# Direct limit of infinite direct products mapped onto each other via shift maps

While working on a project, I ended up having to take direct limits, for which I admit I don't have a good intuition. Hoping that it is a simple problem for those who have more experience with direct limits than I do, I decided to ask it here.

Let $(G_i)_{i \in \mathbb{N}}$ be a sequence of abelian groups such that $G_i \subseteq G_{i+1}$ and consider the directed system

$G_0 \times G_1 \times \dots \rightarrow_{\varphi_0} G_1 \times G_2 \times \dots \rightarrow_{\varphi_1} \dots$

where each homomorphism $\varphi_k$ is given by $(\alpha,\beta,\gamma,\delta,\dots) \mapsto (\alpha\beta, \gamma, \delta, \dots)$. Is it possible to describe the direct limit of this system in terms of standard group theoretic constructions? I am aware of the standard construction of direct limits by taking an appropriate quotient of the disjoint union. I was hoping that there is a "simpler" description which avoids disjoint unions.

Here is what I was able to think so far. Consider the system

$G_1 \times G_2 \dots \rightarrow_{\psi_0} G_2 \times G_3 \dots \rightarrow_{\psi_1} \dots$

where each homomorphism $\psi_k$ is the left shift map given by $(\alpha,\beta,\gamma,\delta,\dots) \mapsto (\beta, \gamma, \delta, \dots)$. If I am not mistaken, the direct limit of this system should be the reduced product of the groups $G_i$ along the cofinite filter, where the isomorphism takes any element in the direct limit to the equivalence class of the appropriate sequence.

This suggests that the reduced product should be a part of the original direct limit I am considering. However, I can't really figure out how the first component that I discarded is going to interact with the reduced product.

## 1 Answer

First of all, there is a much simpler description of direct limits in the case where all of the bonding maps are surjective. In particular, if $$G_0 \,\xrightarrow{\varphi_0}\, G_1 \,\xrightarrow{\varphi_1}\, G_2 \,\xrightarrow{\varphi_2}\, \cdots$$ is a directed system of groups and epimorphisms, then the direct limit is a quotient $G_0/N$, where $N$ is the following normal subgroup of $G_0$: $$N \,=\, \{g\in G_0 \mid \varphi_n\cdots\varphi_1\varphi_0(g) = 1\text{ for some }n\in\mathbb{N}\} \,=\, \bigcup_{n\in\mathbb{N}} \ker(\varphi_n\cdots \varphi_1\varphi_0).$$

For the directed system you have given, it follows that the direct limit is the quotient $$(G_0\times G_1 \times \cdots )\,\bigr/\,N$$ where $N$ is the subgroup of the infinite direct sum $G_0 \oplus G_1 \oplus \cdots$ consisting of all tuples $(g_0,g_1,\ldots,g_n,1,1,1\ldots)$ for which $g_0g_1\cdots g_n = 1$.