  Let $\mathfrak{a}$ be an ideal of Noetherian ring $R$ and let $M$ be an
$R$-module such that $\mathrm{Ext}^i_R(R/\mathfrak{a},M)$ is finite $R$-module
for every $i$. If $s$ is the first integer such that the local cohomology
module $\mathrm{H}^s_\mathfrak{a}(M)$ is non $\mathfrak{a}$-cofinite, then we
show that $\mathrm{Hom}_{R}(R/\mathfrak{a}, \mathrm{H}^s_\mathfrak{a}(M))$ is
finite. Specially, the set of associated primes of
$\mathrm{H}^s_\mathfrak{a}(M)$ is finite.
  Next assume $(R,\mathfrak{m})$ is a local Noetherian ring and $M$ is a
finitely generated module. We study the last integer $n$ such that the local
cohomology module $\mathrm{H}^n_\mathfrak{a}(M)$ is not $\mathfrak{m}$-cofinite
and show that $n$ just depends on the support of $M$.
