# Purity and skyscraper sheaves

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?

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.
• 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. – LarryFisherman Jul 10 at 9:36