# Integral Leray Number?

The Leray number of a finite simplicial complex $K$ relative to a field $\Bbbk$ is defined to be the least $d\geq 0$ such that $\widetilde H^n(C,\Bbbk)=0$ for all $n\geq d$ and all induced subcomplexes $C$ of $K$.

The Leray number is the Castelnuovo-Mumford regularity of the Stanley-Reisner ring of $K$ with base field $\Bbbk$ and is also relevant to Helly problems in combinatorial geometry.

Question. Do people consider the Leray number with respect to arbitrary commutative rings (defined analogously), and in particular, with respect to $\mathbb Z$? Does it still have meaning with respect to Castelnuovo-Mumford regularity and Helly theorems?

-