Hatcher's definition of direct limit of a sequence of homomorphisms of abelian groups

Question

In section 3.F of Algebraic Topology, Hatcher writes

... the direct limit $\displaystyle\lim_{\longrightarrow}G_i$ of a sequence of homomorphisms of abelian groups $G_1\xrightarrow{\alpha_1}G_2\xrightarrow{\alpha_2}G_3\to\cdots$ is defined to be the quotient of the direct sum $\bigoplus_i G_i$ by the subgroup consisting of elements of the form $(g_1, g_2-\alpha_1(g_1), g_3-\alpha_2(g_2), \cdots)$.

Perhaps I misunderstand. However, by letting $g_1=\cdots=g_{m-1}=0$, $g_m\in G_m$ be arbitrary and $g_n=\alpha_{n-1}(\alpha_{n-2}(\cdots(\alpha_m(g_m))\cdots))$ for $n>m$, don't we have $(g_1, g_2-\alpha_1(g_1), g_3-\alpha_2(g_2), \cdots)=(0, \ldots, 0, g_m, 0, \ldots)$? If this is the case, the subgroup to which Hatcher refers is the whole direct sum. Where have I made a mistake?

Answer 1

You are right; if Hatcher wrote that, then that is poor writing.

The subgroup is made up of such elements with the property that only finitely many of the $g_i$ are nonzero.

I would describe the subgroup as that generated by elements $(0,0,\ldots,0,g_m,-\alpha(g_m),0,0,\ldots)$.

Answer 2

Hatcher is in fact a little imprecise because he does't say anything about the sequence $(g_1,g_2,g_3,..)$. What he means is the subgroup of elements of the form $(g_1, g_2-\alpha_1(g_1), g_3-\alpha_2(g_2), \cdots)$ with $(g_1,g_2,g_3,..) \in \bigoplus_i G_i$. Your example is based on a sequence which in general does not belong to the direct sum.

Formally, define $\alpha : \bigoplus_i G_i \to \bigoplus_i G_i, \alpha(g_1,g_2,g_3,...) = (0, \alpha_1(g_1), \alpha_2(g_2), \alpha_3(g_3),...)$. This is a group homomorphism and the above subgroup is the image of the group homomorphism $id - \alpha$.