Associated primes and cofiniteness of local cohomology modules

Let $\mathfrak{a}$ be an ideal of Noetherian ring $R$ and let $M$ be an $R$-module such that ${\mathrm{E}\mathrm{x}\mathrm{t}}_R^i(R\mathrm{/}\mathfrak{a},M)$ is finite $R$-module for every $i$. If $s$ is the first integer such that the local cohomology module ${\mathrm{H}}_{\mathfrak{a}}^s(M)$ is non $\mathfrak{a}$-cofinite, then we show that ${\mathrm{H}\mathrm{o}\mathrm{m}}_R(R\mathrm{/}\mathfrak{a},{\mathrm{H}}_{\mathfrak{a}}^s(M))$ is finite. Specially, the set of associated primes of ${\mathrm{H}}_{\mathfrak{a}}^s(M)$ 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 ${\mathrm{H}}_{\mathfrak{a}}^n(M)$ is not $\mathfrak{m}$-cofinite and show that $n$ just depends on the support of $M$.