We know that for a curve $X$, any object $\mathcal{E}^{\bullet}$ in the derived category $D^b\_{\text{coh}}(X)$ is formal, i.e. $\mathcal{E}^{\bullet}$ is quasi-isomporphic to the direct sum of its cohomology sheaves. The reason is that the cohomological dimension of $X$ is $1$. We can see Corollary 3.15 of Daniel Huybrechts' book "Fourier–Mukai transforms in algebraic geometry" for details.


Now could we find an "easy" example of object in $D^b\_{\text{coh}}(\mathbb{P}^2)$ which is not formal? In particular, could we find a complex of sheaves on $\mathbb{P}^2$ of length $2$ with coherent cohomology which is not quasi-isomorphic to the direct sum of its cohomology sheaves?



An example of an object in $D^b_{\text{coh}}(\mathbb{P}^2)$ which is not formal
Yes; whenever you have two objects in an abelian category such that $Ext^2(M,N)$ is not equal to 0, we have a nonformal object given by coning with this morphism. More down-to-earthly, the element of $Ext^2(M,N)$ is given by some complex $N \to K \to L\to M$; the non-formal complex is just $\cdots \to 0\to K \to L \to 0\to \cdots$.


**EDIT:** Thanks for the example below. I was too lazy to provide one, but I also think it risks camouflaging the actual point here, since there's nothing special about coherent sheaves on $\mathbb{P}^2$, this happens in any abelian category with global dimension $>1$.


