Question: Does it follow from the axioms for a direct limit that if $\mu_i(x_i)=0$ then there exists $j \geq i$ such that $\mu_{ij}(x_i)=0$?
Definitions and notation:
(Atiyah MacDonald, chapter 2, question 14 and 15 give the following construction for a direct limit of modules over a ring):
Begin with a directed set $I$. This is a partially ordered set $(I,\leq)$ such that for every $i,j \in I$, there exists $k \in I$ such that $i \leq k$ and $j \leq k$.
If $A$ is a ring, $I$ is a directed set, and $(M_i)_{i \in I}$ a family of modules with $A$-module homomorphisms $\mu_{ij}: M_i \rightarrow M_j$ for each pair $i \leq j$ such that the following axioms hold:
- $\mu_{ii}$ is the identity homomorphism on $M_i$ for each $i \in I$
- if $i \leq j \leq k$, then $\mu_{ik}=\mu_{jk}\circ\mu_{ij}$
then $(M_i,\mu_{ij})$ is called a direct system over $I$.
The direct limit of $(M_i,\mu_{ij})$ is constructed as follows:
Take $C=\bigoplus_{i \in I} M_i$, and let $D = \langle x_i - \mu_{ij}(x_i) | x_i \in M_i, i \leq j \rangle \leq C$. Identify each $M_i$ with its image in $C$. Form the quotient via $\mu: C \rightarrow M:=C/D$, and let $\mu_i$ be the restriction of $\mu$ to $M_i$.
The module $M$, together with the homomorphisms $\mu_i$, is the direct limit of the direct system $(M_i,\mu_{ij})$.
Note: This is part of problem 2.15 from Atiyah-MacDonald. There are several attempts at this problem available; for example,
The better solutions of these rewrite the axioms in terms of the construction of a stalk (for example, in Hartshorne II.1, or the second paragraph of the answer here).
I would like to know whether the property of a direct limit that if $\mu_i(x_i)=0$ then there exists $j \geq i$ such that $\mu_{ij}(x_i)=0$ follows directly from the axioms given, or whether the stalk construction can be shown to be equivalent to the axioms given above?