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I have a question about the definition of Q-Gorenstein variety.

I saw a definition of Q-Gorenstein variety：for a normal variety $X$, it's Q-Gorenstein if the canonical divisor is Q Cartier. I wonder how to use this definition (about canonical divisor) to see the structure ring is Q-Gorenstein. Through some duality?

Maybe there's no such thing like $\mathbb Q$-Gorenstein ring? Basically, I want to know from canonical divisor is $\mathbb Q$ Cartier, what can we say about the structure ring R. Here I mean locally X=Spec(R).

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    $\begingroup$ The structure ring is $\Bbb{Q}$-Gorenstein, not Gorenstein. $\endgroup$
    – abx
    Commented Sep 2, 2020 at 6:48
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    $\begingroup$ @abx Thanks,edited it $\endgroup$
    – xin fu
    Commented Sep 2, 2020 at 16:41
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    $\begingroup$ What is your definition of $\Bbb{Q}$-Gorenstein? For me, it is the property that you quote. $\endgroup$
    – abx
    Commented Sep 2, 2020 at 18:33
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    $\begingroup$ Again: what is, for you, the meaning of "the ring R is Q-Gorenstein"? What definition do you have in mind? $\endgroup$
    – abx
    Commented Sep 3, 2020 at 4:25
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    $\begingroup$ I do not know of any other definition of $\Bbb{Q}$-Gorenstein. For Gorenstein it is equivalent to finite injective dimension, as you will find in any textbook in commutative algebra. $\endgroup$
    – abx
    Commented Sep 3, 2020 at 6:10

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