## 10.8 Colimits

Some of the material in this section overlaps with the general discussion on colimits in Categories, Sections 4.144.21. The notion of a preordered set is defined in Categories, Definition 4.21.1. It is a slightly weaker notion than a partially ordered set.

Definition 10.8.1. Let $(I, \leq )$ be a preordered set. A system $(M_ i, \mu _{ij})$ of $R$-modules over $I$ consists of a family of $R$-modules $\{ M_ i\} _{i\in I}$ indexed by $I$ and a family of $R$-module maps $\{ \mu _{ij} : M_ i \to M_ j\} _{i \leq j}$ such that for all $i \leq j \leq k$

$\mu _{ii} = \text{id}_{M_ i}\quad \mu _{ik} = \mu _{jk}\circ \mu _{ij}$

We say $(M_ i, \mu _{ij})$ is a directed system if $I$ is a directed set.

This is the same as the notion defined in Categories, Definition 4.21.2 and Section 4.21. We refer to Categories, Definition 4.14.2 for the definition of a colimit of a diagram/system in any category.

Lemma 10.8.2. Let $(M_ i, \mu _{ij})$ be a system of $R$-modules over the preordered set $I$. The colimit of the system $(M_ i, \mu _{ij})$ is the quotient $R$-module $(\bigoplus _{i\in I} M_ i) /Q$ where $Q$ is the $R$-submodule generated by all elements

$\iota _ i(x_ i) - \iota _ j(\mu _{ij}(x_ i))$

where $\iota _ i : M_ i \to \bigoplus _{i\in I} M_ i$ is the natural inclusion. We denote the colimit $M = \mathop{\mathrm{colim}}\nolimits _ i M_ i$. We denote $\pi : \bigoplus _{i\in I} M_ i \to M$ the projection map and $\phi _ i = \pi \circ \iota _ i : M_ i \to M$.

Proof. This lemma is a special case of Categories, Lemma 4.14.12 but we will also prove it directly in this case. Namely, note that $\phi _ i = \phi _ j\circ \mu _{ij}$ in the above construction. To show the pair $(M, \phi _ i)$ is the colimit we have to show it satisfies the universal property: for any other such pair $(Y, \psi _ i)$ with $\psi _ i : M_ i \to Y$, $\psi _ i = \psi _ j\circ \mu _{ij}$, there is a unique $R$-module homomorphism $g : M \to Y$ such that the following diagram commutes:

$\xymatrix{ M_ i \ar[rr]^{\mu _{ij}} \ar[dr]^{\phi _ i} \ar[ddr]_{\psi _ i} & & M_ j\ar[dl]_{\phi _ j} \ar[ddl]^{\psi _ j} \\ & M \ar[d]^{g}\\ & Y }$

And this is clear because we can define $g$ by taking the map $\psi _ i$ on the summand $M_ i$ in the direct sum $\bigoplus M_ i$. $\square$

Lemma 10.8.3. Let $(M_ i, \mu _{ij})$ be a system of $R$-modules over the preordered set $I$. Assume that $I$ is directed. The colimit of the system $(M_ i, \mu _{ij})$ is canonically isomorphic to the module $M$ defined as follows:

1. as a set let

$M = \left(\coprod \nolimits _{i \in I} M_ i\right)/\sim$

where for $m \in M_ i$ and $m' \in M_{i'}$ we have

$m \sim m' \Leftrightarrow \mu _{ij}(m) = \mu _{i'j}(m')\text{ for some }j \geq i, i'$
2. as an abelian group for $m \in M_ i$ and $m' \in M_{i'}$ we define the sum of the classes of $m$ and $m'$ in $M$ to be the class of $\mu _{ij}(m) + \mu _{i'j}(m')$ where $j \in I$ is any index with $i \leq j$ and $i' \leq j$, and

3. as an $R$-module define for $m \in M_ i$ and $x \in R$ the product of $x$ and the class of $m$ in $M$ to be the class of $xm$ in $M$.

The canonical maps $\phi _ i : M_ i \to M$ are induced by the canonical maps $M_ i \to \coprod _{i \in I} M_ i$.

Proof. Omitted. Compare with Categories, Section 4.19. $\square$

Lemma 10.8.4. Let $(M_ i, \mu _{ij})$ be a directed system. Let $M = \mathop{\mathrm{colim}}\nolimits M_ i$ with $\mu _ i : M_ i \to M$. Then, $\mu _ i(x_ i) = 0$ for $x_ i \in M_ i$ if and only if there exists $j \geq i$ such that $\mu _{ij}(x_ i) = 0$.

Proof. This is clear from the description of the directed colimit in Lemma 10.8.3. $\square$

Example 10.8.5. Consider the partially ordered set $I = \{ a, b, c\}$ with $a < b$ and $a < c$ and no other strict inequalities. A system $(M_ a, M_ b, M_ c, \mu _{ab}, \mu _{ac})$ over $I$ consists of three $R$-modules $M_ a, M_ b, M_ c$ and two $R$-module homomorphisms $\mu _{ab} : M_ a \to M_ b$ and $\mu _{ac} : M_ a \to M_ c$. The colimit of the system is just

$M := \mathop{\mathrm{colim}}\nolimits _{i \in I} M_ i = \mathop{\mathrm{Coker}}(M_ a \to M_ b \oplus M_ c)$

where the map is $\mu _{ab} \oplus -\mu _{ac}$. Thus the kernel of the canonical map $M_ a \to M$ is $\mathop{\mathrm{Ker}}(\mu _{ab}) + \mathop{\mathrm{Ker}}(\mu _{ac})$. And the kernel of the canonical map $M_ b \to M$ is the image of $\mathop{\mathrm{Ker}}(\mu _{ac})$ under the map $\mu _{ab}$. Hence clearly the result of Lemma 10.8.4 is false for general systems.

Definition 10.8.6. Let $(M_ i, \mu _{ij})$, $(N_ i, \nu _{ij})$ be systems of $R$-modules over the same preordered set $I$. A homomorphism of systems $\Phi$ from $(M_ i, \mu _{ij})$ to $(N_ i, \nu _{ij})$ is by definition a family of $R$-module homomorphisms $\phi _ i : M_ i \to N_ i$ such that $\phi _ j \circ \mu _{ij} = \nu _{ij} \circ \phi _ i$ for all $i \leq j$.

This is the same notion as a transformation of functors between the associated diagrams $M : I \to \text{Mod}_ R$ and $N : I \to \text{Mod}_ R$, in the language of categories. The following lemma is a special case of Categories, Lemma 4.14.8.

Lemma 10.8.7. Let $(M_ i, \mu _{ij})$, $(N_ i, \nu _{ij})$ be systems of $R$-modules over the same preordered set. A morphism of systems $\Phi = (\phi _ i)$ from $(M_ i, \mu _{ij})$ to $(N_ i, \nu _{ij})$ induces a unique homomorphism

$\mathop{\mathrm{colim}}\nolimits \phi _ i : \mathop{\mathrm{colim}}\nolimits M_ i \longrightarrow \mathop{\mathrm{colim}}\nolimits N_ i$

such that

$\xymatrix{ M_ i \ar[r] \ar[d]_{\phi _ i} & \mathop{\mathrm{colim}}\nolimits M_ i \ar[d]^{\mathop{\mathrm{colim}}\nolimits \phi _ i} \\ N_ i \ar[r] & \mathop{\mathrm{colim}}\nolimits N_ i }$

commutes for all $i \in I$.

Proof. Write $M = \mathop{\mathrm{colim}}\nolimits M_ i$ and $N = \mathop{\mathrm{colim}}\nolimits N_ i$ and $\phi = \mathop{\mathrm{colim}}\nolimits \phi _ i$ (as yet to be constructed). We will use the explicit description of $M$ and $N$ in Lemma 10.8.2 without further mention. The condition of the lemma is equivalent to the condition that

$\xymatrix{ \bigoplus _{i\in I} M_ i \ar[r] \ar[d]_{\bigoplus \phi _ i} & M \ar[d]^\phi \\ \bigoplus _{i\in I} N_ i \ar[r] & N }$

commutes. Hence it is clear that if $\phi$ exists, then it is unique. To see that $\phi$ exists, it suffices to show that the kernel of the upper horizontal arrow is mapped by $\bigoplus \phi _ i$ to the kernel of the lower horizontal arrow. To see this, let $j \leq k$ and $x_ j \in M_ j$. Then

$(\bigoplus \phi _ i)(x_ j - \mu _{jk}(x_ j)) = \phi _ j(x_ j) - \phi _ k(\mu _{jk}(x_ j)) = \phi _ j(x_ j) - \nu _{jk}(\phi _ j(x_ j))$

which is in the kernel of the lower horizontal arrow as required. $\square$

Lemma 10.8.8. Let $I$ be a directed set. Let $(L_ i, \lambda _{ij})$, $(M_ i, \mu _{ij})$, and $(N_ i, \nu _{ij})$ be systems of $R$-modules over $I$. Let $\varphi _ i : L_ i \to M_ i$ and $\psi _ i : M_ i \to N_ i$ be morphisms of systems over $I$. Assume that for all $i \in I$ the sequence of $R$-modules

$\xymatrix{ L_ i \ar[r]^{\varphi _ i} & M_ i \ar[r]^{\psi _ i} & N_ i }$

is a complex with homology $H_ i$. Then the $R$-modules $H_ i$ form a system over $I$, the sequence of $R$-modules

$\xymatrix{ \mathop{\mathrm{colim}}\nolimits _ i L_ i \ar[r]^\varphi & \mathop{\mathrm{colim}}\nolimits _ i M_ i \ar[r]^\psi & \mathop{\mathrm{colim}}\nolimits _ i N_ i }$

is a complex as well, and denoting $H$ its homology we have

$H = \mathop{\mathrm{colim}}\nolimits _ i H_ i.$

Proof. It is clear that $\xymatrix{ \mathop{\mathrm{colim}}\nolimits _ i L_ i \ar[r]^\varphi & \mathop{\mathrm{colim}}\nolimits _ i M_ i \ar[r]^\psi & \mathop{\mathrm{colim}}\nolimits _ i N_ i }$ is a complex. For each $i \in I$, there is a canonical $R$-module morphism $H_ i \to H$ (sending each $[m] \in H_ i = \mathop{\mathrm{Ker}}(\psi _ i) / \mathop{\mathrm{Im}}(\varphi _ i)$ to the residue class in $H = \mathop{\mathrm{Ker}}(\psi ) / \mathop{\mathrm{Im}}(\varphi )$ of the image of $m$ in $\mathop{\mathrm{colim}}\nolimits _ i M_ i$). These give rise to a morphism $\mathop{\mathrm{colim}}\nolimits _ i H_ i \to H$. It remains to show that this morphism is surjective and injective.

We are going to repeatedly use the description of colimits over $I$ as in Lemma 10.8.3 without further mention. Let $h \in H$. Since $H = \mathop{\mathrm{Ker}}(\psi )/\mathop{\mathrm{Im}}(\varphi )$ we see that $h$ is the class mod $\mathop{\mathrm{Im}}(\varphi )$ of an element $[m]$ in $\mathop{\mathrm{Ker}}(\psi ) \subset \mathop{\mathrm{colim}}\nolimits _ i M_ i$. Choose an $i$ such that $[m]$ comes from an element $m \in M_ i$. Choose a $j \geq i$ such that $\nu _{ij}(\psi _ i(m)) = 0$ which is possible since $[m] \in \mathop{\mathrm{Ker}}(\psi )$. After replacing $i$ by $j$ and $m$ by $\mu _{ij}(m)$ we see that we may assume $m \in \mathop{\mathrm{Ker}}(\psi _ i)$. This shows that the map $\mathop{\mathrm{colim}}\nolimits _ i H_ i \to H$ is surjective.

Suppose that $h_ i \in H_ i$ has image zero on $H$. Since $H_ i = \mathop{\mathrm{Ker}}(\psi _ i)/\mathop{\mathrm{Im}}(\varphi _ i)$ we may represent $h_ i$ by an element $m \in \mathop{\mathrm{Ker}}(\psi _ i) \subset M_ i$. The assumption on the vanishing of $h_ i$ in $H$ means that the class of $m$ in $\mathop{\mathrm{colim}}\nolimits _ i M_ i$ lies in the image of $\varphi$. Hence there exists a $j \geq i$ and an $l \in L_ j$ such that $\varphi _ j(l) = \mu _{ij}(m)$. Clearly this shows that the image of $h_ i$ in $H_ j$ is zero. This proves the injectivity of $\mathop{\mathrm{colim}}\nolimits _ i H_ i \to H$. $\square$

Example 10.8.9. Taking colimits is not exact in general. Consider the partially ordered set $I = \{ a, b, c\}$ with $a < b$ and $a < c$ and no other strict inequalities, as in Example 10.8.5. Consider the map of systems $(0, \mathbf{Z}, \mathbf{Z}, 0, 0) \to (\mathbf{Z}, \mathbf{Z}, \mathbf{Z}, 1, 1)$. From the description of the colimit in Example 10.8.5 we see that the associated map of colimits is not injective, even though the map of systems is injective on each object. Hence the result of Lemma 10.8.8 is false for general systems.

Lemma 10.8.10. Let $\mathcal{I}$ be an index category satisfying the assumptions of Categories, Lemma 4.19.8. Then taking colimits of diagrams of abelian groups over $\mathcal{I}$ is exact (i.e., the analogue of Lemma 10.8.8 holds in this situation).

Proof. By Categories, Lemma 4.19.8 we may write $\mathcal{I} = \coprod _{j \in J} \mathcal{I}_ j$ with each $\mathcal{I}_ j$ a filtered category, and $J$ possibly empty. By Categories, Lemma 4.21.5 taking colimits over the index categories $\mathcal{I}_ j$ is the same as taking the colimit over some directed set. Hence Lemma 10.8.8 applies to these colimits. This reduces the problem to showing that coproducts in the category of $R$-modules over the set $J$ are exact. In other words, exact sequences $L_ j \to M_ j \to N_ j$ of $R$ modules we have to show that

$\bigoplus \nolimits _{j \in J} L_ j \longrightarrow \bigoplus \nolimits _{j \in J} M_ j \longrightarrow \bigoplus \nolimits _{j \in J} N_ j$

is exact. This can be verified by hand, and holds even if $J$ is empty. $\square$

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