the natural map from $N_1$ to $\varinjlim N_i$ is injective?

Question

Direct limit $\varinjlim N_i$ of a direct system $\{N_i\}_{i \in I}$ is defined to the union of all $N_i$ modulo certain equivalence relations. From the definition it seems that if $I=\mathbb{Z}_{>0}$ the set of integers greater than zero, then the natural map from $N_1$ to $\varinjlim N_i$ is injective. Is this correct? If so more generally, can I say that there is an inclusion map from each $N_i$ to the direct limit $\varinjlim N_i$?

Answer 1

The fact that you are taking the "union" is a bit strange, in fact, if the maps in the direct system are not injective, then it is not easy to make sense of this union.

Let me fix some notation:

-- we can let, as in your question, $I=\mathbb Z_{>0}$;

-- call $\phi_{j,i}:N_i\to N_j$, where $i\leq j$, the connecting map;

-- let $\phi_i:N_i\to \varinjlim_IN_i$.

Now, a construction of the direct limit can be given as follows: first take the direct sum $\bigoplus_IN_i$ with the natural maps $\varphi_i:N_i\to \bigoplus_IN_i$. Now, consider the following submodule: $$K:=\left\langle \varphi_j(x)-\varphi_i\phi_{j,i}(x): x\in N_j,\text{ and } i\leq j\right\rangle $$ Then, $\varinjlim_IN_i$ is isomorphic to $\bigoplus_IN_i/K$ and the maps $\phi_j:N_j\to \varinjlim_IN_i$ are conjugated to the composition $N_j\to \bigoplus_IN_i\to \bigoplus_IN_i/K$.

Thus, you can see that the kernel of $\phi_j$ corresponds exactly to those element $x\in N_j$ such that $\varphi_j(x)$ belongs to $K$.

After some manipulation of the above constructions, you can see that this means exactly that $x\in \mathrm{Ker}(\phi_j)$ if and only if there exists $k>j$ such that $\phi_{k,j}(x)=0$.

Thus, for example, if all the maps $\phi_{j,1}$ are injective, then $\phi_1$ will be injective.

Similarly, a natural condition to ensure that all the $\phi_j$ are injective is to assume that $\phi_{i+1,i}$ is injective for any $i\in \mathbb Z_{>0}$.