Given a simplicial complex $X$ on vertex set $V$, $X$ is called $d$-Leray if the reduced homology group satisfying $\tilde{H}_i(Y)=0$ for all induced subcomplexes $Y\subseteq X$ (i.e., Y=X[S] for some $S\subseteq V$) and for all $i\ge d$.
One claims that "Equivalently $X$ is $d$-Leray if $\tilde{H}_i(lk(X,\sigma))=0$ for all $\sigma\in X$ and $i\ge d$", where $$lk(X,\sigma):=\{\tau\in X: \tau\cap\sigma=\emptyset, \tau\cup\sigma\in X\}$$ is the link of a simplex $\sigma \in X$. (See, e.g., page 3 of this article)
I am wondering why they are equivalent?
I am trying to see if $lk(X,\sigma)$ runs over all induced subcomplex of $X$, but it is not clear to me, since I think $lk(X,\sigma)\neq X[V-\sigma]$. ($X[V-\sigma]$ may include some simplex whose union with $\sigma$ is not in $X$?)
There's a proof in Proposition 3.1 of [1] by the same authors. They say it's "well-known".
[1] Kalai, Gil; Meshulam, Roy, Intersections of Leray complexes and regularity of monomial ideals, J. Comb. Theory, Ser. A 113, No. 7, 1586-1592 (2006). ZBL1105.13026.