The direct limit of modules is a module

Let R be a ring. In a previous exercise, we constructed the direct limit \varinjlim_I M_i of abelian groups indexed by a directed, partially ordered set I. Verify that if the M_i are left R-modules, then \varinjlim_I M_i is naturally a left R-module, that the natural injections M_i \rightarrow \varinjlim_I M_i are R-module homomorphisms, and that \varinjlim_I M_i satisfies an appropriate universal property.

Let I be a nonempty, directed, partially ordered set and let \{M_i\}_I be a family of left R-modules indexed by I. Suppose that for all i,j \in I with i \leq j, there exists an R-module homomorpism \rho_{i,j} : M_i \rightarrow M_j such that (1) \rho_{j,k} \circ \rho_{i,j} = \rho_{i,k} whenever i \leq j \leq k and (2) \rho_{i,i} = 1 (the identity mapping) for all i \in I.

Let X = \bigcup_I M_i \times \{i\}, and define a relation \sigma on X by (a,i) \sigma (b,j) if and only if there exists k \geq i,j such that \rho_{i,k}(a) = \rho_{j,k}(b).

Previously, we showed that \sigma is an equivalence relation and that via the operator [(a,i)]_\sigma + [(b,j)]_\sigma = [(\rho_{i,k}(a) + \rho_{j,k}(b),k)] (where k is any element such that k \geq i,j) the set X/\sigma is an abelian group which we denote \varinjlim_I M_i. Moreover, for each i \in I, the mapping \rho_i : M_i \rightarrow \varinjlim_I M_i given by \rho_i(a) = [(a,i)]_\sigma is a group homomorphism.

Define an action of R on \varinjlim M_i by r \cdot [(a,i)]_\sigma = [(r \cdot a, i)]_\sigma. We claim that this action is well defined and makes \varinjlim M_i into a left R-module.

  1. (Well-defined) Suppose [(a,i)]_\sigma = [(b,j)]_\sigma. Then there exists k \geq i,j such that \rho_{i,k}(a) = \rho_{j,k}(b). Now r \cdot \rho_{i,k}(a) = r \cdot \rho_{j,k}(b), and since the \rho_{i,j} are module homomorphisms, \rho_{i,k}(r \cdot a) = \rho_{j,k}(r \cdot b). Thus [(r \cdot a,i)]_\sigma = [(r \cdot b,j)]_\sigma, and we have r \cdot [(a,i)]_\sigma = r \cdot [(b,j)]_\sigma. Hence this action is well-defined.
  2. Now let [(a,i)]_\sigma, [(b,j)]_\sigma \in \varinjlim M_i and let r \in R. Let k \geq i,j. Then we have r \cdot \left( [(a,i)]_\sigma + [(b,j)]_\sigma \right) = r \cdot [(\rho_{i,k}(a) + \rho_{j,k}(b), k)]_\sigma = [(r \cdot (\rho_{i,k}(a) + \rho_{j,k}(b)), k)]_\sigma = [(r \cdot \rho_{i,k}(a) + r \cdot \rho_{j,k}(b), k)]_\sigma = [(\rho_{i,k}(r \cdot a) + \rho_{j,k}(r \cdot b), k)]_\sigma = [(r \cdot a, i)]_\sigma + [(r \cdot b, j)]_\sigma = r \cdot [(a,i)]_\sigma + r \cdot [(b,j)]_\sigma.
  3. Let [(a,i)]_\sigma \in \varinjlim M_i and r,s \in R. Then (rs) \cdot [(a,i)]_\sigma = [(rs \cdot a, i)]_\sigma = [(r \cdot (s \cdot a), i)]_\sigma = r \cdot [(s \cdot a, i)]_\sigma = r \cdot (s \cdot [(a,i)]_\sigma).
  4. Let [(a,i)]_\sigma \in \varinjlim M_i and let r,s \in R. Note that i \geq i. Then (r+s) \cdot [(a,i)]_\sigma = [((r+s) \cdot a, i)]_\sigma = [(r \cdot a + s \cdot a, i)]_\sigma = [(r \cdot \rho_{i,i}(a) + s \cdot \rho_{i,i}(b), i)]_\sigma = [(r \cdot a, i)]_\sigma + [(s \cdot a, i)]_\sigma = r \cdot [(a,i)]_\sigma + s \cdot [(a,i)]_\sigma.

So \varinjlim M_i is a left R-module. Further, suppose that R has a 1 and the M_i are all unital. Then 1 \cdot [(a,i)]_\sigma = [(1 \cdot a, i)]_\sigma = [(a,i)]_\sigma, so that \varinjlim M_i is unital. Finally, note also that for all i \in I, r \in R, and m \in M_i, we have r \cdot \rho_i(m) = r \cdot [(m,i)]_\sigma = [(r \cdot m,i)]_\sigma = \rho_i(r \cdot m). Thus the injections \rho_i are all R-module homomorphisms. Certainly we have \rho_j \circ \rho_{i,j} = \rho_i for all i,j \in I with i \leq j.

We claim that \varinjlim M_i, together with the \rho_i, is universal with respect to this property in the following sense: If N is a left R-module, and for each i \in I, \psi_i : M_i \rightarrow N a module homomorphism, such that \psi_j \circ \rho_{i,j} = \psi_i whenever i \leq j, then there exists a unique R-module homomorphism \Psi : \varinjlim M_i \rightarrow N such that \psi_i = \Psi \circ \rho_i for all i. That is, there exists a unique R-module homomorphism \Psi such that the following diagram commutes for all i \leq j.

Universal property of the direct limit of modules

We will first prove existence and then uniqueness.

A group homomorphism \Psi : \varinjlim_I M_i \rightarrow N given by \Psi([(a,i)]_\sigma) = \psi_i(a) exists such that \Psi \circ \rho_i = \psi_i for all i \in I by the universal property of direct limits of abelian groups. It remains to be seen that \Psi is a module homomorphism- that is, that it preserves scalars. To that end, let r \in R and let [(a,i)]_\sigma \in \varinjlim M_i. Then r \cdot \Psi([(a,i)]_\sigma) = r \cdot \psi_i(a) = \psi_i(r \cdot a) = \Psi([(r \cdot a,i)]_\sigma) = \Psi( r \cdot [(a,i)]_\sigma); so \Psi is indeed an R-module homomorphism.

For uniqueness, suppose that \Theta : \varinjlim M_i \rightarrow N is another R-module homomorphism such that \Theta \circ \rho_i = \psi_i for all i. In particular, \Theta is a group homomorphism. By the universal property of direct limits of abelian groups, \Theta = \Psi, so that \Psi is unique as desired.

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