Torsion subsheaf of coherent sheaf on locally Noetherian scheme Let $X$ be a locally Noetherian scheme and let $\mathscr{F}$ be a coherent sheaf on $X$. I would like to construct the torsion subsheaf $T(\mathscr{F})$ of $\mathscr{F}$ in the following way: for any affine open $U \subseteq X$, denote $A = \Gamma \left(U, \mathscr{O}_X \right)$ and $M = \Gamma \left(U, \mathscr{F} \right)$. Define
$$ T(\mathscr{F})(U) = T(M) = \left\{ m \in M \,|\, \exists \,a \in A\text{ non-zero divisor such that } a \cdot m = 0 \right\}.$$
I want to prove that $T(\mathscr{F})$ is a coherent subsheaf of $\mathscr{F}$. To this end, I need to show two things:
(1) Torsion commutes with localisation: for any $f \in A$, $T(M)_f = T(M_f)$.
(2) Local sections glue uniquely: if $A$ is a Noetherian ring, $(f_1,...,f_r) = A$, $m \in M$ and for every $i \leq r$, $m \in M_{f_i}$ is torsion, then $m \in M$ is torsion.
For (2), it is proven here. I tried to prove (1) myself. Firstly, there is an injective homomorphism
$$ T(M)_f \rightarrow T(M_f).$$
This follows from the fact that an element of $T(M)$ gets sent to an element of $T(M_f)$ under the localisation $M \rightarrow M_f$. I want to prove it is surjective. Let $(m, f^k) \in M_f$ be a torsion element, i.e. there exists a non-zero divisor $(a, f^j) \in A_f$ such that $(a m, f^{j+k}) \sim 0$. By definition of localisation, there exists an $l \in \mathbb{N}$ such that $f^l ( a m) = 0 \in A$. Note that $a$ is a non-zero divisor of $A$, hence $f^l m \in M$ is torsion. But then $(f^l m, f^{l + k}) \in T(M)_f$ and its image in $T(M_f)$ is $(m, f^k)$.
I would like to ask if this is correct. If it is, I am wondering why it is not written anywhere else. I know there is a similar (but non-equivalent) definition of pure sheaf which can be found in Huybrechts-Lehn. Why do people seem to prefer that one?
 A: Question: "Let X be a locally Noetherian scheme and let F be a coherent sheaf on X. I would like to construct the torsion subsheaf T(F) of F in the following way: for any affine open $U⊆X$, denote $A=Γ(U,O_X)$ and $M=Γ(U,F)$. Define
$$T(F)(U)=T(M)=\{m∈M|∃a∈A \text{ non-zero divisor such that $a⋅m=0$} \}.$$
I want to prove that T(F) is a coherent subsheaf of F."
Answer: If $X$ is a locally noetherian integral scheme, there is a canonical map
$$\phi: E \rightarrow E\otimes_{\mathcal{O}_X} K_X$$
where $K_X$ is the "sheaf of quotient fields" of $X$. For any open subscheme $U:=Spec(A) \subseteq X$ it follows $K:=K_X(U)\cong K(A) \cong \mathcal{O}_{X,\eta}$
where $\eta$ is the generic point. Let $\tilde{E}:=ker(\phi)$. It follows by an exercise in Atiyah-Macdonald that if $E(U) \cong M$ that $\tilde{E}(U)\cong T(M) \subseteq M$ is the torsion sub module of $M$. For the more general case you may try to use the sheaf of total quotient rings $K^{tot}_X$. This is claimed to exist in general in Liu's book on arithmetic algebraic geometry. From the discussion above and the reference: The sheaf $K^{tot}_X$ may not exist in complete generality. There is also a construction on the "stack project homepage" (Definition 31.23.1):
https://stacks.math.columbia.edu/tag/01X1
Here they define the sheaf $K^{tot}_X$ for any "locally ringed topological space".
Note: Whenever you find a statement with a "proof" online and you need the statement in your research, you should check the proof: The net is unreliable.
