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In "The Geometry of moduli spaces of sheaves" a coherent sheaf $\mathcal{F}$ is defined to be pure of dimension $d$ if dim$(\mathcal{E})=d$ for all non-trivial proper subsheaves $\mathcal{E} \subset \mathcal{F}$. By this definition, the skyscraper sheaf sky$_p(\mathbb{C}^n)$ should be pure. This is because the only non-trivial proper subsheaves of sky$_p(\mathbb{C}^n)$ are of the form sky$_p(\mathbb{C}^k)$ for $0 < k < n$ and $$dim({\rm sky}(\mathbb{C}^n)) = dim({\rm sky}(\mathbb{C}^k)) = 0.$$ However, I am under the impression that purity is supposed to be a generalization of torsion-freeness and skyscraper sheaves are torsion sheaves. Have I incorrectly classified the non-trivial proper subsheaves of skyscraper sheaves?

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There is nothing wrong here. Let $X$ be irreducible of dimension $n$. Any sheaf of dimension less than $n$ is torsion, since it is annihilated by a function vanishing on the support of the sheaf. Sheaves which are pure of dimension $n$ are torsion free, since if they were not torsion free they would have a torsion subsheaf supported on a proper subvariety.

A pure sheaf with irreducible support is torsion-free when view as a sheaf on its support.

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  • $\begingroup$ Oh, I see. So in the phrase "$\mathcal{F}$ is pure of dimension $d$" where $\mathcal{F}$ is a sheaf on an $n$-equidimensional scheme $X$ one must take $n=d$. Thank you. $\endgroup$ – LarryFisherman Jul 10 at 9:36
  • $\begingroup$ That's if you want to get torsion-free sheaves. It is totally fine to study e.g. sheaves of pure dimension 1 on a surface, but they will all be torsion sheaves supported on curves. $\endgroup$ – Jack Huizenga Jul 10 at 18:19

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