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Suppose having a set of direct systems of abelian groups

$\ldots\{G_{\alpha}\}_{\alpha\in A}$, $\{G_{\beta}\}_{\beta\in B}$, $\{G_{\gamma}\}_{\gamma\in \Gamma}\ldots$

If there is a (long) exact sequence for certain indexes:

$$\cdots\longrightarrow G_\alpha\longrightarrow G_\beta\longrightarrow G_{\gamma}\longrightarrow\cdots\ $$ Then, is the following sequence exact?

$$\cdots\longrightarrow \varinjlim G_\alpha\longrightarrow \varinjlim G_\beta\longrightarrow\varinjlim G_{\gamma}\longrightarrow\cdots\ $$

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up vote 3 down vote accepted

You want this long exact sequence to be compatible with the direct systems; but this cannot be formulated since the index sets are different. In fact, there is no way to define a map $\varinjlim G_{\alpha} \to \varinjlim G_{\beta}$ in your setting.

Probably you would like to know the correct statement: If $I$ is a directed poset and for every $n \in \mathbb{N}$ we have a diagram $G^n : I \to \mathsf{Ab}$, and morphisms of diagrams $G^n \to G^{n+1}$ for all $n$ (i.e. natural transformations), such that for all $i$ and $n$ the induced sequence $G^{n-1}_i \to G^n_i \to G^{n+1}_i$ is exact. Then the induced sequence $\dotsc \to \varinjlim G^n \to \varinjlim G^{n+1} \to \dotsc$ is exact.

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