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Prove that $M_i/M'=ker(M/M_ito (M/M_i)_P_i)$

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As pointed out here, I believe, there is a typo in Eisenbud's Theorem 3.10(b). The modified statement is this:

Let $M'$ be a proper submodule of a f.g. module $M$ over a Noetherian ring $R$ and let $M'=cap_i=1^n M_i$ where each $M_i$ is $P_i$-primary. Suppose the intersection is minimal (as explained here). Then

(a) Each associated prime of $M/M'$ is equal to $P_i$ for exactly one index $i$.

(b) If $P_i$ is minimal over $operatornameAnn(M/M')$, then $M_i/M'=ker(M/M_ito (M/M_i)_P_i)$.

I understand how to prove $(a)$, but I'm not quite sure how to prove $(b)$ (without the assumption $M'=0$). Here is an attempt to adapt the proof from Eisenbud.

Let $N=M/M', N_i= M_i/M'$. Consider the commutative diagram

from Eisenbud and replace $M$ with $N$ and $M_i$ with $N_i$ in it. We need to prove that $ker alpha=N_i$. It suffices to show that $gamma,delta$ are injective.

The injectivity of $delta $ must follow from the fact that $N_i$ is $P_i$-primary. Why is this so? We only know that $M_i$ is $P_i$-primary.

For the injectivity of $gamma$: $gamma$ is the $i$th component of the map $N_P_ito oplus (M/N_i)_P_i$. But the map $Nto oplus N/N_i$ is not injective (it has kernel $cap N_i$) so we cannot apply the exactness of localization. How to prove that $gamma$ is injective?

abstract-algebra commutative-algebra modules primary-decomposition

share|cite|improve this question

edited 2 days ago

Bernard

122k741116

asked 2 days ago

user437309user437309

734313

$endgroup$

add a comment | 

0

$begingroup$

As pointed out here, I believe, there is a typo in Eisenbud's Theorem 3.10(b). The modified statement is this:

Let $M'$ be a proper submodule of a f.g. module $M$ over a Noetherian ring $R$ and let $M'=cap_i=1^n M_i$ where each $M_i$ is $P_i$-primary. Suppose the intersection is minimal (as explained here). Then

(a) Each associated prime of $M/M'$ is equal to $P_i$ for exactly one index $i$.

(b) If $P_i$ is minimal over $operatornameAnn(M/M')$, then $M_i/M'=ker(M/M_ito (M/M_i)_P_i)$.

I understand how to prove $(a)$, but I'm not quite sure how to prove $(b)$ (without the assumption $M'=0$). Here is an attempt to adapt the proof from Eisenbud.

Let $N=M/M', N_i= M_i/M'$. Consider the commutative diagram

from Eisenbud and replace $M$ with $N$ and $M_i$ with $N_i$ in it. We need to prove that $ker alpha=N_i$. It suffices to show that $gamma,delta$ are injective.

The injectivity of $delta $ must follow from the fact that $N_i$ is $P_i$-primary. Why is this so? We only know that $M_i$ is $P_i$-primary.

For the injectivity of $gamma$: $gamma$ is the $i$th component of the map $N_P_ito oplus (M/N_i)_P_i$. But the map $Nto oplus N/N_i$ is not injective (it has kernel $cap N_i$) so we cannot apply the exactness of localization. How to prove that $gamma$ is injective?

abstract-algebra commutative-algebra modules primary-decomposition

share|cite|improve this question

edited 2 days ago

Bernard

122k741116

asked 2 days ago

user437309user437309

734313

$endgroup$

add a comment | 

$begingroup$

As pointed out here, I believe, there is a typo in Eisenbud's Theorem 3.10(b). The modified statement is this:

Let $M'$ be a proper submodule of a f.g. module $M$ over a Noetherian ring $R$ and let $M'=cap_i=1^n M_i$ where each $M_i$ is $P_i$-primary. Suppose the intersection is minimal (as explained here). Then

(a) Each associated prime of $M/M'$ is equal to $P_i$ for exactly one index $i$.

(b) If $P_i$ is minimal over $operatornameAnn(M/M')$, then $M_i/M'=ker(M/M_ito (M/M_i)_P_i)$.

I understand how to prove $(a)$, but I'm not quite sure how to prove $(b)$ (without the assumption $M'=0$). Here is an attempt to adapt the proof from Eisenbud.

Let $N=M/M', N_i= M_i/M'$. Consider the commutative diagram

from Eisenbud and replace $M$ with $N$ and $M_i$ with $N_i$ in it. We need to prove that $ker alpha=N_i$. It suffices to show that $gamma,delta$ are injective.

The injectivity of $delta $ must follow from the fact that $N_i$ is $P_i$-primary. Why is this so? We only know that $M_i$ is $P_i$-primary.

For the injectivity of $gamma$: $gamma$ is the $i$th component of the map $N_P_ito oplus (M/N_i)_P_i$. But the map $Nto oplus N/N_i$ is not injective (it has kernel $cap N_i$) so we cannot apply the exactness of localization. How to prove that $gamma$ is injective?

abstract-algebra commutative-algebra modules primary-decomposition

share|cite|improve this question

edited 2 days ago

Bernard

122k741116

asked 2 days ago

user437309user437309

734313

$endgroup$

share|cite|improve this question

edited 2 days ago

Bernard

122k741116

asked 2 days ago

user437309user437309

734313

share|cite|improve this question

edited 2 days ago

Bernard

122k741116

asked 2 days ago

user437309user437309

734313

edited 2 days ago

Bernard

122k741116

edited 2 days ago

Bernard

122k741116

asked 2 days ago

user437309user437309

734313

asked 2 days ago

user437309user437309

734313