I am thinking about the morphism from the moduli stack of filtered coherent sheaf on $X$ to the moduli stack of coherent sheaves defined by forgetting filtrations,i.e.$Filtcoh(X)$ $\rightarrow$ $Coh(X)$(I am following the notation of the paper “ Components of the stack of torsion-free sheaves of rank 2 on ruled surfaces” by C.Walter ) ,where $X$ is a projective surface over an algebraic closed field $k$.

Is this morphism of algebraic stacks smooth? (at least flat ?)



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