  In this paper, we are concerned with the critical Hartree equation
\begin{equation*} \begin{cases} -\Delta
u=\left(\displaystyle{\displaystyle{\int_{\Omega}}}\frac{u^{2^{*}_{\mu}}(y)}{|x-y|^{\mu}}dy\right)u^{2^{*}_{\mu}-1}+\varepsilon
u,\quad u>0,\quad &\text{in $\Omega$,}\\ u=0,\quad &\text{on $\partial\Omega$,}
\end{cases} \end{equation*} where $\Omega\subset \mathbb{R}^N$ ($N\geq 5$) is a
smooth bounded domain, $\mu\in (0,4)$ and $2^{*}_{\mu}=\frac{2N-\mu}{N-2}$ is
the upper critical exponent in the sense of the Hardy-Littlewood-Sobolev
inequality. Under a non-degeneracy condition on the critical point
$x_0\in\Omega$ of the Robin function $R(x)$, we perform that for
$\varepsilon>0$ sufficiently small, the Morse index of the blow-up solutions
$u_\varepsilon$ concentrating at $x_0$ can be computed in terms of the negative
eigenvalues of the Hessian matrix $D^{2}R(x)$ at $x_0$. Compared with the usual
local cases, our problem is non-local due to the nonlinearity with
Hartree-type, and several difficulties arise and new estimates of the
eigenpairs $\{\left(\lambda_{i,\varepsilon},v_{i,\varepsilon}\right)\}$ to the
associated linearized problem at $u_{\varepsilon}$ should be introduced. To our
knowledge, this seems to be the first paper to consider the qualitative
analysis of a Hartree type Br\'ezis-Nirenberg problem and our results extend
the works established by M. Grossi et al in \cite{GP} and F. Takahashi in
\cite{Ta3} to the non-local case.
