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.
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5$\begingroup$ To define $\mathbb Q$-Gorensteiness you typically needs the ring to be normal domains (so we can talk about class group). But Stanley-Reisner rings are almost never normal domain. $\endgroup$– Hailong DaoCommented May 5, 2020 at 21:06
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$\begingroup$ Thanks for your answer! I was hoping reduced would be enough instead of domain, but I did not know that SR rings are rarely normal... $\endgroup$– equinCommented May 7, 2020 at 13:10
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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.
answered May 7, 2020 at 18:01