# A caveat on the notion of strongly simple

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 \begin{equation} \Hom_{\derived(Y)}(\Phi_{\mathcal{P}}(k(x)),\Phi_{\mathcal{P}}(k(y))[i]) \end{equation}

- 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 $Y$.