Inverse Limits in Galois Theory

I am currently taking a first course in Galois Theory and we are studying finite fields at the moment. In the lectures we have defined the inverse limit of an inverse system of finite groups and had the example of the p-adic integers. However, I am struggling to actually see what an inverse limit actually looks like. Also, it seems implicit from my notes that it is itself a group although I don't know what the operation is (pointwise maybe?). Altogether, this topic seems to have come out of the blue and I don't really understand the basic points of inverse limits. I would be very grateful if someone could give me some basic examples/intuitions.

Many thanks!

Your question is broadly framed, and to give a satisfactory response, I suppose one would have to answer equally broadly. Let’s go back to (maybe) the simplest case and example, the $p$-adic integers $\Bbb Z_p$.

Let me describe $\Bbb Z_p$ in a way that is not commonly seen. Consider the circle group $C\cong\Bbb R/\Bbb Z$. You can think of it as the complex numbers of absolute value $1$ if you like. The important thing is that every finite subgroup is cyclic, and in particular, if I look at all the elements of order a power of $p$, I’m looking at the union of all cyclic subgroups of $C$ that are $p$-groups. Any one of these will be $C_{p^m}$ for some $m$, and I suppose that I could call the union $C_{p^\infty}$.

But when you look at the specific $C_{p^m}$, you know that its ring of endomorphisms is canonically isomorphic to the ring $\Bbb Z/p^m\Bbb Z$. And what about the group $C_{p^\infty}$, and its ring of endomorphisms? Any endomorphism $\gamma$ must restrict to each $C_{p^m}$ as an element $g_m$ of $\Bbb Z/p^m\Bbb Z$, but these various $g_m$’s must fit together in a certain way, namely that when $n<m$, $g_m\equiv g_n\pmod{p^n}$. Do you see the definition of inverse limit here? When you look closely, you see that $\text{End}(C_{p^\infty})\cong\text{InvLim}_m(\Bbb Z/p^m\Bbb Z)$, and this is just the definition of the $p$-adic integers.

Now let’s go to finite fields. Let $k$ be one such, the characteristic does not interest us here. What does interest us is that for each $m$, $k$ has a unique extension $k_m$ of degree $p^m$, and the Galois group is cyclic of that degree, $p^m$. You can take the union (direct limit) of the individual fields $k_m$ if you like, and get $k_\infty$, the maximal $p$-extension of $k$ in an algebraic closure. It’s an infinite Galois extension, and its group? You see that the individual Galois groups, cyclic $p$-groups, have to fit together in exactly the way that the endomorphism rings of the $C_{p^m}$ do above, so that the Galois group of the maximal $p$-extension of $k$ is again $\Bbb Z_p$.

Of course there was nothing special about the prime $p$, it was not the characteristic of $k$ (though it could have been), so the total Galois group of $k$ is the direct product of all the $\Bbb Z_p\,$’s, that’s called $\widehat{\Bbb Z}$.

One can interpret a homomorphic image of an algebraic structure as a "collapsed" or "low-resolution" version of it, since different elements of the original structure get blurred together into becoming the same pixel in the image. Thus if we have a chain of surjective homomorphisms, we are getting higher and higher resolution pictures of ... something, "in the limit." That something is the inverse limit.

Consider real numbers for a moment. What are they? Most people have an intuitive idea of a continuum and maybe have been exposed to the phrase "number line," but only a fraction - people who study math - get to know how we actually construct real numbers. People can cite decimal expansions, for instance that of pi, $\pi=3.1415\dots$, but what does that mean? It means there is a series of approximations of indefinitely increasing precision, $3$, $3.1$, $3.14$, $3.141$, $3.1415$, $\cdots$. In fact one formal construction of the real numbers is to identify a real number with a(n equivalence class) of Cauchy sequences. We can think of these Cauchy sequences (of rationals) as sequences of approximations of "ghosts" that are not really there in the set of rationals.

It is similar with inverse limits. Think about $\Bbb Z/p\Bbb Z\leftarrow\Bbb Z/p^2\Bbb Z\leftarrow\Bbb Z/p^3\Bbb Z\leftarrow\Bbb Z/p^4\Bbb Z\leftarrow\cdots$ with the canonical projection maps. Every positive integer can be written as $a_0+a_1p+\cdots+a_rp^r$, i.e. can be written in base $p$. What does the projection map $\Bbb Z/p^{r+1}\Bbb Z\to\Bbb Z/p^r\Bbb Z$ do to the digital representation of an integer residue? Simple: it simply deletes the $p^r$ digit. Thus, any "sequence of approximations" can be identified with an infinite base $p$ expansion, $a_0+a_1p+a_2p^2+\cdots$ (the sequence of approximations tells us what every digit in the infinite expansion is, and vice-versa).

In general, given a chain $G_0\leftarrow G_1\leftarrow G_2\leftarrow\cdots$, and a sequence $(g_0,g_1,g_2,\cdots)$ which is appropriately "compatible" (i.e. $g_{i+1}\mapsto g_i$) we can think of it as a "convergent sequence" whose limit is an element of the inverse limit $\varprojlim G_i$. The operation(s) can be done pointwise. This construction works just as fine if we have a nonlinear system of surjective homomorphisms.

Now let's think about Galois automorphisms. Given any tower $L/M/K$ in which all three extensions are Galois we know there is a surjection ${\rm Gal}(L/K)\to{\rm Gal}(M/K)$. One can consider the entire system of such projection maps for algebraic extensions of a given field $K$, and then consider $\varprojlim{\rm Gal}(L/K)$ (note that the groups are varying with the field $L$ in this limit). What does an element $\sigma$ of this inverse limit look like, tangibly? Well, for any Galois extension, there is a corresponding automorphism $\sigma|_L$ of the extension $L/K$, so ultimately $\sigma$, via all of these $\sigma|_L$s, "knows" where to send every element which is algebraic over $K$, which means this element "knows" of an automorphism of $\overline{K}/K$. And vice-versa, since any element of ${\rm Gal}(\overline{K}/K)$ restricts to an automorphism of any Galois extension $L/K$.

If we view $\varprojlim{\rm Gal}(L/K)$ as the subgroup of $\prod_L{\rm Gal}(L/K)$ comprised of "compatible" elements (recall, that means our "convergent sequences"), then the obvious map ${\rm Gal}(\overline{K}/K)\to\varprojlim{\rm Gal}(L/K)$ (where the "$L$th" coordinate of where $\sigma$ is sent to is simply the restriction of $\sigma$ to $L$) is a group isomorphism. The corresponding inverse map is simply the process of patching together all of the compatible automorphisms of Galois extensions of $K$ into one giant automorphism of $\overline{K}/K$.

One can also endow these inverse limits with the profinite topology in which case we have an isomorphism of topological groups. Another way to define the inverse limit is via universal properties in category theory. Uniqueness of inverse limits (up to unique isomorphisms) follows from a simple abstract nonsense argument, and existence follows from using the explicit construction I discussed above.