Timeline for Projective & injective dimensions
Current License: CC BY-SA 3.0
11 events
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| May 2, 2012 at 13:08 | history | edited | David White | CC BY-SA 3.0 |
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| Aug 16, 2010 at 21:19 | vote | accept | ashpool | ||
| Aug 8, 2010 at 2:09 | comment | added | ashpool | Thanks. In the last step, what was the condition exactly that allowed you to conclude that $M$ is projective from $M$ being injective? | |
| Aug 7, 2010 at 21:08 | comment | added | Hailong Dao | @kwan: here is a proof from top of my head, but there might be easier ones: replacing $M$ by high syzygy, can assume $M$ is max. Cohen-Macaulay. Then kill a full reg. seq., -> assume $R$ is Artinian. But then $M$ is injective, so projective. | |
| Aug 7, 2010 at 20:33 | comment | added | ashpool | Thanks. The same method (of using minimal free resolution) doesn't seem to work to show the converse, if $A$ is Gorenstein and $M$ has a finite injective dimension then $M$ has a finite projective dimension. I tried to use finite injective resolution instead but it doesn't seem to work either. I would appreciate any suggestions. | |
| Aug 7, 2010 at 18:08 | comment | added | Hailong Dao | @kwan: break the res. into short exact sequences and use the fact that if 2 modules have fi. inj. dim., so is the third one. For ref (without proof), look at "Cohen-Macaulay rings" 1st ed by Bruns-Herzog, Section 3.1, esp. exer. 3.1.25. | |
| Aug 7, 2010 at 3:34 | comment | added | ashpool | I'm sorry, I still have no idea how to proceed. To begin with, if $A$ has a finite injective dimension and $M$ is projective over $A$, why does $M$ have a finite injective dimension? If you could point to any reference that would be great, too. | |
| Aug 6, 2010 at 17:23 | comment | added | Hailong Dao | You can use induction on the length of the min. free res. of $M$, for instance. | |
| Aug 6, 2010 at 16:10 | comment | added | ashpool | Thanks. How do I see that if $R$ has a finite injective dimension then finite projective dimension of $M$ implies finite injective dimension of $M$? | |
| Aug 6, 2010 at 1:03 | history | edited | Hailong Dao | CC BY-SA 2.5 |
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| Aug 6, 2010 at 0:55 | history | answered | Hailong Dao | CC BY-SA 2.5 |