Timeline for the generators of $I$ when $\mathbb C[x,y]/I$ is Gorenstein with zero dimension
Current License: CC BY-SA 3.0
3 events
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| Aug 14, 2012 at 13:31 | comment | added | Mohan | Let me describe one of may ways of seeing this. By the very nature of it, $P$ is torsion free and hence has depth at least one. That it comes from a generator of $Ext^1(I,R)$ also implies $Ext^1(P,R)=0$. If $P$ is not projective, by Auslander-Buchsbaum formula, $pd P$ must be one. Writing a resolution $0\to A\to B\to P\to 0$ with $A,B$ free, using the fact that $Ext^1(P,R)=0$, one easily checks that $P$ must be projective. | |
| Aug 14, 2012 at 0:49 | comment | added | Xingting | Thanks for your reply. Could you explain in details why the identity $1\in Ext^1(I,R)$ corresponds to a ses $0\rightarrow R\rightarrow P\rightarrow I\to 0$ where $P$ is projective? | |
| Aug 10, 2012 at 13:40 | history | answered | Mohan | CC BY-SA 3.0 |