Noetherian Catenary ring and Cohen-Macaulay ring












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Let $A$ be a Noetherian ring. $A$ is called catenary if for any two prime ideals $p$ and $q$ in $A$, $psubset q$, every saturated chain of prime ideals starting at $p$ and ending at $q$ have same length.



It is true that for any two prime ideals $p$ and $q$, $psubset q$, if we have $operatorname{ht}p+operatorname{ht}(q/p)=operatorname{ht}q$, then $A$ is catenary. My question is if the converse is true, that is, if $A$ is catenary then is the above condition satisfied?



Also every Cohen-Macaulay ring is catenary (in fact, universally catenary). What will be an example of a catenary ring which is not Cohen-Macaulay?



Thank you in advance.






abstract-algebra commutative-algebra cohen-macaulay







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    0












    $begingroup$


    Let $A$ be a Noetherian ring. $A$ is called catenary if for any two prime ideals $p$ and $q$ in $A$, $psubset q$, every saturated chain of prime ideals starting at $p$ and ending at $q$ have same length.



    It is true that for any two prime ideals $p$ and $q$, $psubset q$, if we have $operatorname{ht}p+operatorname{ht}(q/p)=operatorname{ht}q$, then $A$ is catenary. My question is if the converse is true, that is, if $A$ is catenary then is the above condition satisfied?



    Also every Cohen-Macaulay ring is catenary (in fact, universally catenary). What will be an example of a catenary ring which is not Cohen-Macaulay?



    Thank you in advance.






    abstract-algebra commutative-algebra cohen-macaulay







    share|cite|improve this question











    $endgroup$















      0












      0








      0





      $begingroup$


      Let $A$ be a Noetherian ring. $A$ is called catenary if for any two prime ideals $p$ and $q$ in $A$, $psubset q$, every saturated chain of prime ideals starting at $p$ and ending at $q$ have same length.



      It is true that for any two prime ideals $p$ and $q$, $psubset q$, if we have $operatorname{ht}p+operatorname{ht}(q/p)=operatorname{ht}q$, then $A$ is catenary. My question is if the converse is true, that is, if $A$ is catenary then is the above condition satisfied?



      Also every Cohen-Macaulay ring is catenary (in fact, universally catenary). What will be an example of a catenary ring which is not Cohen-Macaulay?



      Thank you in advance.






      abstract-algebra commutative-algebra cohen-macaulay







      share|cite|improve this question











      $endgroup$




      Let $A$ be a Noetherian ring. $A$ is called catenary if for any two prime ideals $p$ and $q$ in $A$, $psubset q$, every saturated chain of prime ideals starting at $p$ and ending at $q$ have same length.



      It is true that for any two prime ideals $p$ and $q$, $psubset q$, if we have $operatorname{ht}p+operatorname{ht}(q/p)=operatorname{ht}q$, then $A$ is catenary. My question is if the converse is true, that is, if $A$ is catenary then is the above condition satisfied?



      Also every Cohen-Macaulay ring is catenary (in fact, universally catenary). What will be an example of a catenary ring which is not Cohen-Macaulay?



      Thank you in advance.






      abstract-algebra commutative-algebra cohen-macaulay




      abstract-algebra commutative-algebra cohen-macaulay






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      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Dec 7 '18 at 20:38









      user26857

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      39.3k124183










      asked Dec 7 '18 at 3:47









      Rtk427Rtk427

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          $begingroup$

          A factor ring of a CM ring which is not CM would be an example. For instance, $R=k[x,y,z]/(xz,yz)$ for some field $k$. In the same example, let $q = (x,y,z)R$. Then height of $q$ is $2$. Take $p = (x,y)$. Then height of $p$ is zero, but height of $q/p$ is $1$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            +1 Hi! The OP's question is an opportunity for me to close a gap for that question in the DaRT database. I have two questions: 1) would you consider registering a user account so that I could add this example and attribute it to you? Not a big deal if not. 2) Can you tell, at a glance, if any of the hits listed in the "Consistent with parameters" column is also a satisfactory example? If so, let me know.
            $endgroup$
            – rschwieb
            Dec 7 '18 at 14:52












          • $begingroup$
            Hi @rschwieb, I don't think you should attribute it to me. I am very certain that this was written somewhere by an expert. I appreciate your project!
            $endgroup$
            – Youngsu
            Dec 8 '18 at 9:20










          • $begingroup$
            The attribution isn't to you for inventing it. The attribution is for you bringing it to the database. I would be happy to add any citations you suggest if you find it appearing in the literature :) My hope is that this reinforces viewers impression that it is a community effort.
            $endgroup$
            – rschwieb
            Dec 8 '18 at 14:36













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          1 Answer
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          active

          oldest

          votes









          2












          $begingroup$

          A factor ring of a CM ring which is not CM would be an example. For instance, $R=k[x,y,z]/(xz,yz)$ for some field $k$. In the same example, let $q = (x,y,z)R$. Then height of $q$ is $2$. Take $p = (x,y)$. Then height of $p$ is zero, but height of $q/p$ is $1$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            +1 Hi! The OP's question is an opportunity for me to close a gap for that question in the DaRT database. I have two questions: 1) would you consider registering a user account so that I could add this example and attribute it to you? Not a big deal if not. 2) Can you tell, at a glance, if any of the hits listed in the "Consistent with parameters" column is also a satisfactory example? If so, let me know.
            $endgroup$
            – rschwieb
            Dec 7 '18 at 14:52












          • $begingroup$
            Hi @rschwieb, I don't think you should attribute it to me. I am very certain that this was written somewhere by an expert. I appreciate your project!
            $endgroup$
            – Youngsu
            Dec 8 '18 at 9:20










          • $begingroup$
            The attribution isn't to you for inventing it. The attribution is for you bringing it to the database. I would be happy to add any citations you suggest if you find it appearing in the literature :) My hope is that this reinforces viewers impression that it is a community effort.
            $endgroup$
            – rschwieb
            Dec 8 '18 at 14:36


















          2












          $begingroup$

          A factor ring of a CM ring which is not CM would be an example. For instance, $R=k[x,y,z]/(xz,yz)$ for some field $k$. In the same example, let $q = (x,y,z)R$. Then height of $q$ is $2$. Take $p = (x,y)$. Then height of $p$ is zero, but height of $q/p$ is $1$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            +1 Hi! The OP's question is an opportunity for me to close a gap for that question in the DaRT database. I have two questions: 1) would you consider registering a user account so that I could add this example and attribute it to you? Not a big deal if not. 2) Can you tell, at a glance, if any of the hits listed in the "Consistent with parameters" column is also a satisfactory example? If so, let me know.
            $endgroup$
            – rschwieb
            Dec 7 '18 at 14:52












          • $begingroup$
            Hi @rschwieb, I don't think you should attribute it to me. I am very certain that this was written somewhere by an expert. I appreciate your project!
            $endgroup$
            – Youngsu
            Dec 8 '18 at 9:20










          • $begingroup$
            The attribution isn't to you for inventing it. The attribution is for you bringing it to the database. I would be happy to add any citations you suggest if you find it appearing in the literature :) My hope is that this reinforces viewers impression that it is a community effort.
            $endgroup$
            – rschwieb
            Dec 8 '18 at 14:36
















          2












          2








          2





          $begingroup$

          A factor ring of a CM ring which is not CM would be an example. For instance, $R=k[x,y,z]/(xz,yz)$ for some field $k$. In the same example, let $q = (x,y,z)R$. Then height of $q$ is $2$. Take $p = (x,y)$. Then height of $p$ is zero, but height of $q/p$ is $1$.






          share|cite|improve this answer









          $endgroup$



          A factor ring of a CM ring which is not CM would be an example. For instance, $R=k[x,y,z]/(xz,yz)$ for some field $k$. In the same example, let $q = (x,y,z)R$. Then height of $q$ is $2$. Take $p = (x,y)$. Then height of $p$ is zero, but height of $q/p$ is $1$.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 7 '18 at 8:58









          YoungsuYoungsu

          1,843715




          1,843715












          • $begingroup$
            +1 Hi! The OP's question is an opportunity for me to close a gap for that question in the DaRT database. I have two questions: 1) would you consider registering a user account so that I could add this example and attribute it to you? Not a big deal if not. 2) Can you tell, at a glance, if any of the hits listed in the "Consistent with parameters" column is also a satisfactory example? If so, let me know.
            $endgroup$
            – rschwieb
            Dec 7 '18 at 14:52












          • $begingroup$
            Hi @rschwieb, I don't think you should attribute it to me. I am very certain that this was written somewhere by an expert. I appreciate your project!
            $endgroup$
            – Youngsu
            Dec 8 '18 at 9:20










          • $begingroup$
            The attribution isn't to you for inventing it. The attribution is for you bringing it to the database. I would be happy to add any citations you suggest if you find it appearing in the literature :) My hope is that this reinforces viewers impression that it is a community effort.
            $endgroup$
            – rschwieb
            Dec 8 '18 at 14:36




















          • $begingroup$
            +1 Hi! The OP's question is an opportunity for me to close a gap for that question in the DaRT database. I have two questions: 1) would you consider registering a user account so that I could add this example and attribute it to you? Not a big deal if not. 2) Can you tell, at a glance, if any of the hits listed in the "Consistent with parameters" column is also a satisfactory example? If so, let me know.
            $endgroup$
            – rschwieb
            Dec 7 '18 at 14:52












          • $begingroup$
            Hi @rschwieb, I don't think you should attribute it to me. I am very certain that this was written somewhere by an expert. I appreciate your project!
            $endgroup$
            – Youngsu
            Dec 8 '18 at 9:20










          • $begingroup$
            The attribution isn't to you for inventing it. The attribution is for you bringing it to the database. I would be happy to add any citations you suggest if you find it appearing in the literature :) My hope is that this reinforces viewers impression that it is a community effort.
            $endgroup$
            – rschwieb
            Dec 8 '18 at 14:36


















          $begingroup$
          +1 Hi! The OP's question is an opportunity for me to close a gap for that question in the DaRT database. I have two questions: 1) would you consider registering a user account so that I could add this example and attribute it to you? Not a big deal if not. 2) Can you tell, at a glance, if any of the hits listed in the "Consistent with parameters" column is also a satisfactory example? If so, let me know.
          $endgroup$
          – rschwieb
          Dec 7 '18 at 14:52






          $begingroup$
          +1 Hi! The OP's question is an opportunity for me to close a gap for that question in the DaRT database. I have two questions: 1) would you consider registering a user account so that I could add this example and attribute it to you? Not a big deal if not. 2) Can you tell, at a glance, if any of the hits listed in the "Consistent with parameters" column is also a satisfactory example? If so, let me know.
          $endgroup$
          – rschwieb
          Dec 7 '18 at 14:52














          $begingroup$
          Hi @rschwieb, I don't think you should attribute it to me. I am very certain that this was written somewhere by an expert. I appreciate your project!
          $endgroup$
          – Youngsu
          Dec 8 '18 at 9:20




          $begingroup$
          Hi @rschwieb, I don't think you should attribute it to me. I am very certain that this was written somewhere by an expert. I appreciate your project!
          $endgroup$
          – Youngsu
          Dec 8 '18 at 9:20












          $begingroup$
          The attribution isn't to you for inventing it. The attribution is for you bringing it to the database. I would be happy to add any citations you suggest if you find it appearing in the literature :) My hope is that this reinforces viewers impression that it is a community effort.
          $endgroup$
          – rschwieb
          Dec 8 '18 at 14:36






          $begingroup$
          The attribution isn't to you for inventing it. The attribution is for you bringing it to the database. I would be happy to add any citations you suggest if you find it appearing in the literature :) My hope is that this reinforces viewers impression that it is a community effort.
          $endgroup$
          – rschwieb
          Dec 8 '18 at 14:36




















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