!


  • Please be sure to answer the question. Provide details and share your research!

But avoid



  • Asking for help, clarification, or responding to other answers.

  • Making statements based on opinion; back them up with references or personal experience.


Use MathJax to format equations. MathJax reference.


To learn more, see our tips on writing great answers.





Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


Please pay close attention to the following guidance:


  • Please be sure to answer the question. Provide details and share your research!

But avoid



  • Asking for help, clarification, or responding to other answers.

  • Making statements based on opinion; back them up with references or personal experience.


To learn more, see our tips on writing great answers.




draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3028787%2fquotient-of-monomial-ideals%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown





















































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown

































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown







Popular posts from this blog

Bressuire

Cabo Verde

Gyllenstierna