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Flat twisted sheaves are mentioned in Căldăraru's thesis (Lemma 2.1.2 for example), but I'm confused about how they should be defined. I have in mind some possibilities, given an $\alpha$-twisted sheaf $\mathcal F \in \operatorname{Mod}(X,\alpha)$:

  1. The functor $\mathcal F \otimes - \colon \operatorname{Mod}(X,\beta) \to \operatorname{Mod}(X,\alpha \beta)$ is exact for all $\beta$ in the Brauer group of $X$.
  2. The functor $\mathcal F \otimes - \colon \operatorname{Mod}(X) \to \operatorname{Mod}(X,\alpha)$ is exact.
  3. The functor $\mathcal F \otimes - \colon \operatorname{Mod}(X,\alpha^{-1}) \to \operatorname{Mod}(X)$ is exact.
  4. Writing $\mathcal F = (\mathcal F_i, \varphi_{ij})$ on some cover $\mathfrak U = \{ U_i \}$, $\mathcal F_i$ is a flat $\mathcal O_{U_i}$-module for all $i$.

Are those conditions equivalent? Or perhaps none of them is the correct definition?

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  • $\begingroup$ 2,3,4 seem all equivalent to me, by viewing a-twisted sheaves (and more generally a^k-twisted sheaves) as sheaves on the total stack of the gerbe a $\endgroup$
    – Yosemite Sam
    Feb 20, 2017 at 15:45
  • $\begingroup$ What about the first one? $\endgroup$
    – Francesco Genovese
    Feb 20, 2017 at 16:14

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