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Let $\mathcal{X}$ be an integral algebraic space (or stack). As given in the Stacks Project http://stacks.math.columbia.edu/tag/0AD3 , the global section $\Gamma (\mathcal{X},\mathcal{O_X})$ of the structure sheaf is a domain. But what about the other sections? In short, is the structure sheaf torsion-free?

Thanks in advance!

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  • $\begingroup$ Sections over what? For every $1$-morphism that is represented by open immersions, the domain of that stack is again integral, so the sections are again an integral domain. Even for schemes, for 'etale morphisms, the domain can be disconnected, and thus the global sections would not be a domain. Are you asking about nilpotent elements in the ring of global sections? $\endgroup$
    – Jason Starr
    Commented Dec 3, 2016 at 11:00
  • $\begingroup$ I am asking about those 1-morphisms that are represented by etale maps which are not open immersions. The ring of global sections $\Gamma(\mathcal X_{etale}, \mathcal {O_X})$ of the small etale site of $\mathcal X$ is shown to be an integral domain in the Stack Project link I have shared. But I don't know if $\mathcal O_X(U \xrightarrow{etale} \mathcal X):= \mathcal O_U(U)$ would also be an integral domain since being integral is not an etale local property of schemes. $\endgroup$
    – Sam
    Commented Dec 3, 2016 at 14:43
  • $\begingroup$ Consider the case that $U\xrightarrow{\text{etale}} X$ is a disjoint union of two copies of $X$. $\endgroup$
    – Jason Starr
    Commented Dec 3, 2016 at 14:45
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    $\begingroup$ Now if $\mathcal {O_X}(U \xrightarrow{etale} \mathcal X)$ is not an integral domain in general, then by the usual definition of a torsion-free sheaf, the structure sheaf wouldn't be torsion-free. But it seems fair enough to ask if we have the structure sheaf torsion-free while considering an integral stack...because this happens in case of an integral scheme. $\endgroup$
    – Sam
    Commented Dec 3, 2016 at 14:49
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    $\begingroup$ @Jason...Sorry but I didn't understand what you meant by that case of disjoint union of copies of $X$. Could you kindly explain a little ? $\endgroup$
    – Sam
    Commented Dec 3, 2016 at 14:54

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