What Stanley-Reisner rings are $\mathbb{Q}$-Gorenstein? Let $\Delta$ be a simplicial complex and let $R$ be the associated Stanley-Reisner ring. We can characterize when $R$ is Cohen-Macaulay or when $R$ is Gorenstein in terms of the topology of $\Delta$ (c.f. Stanley's book). Can we similarly characterize when $R$ is $\mathbb{Q}$-Gorenstein in terms of $\Delta$? I would be grateful for an answer even in the case when $\Delta$ is a traingulation of a manifold. 
 A: Perhaps there is hope, despite my original comment. Hartshorne in this paper developed a theory of "generalized divisors" that works on any scheme that is generically Gorenstein and $(S_2)$. 
Going quickly through his definitions and early results, the set of "almost Cartier divisors" (basically locally principal in codimension $1$) form a group. For the canonical ideal $w_R$ to be in this group, you need $R$ to be Gorenstein in codimension $1$.
Okay, so $R=k[\Delta]$ needs to be $(S_2)$ and $(G_1)$ to start with. These conditions are well-studied for Stanley-Reisner ring and can be checked relatively easily.
The last thing we need is $\mathbb Q$-Gorensteiness. This would mean that the class of $w_R$ is torsion in this group of "almost Cartier divisors". Algebraically, it says that $(I^n)^{**}$ is principal for some $n>0$, where $I$ is a fractional ideal representing $w_R$, and $^*$ means $Hom(-,R)$. 
There are formulas for canonical ideals of Stanley-Reisner ring, I believe. So perhaps you can get there with some efforts.  
