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Background:

Exercise: Recall that $F_{n,m}([a]_n)=[a]_m$ defines $F_{n,m}:\mathbb{Z}_n\to \mathbb{Z}_\mathbb{m}$ iff $m\mid n.$ When $p$ is a prime and $j\geq i,$ for convenience set $g_{j,i}=F_{p^j, p^i},$ Consider the group $G=\prod \mathbb{Z}_{p^t}$ over $i\in \mathbb{N}$, and set $H=\{f\in G\mid \text{ whenever } j\geq i, g_{j,i}(f(j))=f(i)\}.$ $H$ is called the inverse limit of $\{\mathbb{Z}_{p^i}\}$. It may be easier to consider $f\in G$ to be given by $f(i)=[f_i]\in \mathbb{Z}_{p^i}$ for representative $f_i\in \{0,1,2,\ldots, p^i - 1\}.$ Thus, $H=\{f\in G\mid \text{ whenever } j\geq i, f_j\equiv f_i \pmod {p^i}\}$.

(i) Show that $H$ makes sense: that is, it unambiguously defines elements of $G.$ (The problem is that $H$ is defined by conditions on all pairs of its coordinates, but it has infinitely many coordinates, so one must check that the conditions are consistent.

Questions:

For the above exercise, I would like to know what i have to do to show that $H$ defines elements of $G$. I don't understand when it says that I have to show that $H$ defines elements of $G$. In the bracket explanation, it says the reader have to check that the conditions are consistent which I do not understand what it means.

Thank you in advance

Arturo Magidin
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Seth
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  • This problem is badly stated, you should complain to your instructor if this is from a class. What you need to check is that the system of maps $g_{ij}$ is an inverse system. – Moishe Kohan Feb 20 '24 at 02:09
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  • @MoisheKohan You mean the following: Let $\mathbb{Z}n=A_j,$ $\mathbb{Z}_m=A_i,$ and $F{n,m}=f_{ij}$ for all $m\leq n,$ and $m\mid n.$ Also let the map $F_{p^i,p^i}: \mathbb{Z}{p^i}\to \mathbb{Z}{p^i}$ be defined by $g_{i,i}=F_{p^i,p^i},$ and $g_{i,i}(f(i))=f(i).$ We define for $i\leq j,$ $g_{i,j}=F_{p^i,p^j},$ $g_{i,h}(f(j))=f(i),$ and for $j\leq k$ $g_{j,k}=F_{p^j,p^k},$ $g_{j,k}(f(j))=f(k).$ I then show that $F_{p^i,p^k}=F_{p^i,p^j}\circ F_{p^j,p^k},$ for $i\leq j\leq k.$ – Seth Feb 20 '24 at 02:44
  • Of course...... – Moishe Kohan Feb 20 '24 at 03:11
  • @MoisheKohan sorry, am I understanding you correctly in what I need to do to show that the $g_{i,j}$ map satisfies the criteria for an inverse system? – Seth Feb 20 '24 at 03:35

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