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How to define a $P$-primary component of $M'ne 0$ in $M$?
The notion of $P$-primary componentIrredundant primary decomposition of a submodule transferred to irredundant primary decomposition of an idealInfinitely many primary decompositions of an idealIntersection of Primary IdealsDummit and Foote $p$-Primary Components of a Module over a Principal Ideal DomainEquivalence of two definitions of primary idealsIs a contracted primary ideal the contraction of a primary ideal?Length of primary component: confusion about definitionWhat is the relationship between primary decomposition and irreducible decomposition?Minimal (primary) decomposition vs. irredundant decompositionA typo in Eisenbud's Theorem 3.10?
$begingroup$
The definition of a $P$-primary component of $0$ in $M$ is given here. But I haven't found a definition of a $P$-primary component of a nonzero submodule $M'$ in $M$. Is there such a notion? If so, what's its definition in terms of the terminology in the question cited above?
abstract-algebra commutative-algebra definition
asked 2 days ago
user437309user437309
734313
$endgroup$
add a comment |
$begingroup$
The definition of a $P$-primary component of $0$ in $M$ is given here. But I haven't found a definition of a $P$-primary component of a nonzero submodule $M'$ in $M$. Is there such a notion? If so, what's its definition in terms of the terminology in the question cited above?
abstract-algebra commutative-algebra definition
asked 2 days ago
user437309user437309
734313
$endgroup$
$begingroup$
I'd change your $M'$ to $N$ in order to avoid some confusion with the linked question. Then consider $P$ minimal over the annihilator of $M/N$. We say that $ker(M/Nto(M/N)_P$ is the $P$-primary component of $N$ in $M$.
$endgroup$
– user26857
yesterday
$begingroup$
Btw, I don't think that Eisenbud's textbook is the best place to learn about primary decomposition of modules. For instance, Matsumura's approach is more natural and the proofs are pretty clear. (Not mentioning Bourbaki which is the best.)
$endgroup$
– user26857
yesterday
add a comment |
$begingroup$
The definition of a $P$-primary component of $0$ in $M$ is given here. But I haven't found a definition of a $P$-primary component of a nonzero submodule $M'$ in $M$. Is there such a notion? If so, what's its definition in terms of the terminology in the question cited above?
abstract-algebra commutative-algebra definition
asked 2 days ago
user437309user437309
734313
$endgroup$
The definition of a $P$-primary component of $0$ in $M$ is given here. But I haven't found a definition of a $P$-primary component of a nonzero submodule $M'$ in $M$. Is there such a notion? If so, what's its definition in terms of the terminology in the question cited above?
abstract-algebra commutative-algebra definition
abstract-algebra commutative-algebra definition
asked 2 days ago
user437309user437309
734313
asked 2 days ago
user437309user437309
734313
asked 2 days ago
user437309user437309
734313
asked 2 days ago
user437309user437309
734313
asked 2 days ago
user437309user437309
734313
734313
$begingroup$
I'd change your $M'$ to $N$ in order to avoid some confusion with the linked question. Then consider $P$ minimal over the annihilator of $M/N$. We say that $ker(M/Nto(M/N)_P$ is the $P$-primary component of $N$ in $M$.
$endgroup$
– user26857
yesterday
$begingroup$
Btw, I don't think that Eisenbud's textbook is the best place to learn about primary decomposition of modules. For instance, Matsumura's approach is more natural and the proofs are pretty clear. (Not mentioning Bourbaki which is the best.)
$endgroup$
– user26857
yesterday
add a comment |
$begingroup$
I'd change your $M'$ to $N$ in order to avoid some confusion with the linked question. Then consider $P$ minimal over the annihilator of $M/N$. We say that $ker(M/Nto(M/N)_P$ is the $P$-primary component of $N$ in $M$.
$endgroup$
– user26857
yesterday
$begingroup$
Btw, I don't think that Eisenbud's textbook is the best place to learn about primary decomposition of modules. For instance, Matsumura's approach is more natural and the proofs are pretty clear. (Not mentioning Bourbaki which is the best.)
$endgroup$
– user26857
yesterday
$begingroup$
I'd change your $M'$ to $N$ in order to avoid some confusion with the linked question. Then consider $P$ minimal over the annihilator of $M/N$. We say that $ker(M/Nto(M/N)_P$ is the $P$-primary component of $N$ in $M$.
$endgroup$
– user26857
yesterday
$begingroup$
I'd change your $M'$ to $N$ in order to avoid some confusion with the linked question. Then consider $P$ minimal over the annihilator of $M/N$. We say that $ker(M/Nto(M/N)_P$ is the $P$-primary component of $N$ in $M$.
$endgroup$
– user26857
yesterday
$begingroup$
Btw, I don't think that Eisenbud's textbook is the best place to learn about primary decomposition of modules. For instance, Matsumura's approach is more natural and the proofs are pretty clear. (Not mentioning Bourbaki which is the best.)
$endgroup$
– user26857
yesterday
$begingroup$
Btw, I don't think that Eisenbud's textbook is the best place to learn about primary decomposition of modules. For instance, Matsumura's approach is more natural and the proofs are pretty clear. (Not mentioning Bourbaki which is the best.)
$endgroup$
– user26857
yesterday
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
From Bourbaki, Commutative Algebra, Ch. IV Associated Prime Ideals and Primary Decomposition, §2:
Let $A$ be a noetherian ring, $M$ an $A$-module, $N$ a submodule of $M$. We call $,$ primary decomposition of $N$ in $M$ a finite family $(Q_i)_iin I$ of submodules of $M$, primary w.r.t. $M$, and such that $;N = bigcaplimits_iin I Q_i$.
$Q$ is said to be primary w.r.t. $M$ if $:operatornameAssM/Q$ contains only one element.
answered 2 days ago
BernardBernard
122k741116
$endgroup$
$begingroup$
But these definitions don't say anything about a $P$-primary component of $M'ne 0$ in $M$. I was hoping to get a definition of that in terms of localizations, generalizing the definition in the cited question.
$endgroup$
– user437309
2 days ago
$begingroup$
With Bourbaki's notations, the $P$_primary component of $N$ in $M$ is the submodule $Q_i$ such that $operatornameAss(M/Q_i)=P$, unless I misunderstand what you're asking..
$endgroup$
– Bernard
2 days ago
add a comment |
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1 Answer
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$begingroup$
From Bourbaki, Commutative Algebra, Ch. IV Associated Prime Ideals and Primary Decomposition, §2:
Let $A$ be a noetherian ring, $M$ an $A$-module, $N$ a submodule of $M$. We call $,$ primary decomposition of $N$ in $M$ a finite family $(Q_i)_iin I$ of submodules of $M$, primary w.r.t. $M$, and such that $;N = bigcaplimits_iin I Q_i$.
$Q$ is said to be primary w.r.t. $M$ if $:operatornameAssM/Q$ contains only one element.
answered 2 days ago
BernardBernard
122k741116
$endgroup$
$begingroup$
But these definitions don't say anything about a $P$-primary component of $M'ne 0$ in $M$. I was hoping to get a definition of that in terms of localizations, generalizing the definition in the cited question.
$endgroup$
– user437309
2 days ago
$begingroup$
With Bourbaki's notations, the $P$_primary component of $N$ in $M$ is the submodule $Q_i$ such that $operatornameAss(M/Q_i)=P$, unless I misunderstand what you're asking..
$endgroup$
– Bernard
2 days ago
add a comment |
$begingroup$
From Bourbaki, Commutative Algebra, Ch. IV Associated Prime Ideals and Primary Decomposition, §2:
Let $A$ be a noetherian ring, $M$ an $A$-module, $N$ a submodule of $M$. We call $,$ primary decomposition of $N$ in $M$ a finite family $(Q_i)_iin I$ of submodules of $M$, primary w.r.t. $M$, and such that $;N = bigcaplimits_iin I Q_i$.
$Q$ is said to be primary w.r.t. $M$ if $:operatornameAssM/Q$ contains only one element.
answered 2 days ago
BernardBernard
122k741116
$endgroup$
$begingroup$
But these definitions don't say anything about a $P$-primary component of $M'ne 0$ in $M$. I was hoping to get a definition of that in terms of localizations, generalizing the definition in the cited question.
$endgroup$
– user437309
2 days ago
$begingroup$
With Bourbaki's notations, the $P$_primary component of $N$ in $M$ is the submodule $Q_i$ such that $operatornameAss(M/Q_i)=P$, unless I misunderstand what you're asking..
$endgroup$
– Bernard
2 days ago
add a comment |
$begingroup$
From Bourbaki, Commutative Algebra, Ch. IV Associated Prime Ideals and Primary Decomposition, §2:
Let $A$ be a noetherian ring, $M$ an $A$-module, $N$ a submodule of $M$. We call $,$ primary decomposition of $N$ in $M$ a finite family $(Q_i)_iin I$ of submodules of $M$, primary w.r.t. $M$, and such that $;N = bigcaplimits_iin I Q_i$.
$Q$ is said to be primary w.r.t. $M$ if $:operatornameAssM/Q$ contains only one element.
answered 2 days ago
BernardBernard
122k741116
$endgroup$
From Bourbaki, Commutative Algebra, Ch. IV Associated Prime Ideals and Primary Decomposition, §2:
Let $A$ be a noetherian ring, $M$ an $A$-module, $N$ a submodule of $M$. We call $,$ primary decomposition of $N$ in $M$ a finite family $(Q_i)_iin I$ of submodules of $M$, primary w.r.t. $M$, and such that $;N = bigcaplimits_iin I Q_i$.
$Q$ is said to be primary w.r.t. $M$ if $:operatornameAssM/Q$ contains only one element.
answered 2 days ago
BernardBernard
122k741116
answered 2 days ago
BernardBernard
122k741116
answered 2 days ago
BernardBernard
122k741116
answered 2 days ago
BernardBernard
122k741116
122k741116
$begingroup$
But these definitions don't say anything about a $P$-primary component of $M'ne 0$ in $M$. I was hoping to get a definition of that in terms of localizations, generalizing the definition in the cited question.
$endgroup$
– user437309
2 days ago
$begingroup$
With Bourbaki's notations, the $P$_primary component of $N$ in $M$ is the submodule $Q_i$ such that $operatornameAss(M/Q_i)=P$, unless I misunderstand what you're asking..
$endgroup$
– Bernard
2 days ago
add a comment |
$begingroup$
But these definitions don't say anything about a $P$-primary component of $M'ne 0$ in $M$. I was hoping to get a definition of that in terms of localizations, generalizing the definition in the cited question.
$endgroup$
– user437309
2 days ago
$begingroup$
With Bourbaki's notations, the $P$_primary component of $N$ in $M$ is the submodule $Q_i$ such that $operatornameAss(M/Q_i)=P$, unless I misunderstand what you're asking..
$endgroup$
– Bernard
2 days ago
$begingroup$
But these definitions don't say anything about a $P$-primary component of $M'ne 0$ in $M$. I was hoping to get a definition of that in terms of localizations, generalizing the definition in the cited question.
$endgroup$
– user437309
2 days ago
$begingroup$
But these definitions don't say anything about a $P$-primary component of $M'ne 0$ in $M$. I was hoping to get a definition of that in terms of localizations, generalizing the definition in the cited question.
$endgroup$
– user437309
2 days ago
$begingroup$
With Bourbaki's notations, the $P$_primary component of $N$ in $M$ is the submodule $Q_i$ such that $operatornameAss(M/Q_i)=P$, unless I misunderstand what you're asking..
$endgroup$
– Bernard
2 days ago
$begingroup$
With Bourbaki's notations, the $P$_primary component of $N$ in $M$ is the submodule $Q_i$ such that $operatornameAss(M/Q_i)=P$, unless I misunderstand what you're asking..
$endgroup$
– Bernard
2 days ago
add a comment |
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$begingroup$
I'd change your $M'$ to $N$ in order to avoid some confusion with the linked question. Then consider $P$ minimal over the annihilator of $M/N$. We say that $ker(M/Nto(M/N)_P$ is the $P$-primary component of $N$ in $M$.
$endgroup$
– user26857
yesterday
$begingroup$
Btw, I don't think that Eisenbud's textbook is the best place to learn about primary decomposition of modules. For instance, Matsumura's approach is more natural and the proofs are pretty clear. (Not mentioning Bourbaki which is the best.)
$endgroup$
– user26857
yesterday