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?