  In this paper, we study normalized ground states for the following critical
fractional NLS equation with prescribed mass: \begin{equation*} \begin{cases}
  (-\Delta)^{s}u=\lambda u
+\mu|u|^{q-2}u+|u|^{2_{s}^{\ast}-2}u,&x\in\mathbb{R}^{N},
  \int_{\mathbb{R}^{N}}u^{2}dx=a^{2},\\ \end{cases} \end{equation*} where
$(-\Delta)^{s}$ is the fractional Laplacian, $0<s<1$, $N>2s$,
$2<q<2_{s}^{\ast}=2N/(N-2s)$ is a fractional critical Sobolev exponent, $a>0$,
$\mu\in \mathbb{R}$. By using Jeanjean's trick in \cite{Jeanjean}, and the
standard method which can be found in \cite{Brezis} to overcome the lack of
compactness, we first prove several existence and nonexistence results for a
$L^{2}$-subcritical (or $L^{2}$-critical or $L^{2}$-supercritical) perturbation
$\mu|u|^{q-2}u$, then we give some results about the behavior of the ground
state obtained above as $\mu\rightarrow 0^{+}$. Our results extend and improve
the existing ones in several directions.
