Intersection of all symbolic powers of a prime ideal Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern)Analogue of Krull intersection theorem for Symbolic powers of idealsDimension inequality for homomorphisms between noetherian local ringsNeed an explanation for homomorphism in commutative algebraIs there a purely category-theoretic description of the total quotient ring?Proof of Krull's intersection theorem with Taylor expansionLocal ring with finitely generated maximal ideal and $cap_n>0mathfrakm^n=0$ is Noetherian?(Dimension theory)How can I derive this corollary from Theorem 5.4.7?$bigcap_ninmathbbN I^n = (0)$ if and only if no zero divisor of $R$ is of the form $1-z$ with $zin I$.Unique prime ideal factorization in noetherian domains?Non Noetherian ring with only one prime idealIf finite product of maximal ideals of ring $R$ is zero, then $R$ is Noetherian$iff R$ is Artinian
How to resize main filesystem
In musical terms, what properties are varied by the human voice to produce different words / syllables?
Weaponising the Grasp-at-a-Distance spell
Where did Ptolemy compare the Earth to the distance of fixed stars?
How to make triangles with rounded sides and corners? (squircle with 3 sides)
Short story about astronauts fertilizing soil with their own bodies
What are some likely causes to domain member PC losing contact to domain controller?
Does the universe have a fixed centre of mass?
Table formatting with tabularx?
New Order #6: Easter Egg
Understanding piped commands in GNU/Linux
Derived column in a data extension
How can I prevent/balance waiting and turtling as a response to cooldown mechanics
Is there a verb for listening stealthily?
Who said what about *meanings*?
latest version of QGIS fails to edit attribute table of GeoJSON file
malloc in main() or malloc in another function: allocating memory for a struct and its members
Did pre-Columbian Americans know the spherical shape of the Earth?
What did Turing mean when saying that "machines cannot give rise to surprises" is due to a fallacy?
2018 MacBook Pro won't let me install macOS High Sierra 10.13 from USB installer
The test team as an enemy of development? And how can this be avoided?
Where and when has Thucydides been studied?
Flight departed from the gate 5 min before scheduled departure time. Refund options
What is a more techy Technical Writer job title that isn't cutesy or confusing?
Intersection of all symbolic powers of a prime ideal
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern)Analogue of Krull intersection theorem for Symbolic powers of idealsDimension inequality for homomorphisms between noetherian local ringsNeed an explanation for homomorphism in commutative algebraIs there a purely category-theoretic description of the total quotient ring?Proof of Krull's intersection theorem with Taylor expansionLocal ring with finitely generated maximal ideal and $cap_n>0mathfrakm^n=0$ is Noetherian?(Dimension theory)How can I derive this corollary from Theorem 5.4.7?$bigcap_ninmathbbN I^n = (0)$ if and only if no zero divisor of $R$ is of the form $1-z$ with $zin I$.Unique prime ideal factorization in noetherian domains?Non Noetherian ring with only one prime idealIf finite product of maximal ideals of ring $R$ is zero, then $R$ is Noetherian$iff R$ is Artinian
$begingroup$
This is Exercise 8.37 in R.Y.Sharp's Steps in Commutative Algebra:
Let $P$ be a prime ideal of the commutative Noetherian ring $R$. Prove that
$$bigcap_n=1^infty P^(n)= rin R mid exists sin R setminus P ,sr=0 $$ in which $P^(n)=(P^n)^textec$ with extension and contraction notation in conjunction with the natural ring homomorphism $Rto R_P$.
There's 2 reasons I got stuck. First, I'm confused with $bigcap_1^infty$, I don't know how to use this notation. Second, the only theorem I know involving with $bigcap_1^infty$ is Krull's Intersection Theorem, stating "If $IsubseteqmathrmJac(R)$ then $bigcap_n=1^inftyI^n=0$", which I believe is useless in this situation.
So help me with this problem. THank you.
commutative-algebra noetherian
edited Apr 2 at 14:58
user26857
39.6k124284
asked Jan 12 '17 at 10:00
chí trung châuchí trung châu
1,0801725
$endgroup$
add a comment |
$begingroup$
This is Exercise 8.37 in R.Y.Sharp's Steps in Commutative Algebra:
Let $P$ be a prime ideal of the commutative Noetherian ring $R$. Prove that
$$bigcap_n=1^infty P^(n)= rin R mid exists sin R setminus P ,sr=0 $$ in which $P^(n)=(P^n)^textec$ with extension and contraction notation in conjunction with the natural ring homomorphism $Rto R_P$.
There's 2 reasons I got stuck. First, I'm confused with $bigcap_1^infty$, I don't know how to use this notation. Second, the only theorem I know involving with $bigcap_1^infty$ is Krull's Intersection Theorem, stating "If $IsubseteqmathrmJac(R)$ then $bigcap_n=1^inftyI^n=0$", which I believe is useless in this situation.
So help me with this problem. THank you.
commutative-algebra noetherian
edited Apr 2 at 14:58
user26857
39.6k124284
asked Jan 12 '17 at 10:00
chí trung châuchí trung châu
1,0801725
$endgroup$
$begingroup$
Your guess that the KIT is useless is wrong. If $xin P^(n)$ then $xin P^nR_P=(PR_P)^n$. Can you take it from here?
$endgroup$
– user26857
Jan 12 '17 at 15:18
$begingroup$
Okay, thanks to your hint, I solved the inclusion. One more thing, now that I recheck, I was wrong about the converse. Now I solved it using your hint, too.
$endgroup$
– chí trung châu
Jan 12 '17 at 15:38
$begingroup$
like this, $r/1=0in (P^n)^e$, so $rin (P^n)^ec$. It's true, right?
$endgroup$
– chí trung châu
Jan 12 '17 at 15:39
$begingroup$
It sounds right.
$endgroup$
– user26857
Jan 12 '17 at 15:43
1
$begingroup$
Ok. Thank you for your help @user26857
$endgroup$
– chí trung châu
Jan 12 '17 at 15:44
add a comment |
$begingroup$
This is Exercise 8.37 in R.Y.Sharp's Steps in Commutative Algebra:
Let $P$ be a prime ideal of the commutative Noetherian ring $R$. Prove that
$$bigcap_n=1^infty P^(n)= rin R mid exists sin R setminus P ,sr=0 $$ in which $P^(n)=(P^n)^textec$ with extension and contraction notation in conjunction with the natural ring homomorphism $Rto R_P$.
There's 2 reasons I got stuck. First, I'm confused with $bigcap_1^infty$, I don't know how to use this notation. Second, the only theorem I know involving with $bigcap_1^infty$ is Krull's Intersection Theorem, stating "If $IsubseteqmathrmJac(R)$ then $bigcap_n=1^inftyI^n=0$", which I believe is useless in this situation.
So help me with this problem. THank you.
commutative-algebra noetherian
edited Apr 2 at 14:58
user26857
39.6k124284
asked Jan 12 '17 at 10:00
chí trung châuchí trung châu
1,0801725
$endgroup$
This is Exercise 8.37 in R.Y.Sharp's Steps in Commutative Algebra:
Let $P$ be a prime ideal of the commutative Noetherian ring $R$. Prove that
$$bigcap_n=1^infty P^(n)= rin R mid exists sin R setminus P ,sr=0 $$ in which $P^(n)=(P^n)^textec$ with extension and contraction notation in conjunction with the natural ring homomorphism $Rto R_P$.
There's 2 reasons I got stuck. First, I'm confused with $bigcap_1^infty$, I don't know how to use this notation. Second, the only theorem I know involving with $bigcap_1^infty$ is Krull's Intersection Theorem, stating "If $IsubseteqmathrmJac(R)$ then $bigcap_n=1^inftyI^n=0$", which I believe is useless in this situation.
So help me with this problem. THank you.
commutative-algebra noetherian
commutative-algebra noetherian
edited Apr 2 at 14:58
user26857
39.6k124284
asked Jan 12 '17 at 10:00
chí trung châuchí trung châu
1,0801725
edited Apr 2 at 14:58
user26857
39.6k124284
asked Jan 12 '17 at 10:00
chí trung châuchí trung châu
1,0801725
edited Apr 2 at 14:58
user26857
39.6k124284
edited Apr 2 at 14:58
user26857
39.6k124284
edited Apr 2 at 14:58
user26857
39.6k124284
39.6k124284
asked Jan 12 '17 at 10:00
chí trung châuchí trung châu
1,0801725
asked Jan 12 '17 at 10:00
chí trung châuchí trung châu
1,0801725
asked Jan 12 '17 at 10:00
chí trung châuchí trung châu
1,0801725
1,0801725
$begingroup$
Your guess that the KIT is useless is wrong. If $xin P^(n)$ then $xin P^nR_P=(PR_P)^n$. Can you take it from here?
$endgroup$
– user26857
Jan 12 '17 at 15:18
$begingroup$
Okay, thanks to your hint, I solved the inclusion. One more thing, now that I recheck, I was wrong about the converse. Now I solved it using your hint, too.
$endgroup$
– chí trung châu
Jan 12 '17 at 15:38
$begingroup$
like this, $r/1=0in (P^n)^e$, so $rin (P^n)^ec$. It's true, right?
$endgroup$
– chí trung châu
Jan 12 '17 at 15:39
$begingroup$
It sounds right.
$endgroup$
– user26857
Jan 12 '17 at 15:43
1
$begingroup$
Ok. Thank you for your help @user26857
$endgroup$
– chí trung châu
Jan 12 '17 at 15:44
add a comment |
$begingroup$
Your guess that the KIT is useless is wrong. If $xin P^(n)$ then $xin P^nR_P=(PR_P)^n$. Can you take it from here?
$endgroup$
– user26857
Jan 12 '17 at 15:18
$begingroup$
Okay, thanks to your hint, I solved the inclusion. One more thing, now that I recheck, I was wrong about the converse. Now I solved it using your hint, too.
$endgroup$
– chí trung châu
Jan 12 '17 at 15:38
$begingroup$
like this, $r/1=0in (P^n)^e$, so $rin (P^n)^ec$. It's true, right?
$endgroup$
– chí trung châu
Jan 12 '17 at 15:39
$begingroup$
It sounds right.
$endgroup$
– user26857
Jan 12 '17 at 15:43
1
$begingroup$
Ok. Thank you for your help @user26857
$endgroup$
– chí trung châu
Jan 12 '17 at 15:44
$begingroup$
Your guess that the KIT is useless is wrong. If $xin P^(n)$ then $xin P^nR_P=(PR_P)^n$. Can you take it from here?
$endgroup$
– user26857
Jan 12 '17 at 15:18
$begingroup$
Your guess that the KIT is useless is wrong. If $xin P^(n)$ then $xin P^nR_P=(PR_P)^n$. Can you take it from here?
$endgroup$
– user26857
Jan 12 '17 at 15:18
$begingroup$
Okay, thanks to your hint, I solved the inclusion. One more thing, now that I recheck, I was wrong about the converse. Now I solved it using your hint, too.
$endgroup$
– chí trung châu
Jan 12 '17 at 15:38
$begingroup$
Okay, thanks to your hint, I solved the inclusion. One more thing, now that I recheck, I was wrong about the converse. Now I solved it using your hint, too.
$endgroup$
– chí trung châu
Jan 12 '17 at 15:38
$begingroup$
like this, $r/1=0in (P^n)^e$, so $rin (P^n)^ec$. It's true, right?
$endgroup$
– chí trung châu
Jan 12 '17 at 15:39
$begingroup$
like this, $r/1=0in (P^n)^e$, so $rin (P^n)^ec$. It's true, right?
$endgroup$
– chí trung châu
Jan 12 '17 at 15:39
$begingroup$
It sounds right.
$endgroup$
– user26857
Jan 12 '17 at 15:43
$begingroup$
It sounds right.
$endgroup$
– user26857
Jan 12 '17 at 15:43
1
1
$begingroup$
Ok. Thank you for your help @user26857
$endgroup$
– chí trung châu
Jan 12 '17 at 15:44
$begingroup$
Ok. Thank you for your help @user26857
$endgroup$
– chí trung châu
Jan 12 '17 at 15:44
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
We have:
$bigcap_n=1^infty P^(n)=bigcap_n=1^infty (P^n)^ec= (bigcap_n=1^infty (P^e)^n)^c=(frac01)^c$
$(frac01)^c=rin R$: $exists snotin P $ such that $sr=0$
edited Jan 13 '17 at 15:30
user26857
39.6k124284
answered Jan 13 '17 at 3:35
SoulostarSoulostar
9918
$endgroup$
add a comment |
Your Answer
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
);
);
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2094537%2fintersection-of-all-symbolic-powers-of-a-prime-ideal%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
We have:
$bigcap_n=1^infty P^(n)=bigcap_n=1^infty (P^n)^ec= (bigcap_n=1^infty (P^e)^n)^c=(frac01)^c$
$(frac01)^c=rin R$: $exists snotin P $ such that $sr=0$
edited Jan 13 '17 at 15:30
user26857
39.6k124284
answered Jan 13 '17 at 3:35
SoulostarSoulostar
9918
$endgroup$
add a comment |
$begingroup$
We have:
$bigcap_n=1^infty P^(n)=bigcap_n=1^infty (P^n)^ec= (bigcap_n=1^infty (P^e)^n)^c=(frac01)^c$
$(frac01)^c=rin R$: $exists snotin P $ such that $sr=0$
edited Jan 13 '17 at 15:30
user26857
39.6k124284
answered Jan 13 '17 at 3:35
SoulostarSoulostar
9918
$endgroup$
add a comment |
$begingroup$
We have:
$bigcap_n=1^infty P^(n)=bigcap_n=1^infty (P^n)^ec= (bigcap_n=1^infty (P^e)^n)^c=(frac01)^c$
$(frac01)^c=rin R$: $exists snotin P $ such that $sr=0$
edited Jan 13 '17 at 15:30
user26857
39.6k124284
answered Jan 13 '17 at 3:35
SoulostarSoulostar
9918
$endgroup$
We have:
$bigcap_n=1^infty P^(n)=bigcap_n=1^infty (P^n)^ec= (bigcap_n=1^infty (P^e)^n)^c=(frac01)^c$
$(frac01)^c=rin R$: $exists snotin P $ such that $sr=0$
edited Jan 13 '17 at 15:30
user26857
39.6k124284
answered Jan 13 '17 at 3:35
SoulostarSoulostar
9918
edited Jan 13 '17 at 15:30
user26857
39.6k124284
edited Jan 13 '17 at 15:30
user26857
39.6k124284
edited Jan 13 '17 at 15:30
user26857
39.6k124284
39.6k124284
answered Jan 13 '17 at 3:35
SoulostarSoulostar
9918
answered Jan 13 '17 at 3:35
SoulostarSoulostar
9918
answered Jan 13 '17 at 3:35
SoulostarSoulostar
9918
9918
add a comment |
add a comment |
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.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2094537%2fintersection-of-all-symbolic-powers-of-a-prime-ideal%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
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
$begingroup$
Your guess that the KIT is useless is wrong. If $xin P^(n)$ then $xin P^nR_P=(PR_P)^n$. Can you take it from here?
$endgroup$
– user26857
Jan 12 '17 at 15:18
$begingroup$
Okay, thanks to your hint, I solved the inclusion. One more thing, now that I recheck, I was wrong about the converse. Now I solved it using your hint, too.
$endgroup$
– chí trung châu
Jan 12 '17 at 15:38
$begingroup$
like this, $r/1=0in (P^n)^e$, so $rin (P^n)^ec$. It's true, right?
$endgroup$
– chí trung châu
Jan 12 '17 at 15:39
$begingroup$
It sounds right.
$endgroup$
– user26857
Jan 12 '17 at 15:43
1
$begingroup$
Ok. Thank you for your help @user26857
$endgroup$
– chí trung châu
Jan 12 '17 at 15:44