The projective cover of the trivial representation for a finite group of Lie type in defining characteristic

We give a lower bound of the Loewy length of the projective cover of the trivial module for the group algebra $kG$ of a finite group $G$ of Lie type defined over a finite field of odd characteristic $p$, where $k$ is an arbitrary field of characteristic $p$; the proof uses Auslander-Reiten theory.