I'm reading Beilinson's paper "Constructible sheaves are holonomic"
In 4.6 of this paper, he used the following fact:
Consider sheaf of $\Lambda$-modules, where $\Lambda=\mathbb{Z}/l^N\mathbb{Z}$ for some $N$ fixed, $l\neq p=\text{char}\,k$.
For $X=\mathbb{P}^N_k$, $\mathcal{G}$ a simple perverse sheaf on $X$,
let $D$ be the minimal closed subset s.t. $\mathcal{G}|_{X-D}$ is locally constant,
then $D$ is of pure codimension 1, i.e. an divisor, called ramification divisor of $\mathcal{G}_{\eta}$
I think $X$ could be replaced by any smooth variety, or something like that.
Then we need to show that given $Y\subset X$ of codimension greater than $2$,
$\mathcal{G}$ simple perverse sheaf on $X$ s.t. $\mathcal{G}|_{X-Y}$ is locally constant, then $\mathcal{G}$ is locally constant
I didn't find a way to prove this.
If we are talking about constructible sheaves of sets, the locally constant constructible sheaf $\mathcal{G}|_{X-D}$ is represented by a finite etale covering over $X-D$. By Zariski-Nagata purity, this can be extended to a finite etale covering of $X$, which should represent $\mathcal{G}$ (by property of intermediate extension) as a locally constant sheaf? I'm not sure whether this is a correct argument, and I don't think this could be generalised to the case for modules.
The author didn't provide any explanation for this fact. I think this should be a well-known theory. Is there any good reference?