In truth, I do not need in the general case.

Let $R$ be a ring (associative, with unit element) and let $\mathrm{Comp}(R)$ the category of complexes over $\mathrm{Mod}\ R$.

If $\mathbb{N}$ is the natural numbers with the usual partial order and $\{X_i^\bullet, \varphi_{i,j}\colon X_j^\bullet \to X_i^\bullet\}$ is an inverse system over $\mathbb{N}$ such that $X_i^\bullet \in \mathrm{Comp}(R)$ is indecomposable, for all $i \in \mathbb{N}$, so is indecomposable $\varprojlim X_i^\bullet$?

Edit. Of course, I'm assuming $\varphi_{i,j} \neq 0$ para todo $j\geq i \in \mathbb{N}$.

Remark. A category $\mathcal{C}$ is complete if $\varprojlim A_i$ exists in $\mathcal{C}$ for every inverse system $\{A_i , \varphi_{i,j}\}$ in $\mathcal{C}$. The categories $\mathcal{C} = \mathrm{Mod} \ R$ and $\mathcal{C} = \mathrm{Comp}(R)$ are examples of complete categories.

In truth, I do not need in the general case.

Let $R$ be a ring (associative, with unit element) and let $\mathrm{Comp}(R)$ the category of complexes over $\mathrm{Mod}\ R$.

If $\mathbb{N}$ is the natural numbers with the usual partial order and $\{X_i^\bullet, \varphi_{i,j}\colon X_j^\bullet \to X_i^\bullet\}$ is an inverse system over $\mathbb{N}$ such that $X_i^\bullet \in \mathrm{Comp}(R)$ is indecomposable, for all $i \in \mathbb{N}$, so is indecomposable $\varprojlim X_i^\bullet$?

Edit. Of course, I'm assuming $\varphi_{i,j} \neq 0$ for all $j\geq i \in \mathbb{N}$.

Remark. A category $\mathcal{C}$ is complete if $\varprojlim A_i$ exists in $\mathcal{C}$ for every inverse system $\{A_i , \varphi_{i,j}\}$ in $\mathcal{C}$. The categories $\mathcal{C} = \mathrm{Mod} \ R$ and $\mathcal{C} = \mathrm{Comp}(R)$ are examples of complete categories.

In truth, I do not need in the general case.

Let $R$ be a ring (associative, with unit element) and let $\mathrm{Comp}(R)$ the category of complexes over $\mathrm{Mod}\ R$.

If $\mathbb{N}$ is the natural numbers with the usual partial order and $\{X_i^\bullet, \varphi_{i,j}\colon X_j^\bullet \to X_i^\bullet\}$ is an inverse system over $\mathbb{N}$ such that $X_i^\bullet \in \mathrm{Comp}(R)$ is indecomposable, for all $i \in \mathbb{N}$, so is indecomposable $\varprojlim X_i^\bullet$?

Remark. A category $\mathcal{C}$ is complete if $\varprojlim A_i$ exists in $\mathcal{C}$ for every inverse system $\{A_i , \varphi_{i,j}\}$ in $\mathcal{C}$. The categories $\mathcal{C} = \mathrm{Mod} \ R$ and $\mathcal{C} = \mathrm{Comp}(R)$ are examples of complete categories.

In truth, I do not need in the general case.

Let $R$ be a ring (associative, with unit element) and let $\mathrm{Comp}(R)$ the category of complexes over $\mathrm{Mod}\ R$.

If $\mathbb{N}$ is the natural numbers with the usual partial order and $\{X_i^\bullet, \varphi_{i,j}\colon X_j^\bullet \to X_i^\bullet\}$ is an inverse system over $\mathbb{N}$ such that $X_i^\bullet \in \mathrm{Comp}(R)$ is indecomposable, for all $i \in \mathbb{N}$, so is indecomposable $\varprojlim X_i^\bullet$?

Edit. Of course, I'm assuming $\varphi_{i,j} \neq 0$ para todo $j\geq i \in \mathbb{N}$.

Remark. A category $\mathcal{C}$ is complete if $\varprojlim A_i$ exists in $\mathcal{C}$ for every inverse system $\{A_i , \varphi_{i,j}\}$ in $\mathcal{C}$. The categories $\mathcal{C} = \mathrm{Mod} \ R$ and $\mathcal{C} = \mathrm{Comp}(R)$ are examples of complete categories.