On $\mathbb{P}^2$, we have two full strong exceptional collections:


$\{\mathcal{O}\_{\mathbb{P}^2},\Omega\_{\mathbb{P}^2}(2),\mathcal{O}\_{\mathbb{P}^2}(1)\}$ and $\{\mathcal{O}\_{\mathbb{P}^2}(-2), \mathcal{O}\_{\mathbb{P}^2}(-1), \mathcal{O}\_{\mathbb{P}^2}\}$. 


We use $B\_1,B\_2$ to denote the endomorphism algebra of the direct sum of the two collections respectively. Consider a quiver $Q$ with three vertices $v\_0,v\_1,v\_2$ and three arrows $a\_i$ from $v\_0$ to $v\_1$ and $b\_i$ from $v\_1$ to $v\_2$. Define two ideals $J\_1$,$J\_2$ where $J\_1$ is generated by $b\_ia\_j+b\_ja\_i$ while $J\_2$ is generated by $b\_ia\_j-b\_ja\_i$. Then $B\_1\simeq \mathbb{C}Q/J\_1$ and $B\_2\simeq \mathbb{C}Q/J\_2$.


In the paper of [Ryo Ohkawa][1]


[1]: <https://www.jstage.jst.go.jp/article/kodaimath/33/2/33_2_329/_article/-char/ja/> he computed that using the equivalence $RHom(\mathcal{O}\_{\mathbb{P}^2}\oplus\Omega\_{\mathbb{P}^2}(2)\oplus\mathcal{O}\_{\mathbb{P}^2}(1),-):D^b(P^2)\to D^b(mod-B\_1)$, the preimage of the abelian category $(mod-B\_1)$ is the extension closure


$\mathcal{A}=<\mathcal{O}\_{\mathbb{P}^2}(-2)[2], \mathcal{O}\_{\mathbb{P}^2}(-1)[1], \mathcal{O}\_{\mathbb{P}^2}>$


Objects in this category will be complexes
$\mathcal{O}\_{\mathbb{P}^2}(-2)^{\oplus a}\to\mathcal{O}\_{\mathbb{P}^2}(-1)^{\oplus b}\to\mathcal{O}\_{\mathbb{P}^2}^{\oplus c}$ which give representations of $Q$ with relation $J\_2$. So I believe $\mathcal{A}\simeq (B\_2-mod)$. (EDIT: This is wrong. I did not account for the fact that d^2=0. When this is noted, we obtain $B\_1-mod$!)


Noting the symmetry of elements in $J\_1$, we see $B\_1\simeq B\_1^{op}$, thus combining all this it seems we have $(B\_2-mod)\simeq (B\_1-mod)$, which is a little surprising to me.


My question: Is this an example of Morita equivalence or I made a mistake somewhere? If this is an example, is there a general theory of such a construction using exceptional collections?


Thanks for the help.



Morita equivalence of quivers from related exceptional collections
The algebras $B\_1$ and $B\_2$ are NOT Morita-equivalent. Indeed, their categories of modules have exactly three simple (i.e., without non-trivial subobjects) objects, that correspond to the vertices of the quivers, and the dimensions of $Ext$-spaces between these simple objects is $(3,3,3)$ for one quiver and $(3,3,6)$ for the other.


However, the categories are obviously DERIVED Morita-equivalent.


