Moduli functors for (flat families of) semistable coherent sheaves over a projective scheme are not representable functors, which is to say that no global universal family exists. However one can construct the coarse moduli spaces. Let $M$ be such a moduli space. Is it true that there always exists a covering in the étale topology $\{Z_i \to M\}_{i \in I}$ such that there exists a universal sheaf $\mathcal{U}_i$ over $Z_I$?
2026-03-26 12:41:02.1774528862
Étale local universal sheaves
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