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What is the idea behind defined a Projective system of Groups/Rings. In our class an example for the Projective system was given by Taking the Ring $\mathbb{Z}/n\mathbb{Z}$ over $\mathbb{N}$. The order defined was:

  • $ m \leq n$ if $m \mid n$.

Moreover then we calculated the Projective Limit of this system and found it to be $\hat{\mathbb{Z}}$. I have not at all understood this concept fully so even while posing this question there might be errors. So people who can understand what i am talking about are welcome to pose answers. If one still has problems with the question please let me know i shall try to improve it or delete it.

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The projective limit of the system you have described is $\hat{\mathbb{Z}}$ and not $\mathbb{Z}_p$. In particular, there was no $p$ in the set-up, so where should it come from? – Alex B. Feb 1 '11 at 8:00
@Alex: Thanks Alex. As i told the question may contain errors and it did. – anonymous Feb 1 '11 at 8:51
up vote 7 down vote accepted

A useful heuristic when thinking about projective systems is that you want to specify a congruence condition modulo lots of numbers (or ideals) in a compatible way. For example, you can specify any congruence condition modulo 3 and any congruence condition modulo 5. No matter, what you specify, the two can be satisfied at the same time, since 3 and 5 are co-prime: this is the Chinese remainder theorem. But if you specify a congruence condition modulo 5 and modulo 25, then the two could turn out to be incompatible. E.g. there is no number that is 0 modulo 5, but 1 modulo 25. Why? Because a number that is 0 modulo 5, is divisible by 5, so it can only by congruent to 0,5,10,15, or 20 modulo 25. Similarly, once you have chosen a congruence class modulo 3 and modulo 5, the congruence class modulo 15 is uniquely determined, while the congruence class modulo 30 is not unique but heavily restricted.

The projective limit of $\mathbb{Z}/n\mathbb{Z}$ that you have described is the set of all legal simultaneous choices of congruence classes modulo all integers (with the obvious ring structure). The projective system $\mathbb{Z}/m\mathbb{Z}\rightarrow \mathbb{Z}/n\mathbb{Z}$ for $n|m$ expresses all the above rules, that I have spelled out for 3 and 5.

Similarly, if you fix a prime $p$, then you can consider the projective system $\mathbb{Z}/p^m\mathbb{Z}\rightarrow \mathbb{Z}/p^n\mathbb{Z}$ for $n\leq m$. The projective limit is denoted by $\mathbb{Z}_p$ and is the set of all possible legal simultaneous choices of congruence classes modulo higher and higher powers of $p$. Thus, the above example shows that $(0,5,\ldots)$ could represent an element of the projective limit, while $(0,1,\ldots)$ cannot.

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Basically, a projective system formalizes the idea of "constructing elements by successive approximations" in an algebraic setting.

If $G$ is a group (abelian, to fix ideas) and $G_n$ a sequence of smaller and smaller subgroups, the projective limit $\Gamma=\lim(G/G_n)$ is the set of elements that are defined by giving them up to an ambiguity that gets smaller and smaller.

When $G$ is also a topological space and the $G_n$ are a local basis around the identity element, an element of $\Gamma$ is morally an element given by assigning a Cauchy sequence converging to it.

Incidentally, this is exactly the case for the $p$-adic integers ${\Bbb Z}_p$ because it is possible to define a metric in $\Bbb Z$ in such a way the ideal $p^n{\Bbb Z}$ are open spheres.

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