Timeline for An example of an object in $D^b_{\text{coh}}(\mathbb{P}^2)$ which is not formal

8 events

when toggle format	what		by	license	comment
May 13, 2015 at 19:32	history	edited	Ben Webster♦	CC BY-SA 3.0	edited body
Feb 7, 2015 at 20:22	vote	accept	Zhaoting Wei
Feb 7, 2015 at 18:18	comment	added	abx		Oops -- right, thanks, sorry for the typo.
Feb 7, 2015 at 18:15	history	edited	Ben Webster♦	CC BY-SA 3.0	added 274 characters in body
Feb 7, 2015 at 18:04	comment	added	Sasha		@abx: $O^3$ should be $O(-1)^3$.
Feb 7, 2015 at 16:59	comment	added	abx		Just take the Koszul complex associated to the surjective map $\ \mathscr{O}_{\mathbb{P}^2}(-1)^3\rightarrow \mathscr{O}_{\mathbb{P}^2}\ $ given by multiplication by $X,Y,Z$. The corresponding morphism is $\ \mathscr{O}_{\mathbb{P}^2}(-1)^3\rightarrow \mathscr{O}_{\mathbb{P}^2}^3\;$, given by the matrix $\pmatrix{0 & Z & -Y\\ -Z & 0 & X\\ Y & -X & 0}$.
Feb 7, 2015 at 16:46	comment	added	Zhaoting Wei		Thank you! Maybe I need a more explicit construction. For example we know that $Ext^2_{\mathbb{P}^2}(\mathcal{O},\mathcal{O}(-3))\neq 0$ hence as you pointed out we have a complex $\mathcal{O}(-3)\rightarrow K \rightarrow L\rightarrow \mathcal{O}$. Now could we find an explicit expression of the $K$ and $L$?
Feb 7, 2015 at 16:33	history	answered	Ben Webster♦	CC BY-SA 3.0