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
edited Dec 7 '18 at 20:38
user26857
39.4k124183
asked Dec 7 '18 at 3:47
Rtk427Rtk427
245
$endgroup$
add a comment |
$begingroup$
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
edited Dec 7 '18 at 20:38
user26857
39.4k124183
asked Dec 7 '18 at 3:47
Rtk427Rtk427
245
$endgroup$
add a comment |
$begingroup$
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
edited Dec 7 '18 at 20:38
user26857
39.4k124183
asked Dec 7 '18 at 3:47
Rtk427Rtk427
245
$endgroup$
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
edited Dec 7 '18 at 20:38
user26857
39.4k124183
asked Dec 7 '18 at 3:47
Rtk427Rtk427
245
edited Dec 7 '18 at 20:38
user26857
39.4k124183
asked Dec 7 '18 at 3:47
Rtk427Rtk427
245
edited Dec 7 '18 at 20:38
user26857
39.4k124183
edited Dec 7 '18 at 20:38
user26857
39.4k124183
edited Dec 7 '18 at 20:38
user26857
39.4k124183
39.4k124183
asked Dec 7 '18 at 3:47
Rtk427Rtk427
245
asked Dec 7 '18 at 3:47
Rtk427Rtk427
245
asked Dec 7 '18 at 3:47
Rtk427Rtk427
245
245
add a comment |
add a comment |
1 Answer
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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 $1$.
answered Dec 7 '18 at 8:58
YoungsuYoungsu
1,823715
$endgroup$
$begingroup$
+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
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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
add a comment |
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1 Answer
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1 Answer
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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 $1$.
answered Dec 7 '18 at 8:58
YoungsuYoungsu
1,823715
$endgroup$
$begingroup$
+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
add a comment |
$begingroup$
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 $1$.
answered Dec 7 '18 at 8:58
YoungsuYoungsu
1,823715
$endgroup$
$begingroup$
+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
add a comment |
$begingroup$
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 $1$.
answered Dec 7 '18 at 8:58
YoungsuYoungsu
1,823715
$endgroup$
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 $1$.
answered Dec 7 '18 at 8:58
YoungsuYoungsu
1,823715
answered Dec 7 '18 at 8:58
YoungsuYoungsu
1,823715
answered Dec 7 '18 at 8:58
YoungsuYoungsu
1,823715
answered Dec 7 '18 at 8:58
YoungsuYoungsu
1,823715
1,823715
$begingroup$
+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
add a comment |
$begingroup$
+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
add a comment |
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