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It is known that in general, the moduli space $M$ of (Gieseker) stable sheaves do not have a universal family. Therefore, a map $f \colon S \to M$ does not necessarily induce a family of sheaves.

On the other hand, there are references (the book by Huybrechts and Lehn 2nd edition on p.139 or Simpson's paper Theorem 1.21) that state that a universal family exists étale locally.

I was wondering what this meant precisely.

Does it mean that for any scheme $S$ and any map $f \colon S \to M$:

  • There exists an étale surjective map $\alpha \colon U \to M$ and our "étale-local universal family" $\mathcal{E}$ on $U \times X.$
  • There exists an étale surjective map $\beta \colon T \to S$ and a map $g \colon T \to U$ such that the map $f\beta \colon T \to S \to M$ corresponds to the family $g^*\mathcal{E},$ i.e., the morphism induced by $g^*\mathcal{E}$ is isomorphic to $f\beta?$
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    $\begingroup$ I think it means that the moduli stack $\mathcal{M}$ (the "functor" associating to every $S$ the groupoid of suitable stable sheaves on $S\times X$) admits an etale surjection from a scheme $U$. That is, a natural transformation ${\rm Hom}(-, U)\to \mathcal{M}$ (corresponding by Yoneda to a suitable family of stable sheaves parametrized by $U$) such that every map $X\to \mathcal{M}$ etale locally on $X$ lifts to $U$. This means that every family parametrized by $X$ is etale locally on $X$ the pullback of the family on $U$ via some map. The map to the coarse moduli space $M$ need not be etale. $\endgroup$
    – Piotr Achinger
    Commented Nov 22 at 8:55
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    $\begingroup$ The local maps $X\to U$ need not be unique (otherwise they would glue to a global map $X\to U$, and then $U\to\mathcal{M}$ would be an isomorphism). $\endgroup$
    – Piotr Achinger
    Commented Nov 22 at 8:56
  • $\begingroup$ @PiotrAchinger I see, thank you! $\endgroup$
    – quasicoherent_drunk
    Commented Nov 24 at 23:43

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