Timeline for What is the $Ass(Ext^p_R(M,R))$?
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Mar 6, 2019 at 14:38 | comment | added | Jason Starr | You are welcome. | |
| Mar 6, 2019 at 14:36 | comment | added | Tri Nguyen | Thanks you very much. @Jason Starr | |
| Mar 6, 2019 at 14:33 | comment | added | Jason Starr | No, you do not have that. Again, consider the example that $M$ is the maximal ideal $\langle x_0,\dots,x_n \rangle$ in the polynomial ring $k[x_0,\dots,x_n]$ for $n>0$. Then $\text{Ass}(M)$ equals the singleton set consisting only of the zero ideal $\{0\}$. Yet $\text{Ass}(\text{Ext}^n_R(M,R))$ equals the singleton set consisting only of the ideal $M$. | |
| Mar 6, 2019 at 14:26 | history | edited | Tri Nguyen | CC BY-SA 4.0 |
added 25 characters in body
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| Mar 6, 2019 at 14:24 | comment | added | Tri Nguyen | Do we have $Ass(M)\subseteq Ass(Ext^p_R(M,R))$ ? | |
| Mar 6, 2019 at 14:09 | comment | added | Jason Starr | Unfortunately, that does not clarify the question. Certainly all associated primes of $\text{Ext}^p_R(M,R)$ contain the annihilator ideal of $M$. Thus, each contains one of the minimal associated primes of $M$. Beyond that, what is your expectation? Consider, for instance, the special case that $R$ equals $k[x_0,\dots,x_n]$ and $M$ equals the maximal ideal $\langle x_0,\dots,x_n \rangle$. Then $p$ equals $n>0$, and $\text{Ext}^n_R(M,R)$ equals $R/M=k.$ The unique associated prime of $R/M$ equals $M$. The unique associated prime of $M$ equals $\{0\}$. Direct sums give more examples. | |
| Mar 6, 2019 at 13:50 | comment | added | Tri Nguyen | I want to know the relation of $Ass(Ext^p_R(M,R))$ with $Ass M.$ | |
| Mar 6, 2019 at 11:46 | comment | added | Jason Starr | Could you please clarify your question. What do you want to know about the set of associated primes. These primes are contained in the support of $M$. Are you asking about (upper) bounds on the heights of these primes? | |
| Mar 6, 2019 at 6:54 | history | asked | Tri Nguyen | CC BY-SA 4.0 |