I'd like to point out a small caveat in the statement of a particularly useful form of the Bondal–Orlov criterion, as the literature is a bit sloppy on this it seems. Let's be overly wordy in our explanation, so that it is clear where the issue lies. $\DeclareMathOperator\derived{\mathbf{D}} \newcommand\bounded{\mathrm{b}} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Ext{Ext}$

Recall that the Bondal–Orlov criterion is a very useful criterion to check fully faithfulness of such a functor, by checking it on the spanning class given by skyscraper sheaves at closed points, but omitting the complicated part where the criterion involving an arbitrary spanning class would require checking that the natural map \begin{equation*} \Hom_{\derived^\bounded(X)}(k(x),k(x)[i])\to\Hom_{\derived(Y)}(\Phi_{\mathcal{P}}(k(x)),\Phi_{\mathcal{P}}(k(x))[i]) \end{equation*} is an isomorphism, where the domain is ${\dim X \choose i}$-dimensional.

Rather, it suffices to show for $x,y\in X$ closed points that $$\Hom_{\derived(Y)}(\Phi_{\mathcal{P}}(k(x)),\Phi_{\mathcal{P}}(k(y))[i])$$

• is zero for $x\neq y$
• if $x=y$, then
• it is 1-dimensional for $i=0$
• it is zero for $i\notin[0,\dim X]$

Now, if $\mathcal{P}$ is a coherent sheaf, flat over $X$, then one can show that $\Phi_{\mathcal{P}}(k(x))$ is also a coherent sheaf (on $Y$), namely $\mathcal{P}_x$. In this setting, one then says that $\mathcal{P}$ is strongly simple if $\mathcal{P}_y$ is simple for all $x\in X$, and for all $x\neq y$ we have that $\Ext_Y^i(\mathcal{P}_x,\mathcal{P}_y)=0$ (= complete orthogonality).

It is said in a few places that strong simplicity is enough to apply the Bondal–Orlov criterion, but it it important to note that in Bridgeland's original formulation he takes $\dim X=\dim Y$. If $\dim X<\dim Y$ then the vanishing of $\Ext_Y^i$ for $i\geq\dim X+1$ is good enough to ensure vanishing outside the range $[0,\dim Y]$, but not necessarily in $[\dim X+1,\dim Y]$, where for $x=y$ the vanishing is a non-trivial condition to check.

Hence the notion of strongly simple should in general be dependent on the dimension of the target. This seems to be omitted in quite a few places. Exercise for the reader: construct a strongly simple (in the original definition) functor which is not strongly simple in the definition dependent on $Y$.