A counterexample to a conjecture of Lynch

Let $R$ be a commutative Noetherian ring, $mathfrak a$ a proper ideal of $R$ and $N$ a non-zero finitely generated $R$-module with $Neq mathfrak a N$. Let $c$ be the greatest non-negative integer $i$ such that the local cohomology $operatorname{H}^i_{mathfrak a}(N)$ is non-zero. In this paper, we provide a sharp bound under inclusion for the annihilator of the local cohomology module $operatorname{H}^c_{mathfrak a}(N)$ and, by using this bound, we construct a counterexample to Lynch s conjecture.