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?
Best Answer
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 $(0,0,\ldots,0,g_m,-\alpha(g_m),0,0,\ldots)$.