# Characterization of torsion free sheaves

In "The Geometry of Moduli Spaces of Sheaves" by Huybrechts and Lehn a torsion free sheaf is defined as coherent sheaf $$E$$ on an integral Noetherian scheme $$X$$ s.t. for every $$x\in X$$ and every non-zero germ $$s\in O_{X,x}$$, multiplication by $$s$$ $$E_x\to E_x$$ is injective.

It is then stated that this definition is equivalent to $$T(E)=T_{d-1}(E)=0$$ where $$d=\dim X$$ and $$T_{d-1}(E)$$ is the maximal subsheaf of $$E$$ of dimension $$\leq d-1$$.

I am trying to prove this equivalence. I first restricted to the case of $$X=\text{Spec} A$$ affine and so $$E=\tilde{M}$$ for $$M$$ a finitely generated $$A$$-module. In this case I think I was able to prove that $$E$$ is torsion free if and only if $$\forall m\in M$$ we have $$\text{Ann}(m)={0}$$, which is equivalent to $$V(\text{Ann}(m))=X$$ (since $$A$$ is an integral domain) and thus to $$\text{Supp}({mM})=X$$, which amounts to say $$\dim \tilde{mM}=d$$ (since $$\text{Supp}({mM})$$ is closed). Since every submodule of $$M$$ contains a cyclic submodule of the form $$mM$$ we should be done.

Is this argument correct? And is it enough restricting to the affine case? Of course the definition of torsion-freeness is local but I am not sure about the vanishing of $$T(E)$$.

$$T(E)$$ is local in the sense that for any $$U\subset X$$ open, we have $$T(E|_U)=T(E)|_U$$. Why? $$T(E)|_U$$ is a subsheaf of $$E|_U$$ of dimension $$\leq d-1$$, so it necessarily injects into the maximal such subsheaf, and this injection is an isomorphism because the stalks are the same - we don't notice restriction to an open subset when taking stalks. So your argument should be fine (assuming I didn't miss something else).
Another idea to show this without resorting to the affine case is the following. Suppose $$x\in X$$ is contained in the support of $$T(E)$$. Since $$T(E)$$ is supported on a closed subset of codimension at least one, that means that in $$\mathcal{O}_{X,x}$$, there exists a germ which vanishes along the support of $$T(E)$$, and this germ produces a non-injective multiplication map on the stalk of $$E$$ at $$x$$.