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For $X$ a smooth quasi-projective variety and $X^{[n]}$ its associatedHilbert scheme of $n$ points, we study two canonical Fourier--Mukai transforms$D(X)\to D(X^{[n]})$, the one along the structure sheaf and the one along theideal sheaf of the universal family. For $\dim X\ge 2$, we prove that bothfunctors admit a left-inverse. This means in particular that both functors arefaithful and injective on isomorphism classes of objects. Using another method,we also show in the case of an elliptic curve that the Fourier--Mukai transformalong the structure sheaf of the universal family is faithful and injective onisomorphism classes. Furthermore, we prove that the universal family of$X^{[n]}$ is always flat over $X$, which implies that the Fourier--Mukaitransform along its structure sheaf maps coherent sheaves to coherent sheaves.

Some ways to reconstruct a sheaf from its tautological image on a Hilbert scheme of points