Direct limit

Direct Limit

Part 1: Direct Limit

Definition 1.1. (Direct Limit)

1. A partially ordered set $I$ is said to be a direct set if for all pairs $i , j \in I$, there exist $k \in I$ such that $i \leqslant k$ and $j \leqslant k$.
2. Let $\mathfrak{C}$ be a category. A direct system is a pair $(X_i, \mu_{ij})$, where $\{X_i\}_{i \in I}$ is a family of objects in $\mathfrak{C}$ indexed by a direct set $I$, and for all $i \leqslant j$, the morphism $\mu_{ij} \colon X_i \to X_j$ satisfies the property:
• $\mu_{ii} = \text{Id}_{X_i}$.
• For $i \leqslant j \leqslant k$, we have $\mu_{ik} = \mu_{jk} \circ \mu_{ij}$.
3. Given an direct system $(X_i, \mu_{ij})$ in $\mathfrak{C}$, a direct limit of the direct system is a pair $(X, \mu_i)$, where $X \in \text{Ob}(\mathfrak{C})$ and morphism $\mu_i \colon X_i \to X$ satisfying $\mu_i = \mu_j \circ \mu_{ij}$ for all $i \leqslant j$. $X$ satisfies the following universal property:
Given an object $Y$ and morphisms $\alpha_i \colon X_i \to Y$ satisfying $\alpha_i = \alpha_j \circ \mu_{ij}$ for all $i \leqslant j$, there exist a unique morphism $\alpha \colon X \to Y$ satisfying $\alpha_i = \alpha \circ \mu_i$ for all $i \in I$.
 
We denote such direct limit by $\displaystyle X = \varinjlim_{i \in I} X_i$.

Proposition 1.2. (Uniqueness)

A direct limit of a direct system is unique if exist.

Proof.

Suppose both $(X, \mu_i)$ and $(X', \mu_i')$ satisfy the universal property. Note that $\mu_i' = \mu_j' \circ \mu_{ij}$ by definition, then by universal property of $X$, there exist an unique morphism $\mu' \colon X \to X'$ such that $\mu_i' = \mu' \circ \mu_i$ for all $i \in I$. Similarly, there exist an unique morphism $\mu \colon X' \to X$ such that $\mu_i = \mu \circ \mu_i'$ for all $i \in I$.

Observe that: $$(\mu \circ \mu') \circ \mu_i = \mu \circ (\mu' \circ \mu_i) = \mu \circ \mu_i' = \mu_i$$, but also $\text{Id}_X$ satisfies $\text{Id}_X \circ \mu_i = \mu_i$, by uniqueness (universal property of $X$), $\mu \circ \mu' = \text{Id}_X$. The same argument works on $\mu' \circ \mu = \text{Id}_{X'}$, hence $X \simeq X'$.
$\square$

Part 2: Direct Limit of Module

Example 2.1. (Direct Limit of Module)

• Let $R$ be a commutative ring with identity and $(M_i, \mu_{ij})$ a direct system of modules of $R$. Let $\displaystyle C = \bigoplus_{i \in I}M_i$ be the direct sum of $M_i$, with the natural embedding: $$\iota_i \colon M_i \to C = \bigoplus_{i \in I}M_i$$, for all $i \in I$. moreover, define $$D = \langle \iota_i(x_i) - \iota_j(\mu_{ij}(x_i)) \mid i \leqslant j, x_i \in M_i\rangle$$, with quotient map $q \colon C \to C/D$
• Claim: $\displaystyle (C/D, \mu_i) = \varinjlim_{i \in I}M_i$, with $\mu_i = q \circ \iota_i$. $$M_i \overset{\iota_i}{\longrightarrow} C = \bigoplus_{i \in I}M_i \overset{q}{\longrightarrow} C/D$$
• Proof. We need to show that it satisfy the universal property, let $N \in \mathfrak{Mod}(R)$, with morphisms $\alpha_i \colon M_i \to N$ satisfing $\alpha_i = \alpha_j \circ \mu_{ij}$ for all $i \leqslant j$. By universal property of direct sum, there exist a unique morphism $\alpha' \colon C \to N$ satisfying $\alpha_i = \alpha' \circ \pi_i$ for all $i \in I$.
  
(Namely, we may check the mapping $ \displaystyle (x_i)_{i \in I} \mapsto \sum_{i \in I}\alpha_i(x_i)$). Moreover, note that: $$\alpha'(x_i - \mu_{ij}(x_j)) = \alpha_i(x_i) - (\alpha_j \circ \mu_{ij})(x_j) = \alpha_i(x_i) - \alpha_i(x_i) = 0$$ by preconditions of morphisms $\alpha_i$'s. Hence, $D<\ker(\alpha')$ and $\alpha'$ induce a unique morphism $\alpha \colon C/D \to N$ by universal property of quotient, 

the rest conditions of $\alpha$ follows by our construction.
([1], Chapter 2, Exercise 14. 16.)

Lemma 2.2. (Intuition of direct limit)

Fix notation as in Example 2.1..
1. For every $x \in M$, there exist $i \in I$ and $x_i \in M_i$ such that $\mu_i(x_i) = x$.
2. If $\mu_i(x_i) = 0$ in $M$, there exist $j \geqslant i$ such that $\mu_{ij}(x_i) = 0$
([1], Chapter 2, Exercise 15.)

Proof.