LAUSR

Let $R$ be a commutative Noetherian ring, $\mathfrak a$ a proper ideal of $R$ and $N$ a nonzero finitely generated $R$-module with $N\neq \mathfrak a N$. Let $c$ be the greatest nonnegative integer $i$ such that the local cohomology $\operatorname{H}^i_{\mathfrak a}(N)$ is nonzero. In this paper, we provide a sharp bound under inclusion for the annihilator of the top local cohomology module $\operatorname{H}^c_{\mathfrak a}(N)$ and this annihilator is computed in certain cases. Also, using this bound, we construct a counterexample to Lynch's conjecture.