On the highest Lyubeznik number of a local ring

Abstract

Let $A$ be a $d$-dimensional local ring containing a field. We will prove that the highest Lyubeznik number ${\lambda }_{d,d}(A)$ (defined in \cite{l1}) is equal to the number of connected components of the Hochster-Huneke graph (defined in \cite{hh}) associated to $B$, where $B=\widehat{{\hat{A}}^{sh}}$ is the completion of the strict Henselization of the completion of $A$. This was proven by Lyubeznik in characteristic $p>0$. Our statement and proof are characteristic-free.