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If $F$ is a sheaf on usual Zariski site of an integral scheme $X$ (each $O_U(U)$ is a domain), then it is called torsion-free if $F(U)$ is torsion-free $O_U(U)$ module for every $U\in X_{Zar}$.

Now how to define a torsion-free sheaf on the small etale site of a integral scheme? The problem is unlike the Zariski case, $O_U(U)$ are not integral domains ( reduced but not irreducible/connected) when $U \in X_{et}$. Moreover, this also shows that even the structure sheaf for $X_{et}$ is not torsion-free.

So my first guess was to define $F$ in $X_{et}$ as torsion-free if $F|_{X_{zar}}$ is torsion-free. However, it seems to me that I am making some mistake.

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    $\begingroup$ It just means a torsion free abelian group for all $U$. $\endgroup$ Dec 8, 2016 at 9:40
  • $\begingroup$ We are looking for torsion free as $\mathcal O_U(U)$-modules. More generally, we would like to do this by setting the torsion submodule to $0$. If $M$ is an $A$-module, its torsion submodule is not defined unless $A$ is an integral domain. However, if $U=Spec(A)\longrightarrow X$ is an etale morphism, then $A$ is not necessarily an integral domain (it may be shown that $A$ is reduced). $\endgroup$
    – Sam
    Dec 8, 2016 at 15:59
  • $\begingroup$ @sam: torsion free makes sense for reduced rings $A$: a torsion element is one which is annihilated by some non-zero divisor in $A$. More geometrically, a f.g. module is torsion free if the support of the corresponding coherent sheaf is a union of irred. comps. of Spec $A$. $\endgroup$
    – tracing
    Dec 12, 2016 at 20:08
  • $\begingroup$ @tracing: Isn't some condition like $A$ Noetherian (or that the irreducible components of Spec $A$ are locally finite) required? Could you help me with a reference to read about this torsion free modules for reduced rings ? $\endgroup$
    – Sam
    Dec 14, 2016 at 8:16
  • $\begingroup$ @sam: I think the definition makes sense generally (even for non-reduced rings), but you are right that for the theory of associated primes to behave well under localization one needs something like a Noetherian hypothesis. (In the non-Noetherian context, the formation of associated primes need not commute with localization.) I've not thought about things like the stack of torsion free sheaves in a non-Noetherian context (since people usually consider such things for varieties and other highly Noetherian situations); does the Stacks project say anything? $\endgroup$
    – tracing
    Dec 14, 2016 at 12:15

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