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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:

  1. $\mu_{ii}$ is the identity homomorphism on $M_i$ for each $i \in I$
  2. 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?

  • 2
    In mathematics everything *always* follow from the axioms (note that a definition of a mathematical notion is actually a scheme of axioms, or equivalent axioms).2011-05-31
  • 0
    my previous response was incorrect: I'm sorry to say that I didn't carefully read the question. I'll try again later on...2011-05-31
  • 1
    It seems to me that the question has been misunderstood so far. I think that it is asked for a proof for this property "an element is zero in the direct limit iff it is eventually zero" which only uses the *defining universal property* of the direct limit. So no explicit construction (or cheating like in Dylan's answer) is allowed. For related questions, see http://mathoverflow.net/questions/10930 and http://mathoverflow.net/questions/869232012-04-14

2 Answers 2