I came across this thing related to 'inverse limits' while going through the discussion on p-adic Tate Modules in the book 'Arithemetic of Elliptic Curves' and I would appreciate it someone could point me towards a reference where I could find the relevant theory (appropriate for someone with no background in category theory, homological Algebra) I only know the definition of inverse limits and how the ring of p-adic integers can be constructed from rings of the form $\mathbb{Z} / l^n \mathbb{Z}$
1) Does the inverse limit commutes with direct product? 2) Say, $B_i$ are $A_i$-module, and $A= \varprojlim A_i, B= \varprojlim B_i$ then $B$ is a $A$-module.
3) With $A, A_i, B, B_i$ as above, if. $G$ is a group that acts on $A_i$ then it also acts on their inverse limit, $A$.
It'd be great if someone could suggest a reference (notes, books whatever) which are suitable for someone without prior experience in category theory things and let me prove above results.
Also, I think we might have to just play with the involved morphisms which give the inverse limit and the homomorphism maps, consider their composition to get the required map that proves our claim, but given that there are infinitely many morphisms, I don't know how to go on about this.
Any help, hints would be appreciated, thanks!
Short answers :
1) yes
2) yes, because of 1)
3) yes, because inverse limit is functorial.
Your last paragraph is correct : each time you can apply various abstract results, but you can also do it by hand (which is how the abstract results are proved anyway - they're just more general) by playing around with the universal property of the inverse limit.
For references, any textbook on category theory should contain statements such as "limits commute with limits" and also basic training regarding limits (to get used to how they work and be able to understand and prove these things on your own); you can try MacLane's Categories for the working mathematician, and probably Riehl's Category Theory in context.
In the cases you asked, the strategy is always the same : I have certain structures on $A_i, B_i$, and I want the same on $A,B$. I try to describe that structure as "maps into $A,B$ from something constructed from $A,B$", then use the universal property to describe it as "maps into $A_i, B_i$ with certain compatibilities" and then use the projections and structure on $A_i, B_i$ to get that map.
That's not very concrete so here's, as an example, a sketch for 1) :
We want to prove that $\lim (A_i\times B_i)\cong A\times B$.
So first we want a map $\lim (A_i\times B_i)\to A\times B$. This amounts to two maps : one $\lim (A_i\times B_i)\to A$, the other $\lim (A_i\times B_i)\to B$.
Ah ! but $A$ is a limit so a map $\lim (A_i\times B_i)\to A$ amounts to compatible maps $\lim (A_i\times B_i) \to A_j$.
But I know an easy such map : $\lim( A_i\times B_i)\to A_j\times B_j \to A_j$ where the first map is the canonical one from the limit to one of its terms, and the second one is the canonical projection from the product.
Now you have to check that this does provide compatible maps (I only gave you apparently unrelated maps to the $A_j$'s); but I'll let that verification to you.
You do the same to get a map to $B$, and all in all you get a map to $A\times B$. This will be the isomorphism. To prove it is an isomorphism, we construct its inverse isomorphism.
So we want a map $A\times B\to \lim (A_i\times B_i)$. Ah ! the codomain is a limit, so this amounts to a bunch of compatible maps $A\times B\to A_i\times B_i$. But we know a map $A\to A_i$ and a map $B\to B_i$ : the canonical projections from the limit. By taking their product, we get a map $A\times B\to A_i\times B_i$.
Again, we need to prove that these maps are compatible. I'll leave that to you. So we get a map $A\times B\to \lim (A_i\times B_i)$.
Final step : check that these are in fact inverse isomorphisms. By composing them we get a map, say $A\times B\to A\times B$. To check that this is the identity, you unravel the definition we gave by universal properties, and check that if you compose it with the projection to $A$, you get the projection to $A$, and similarly with $B$.
I'll leave that verification to you as well, it's just a matter of staring at the definitions in the correct angle, and using uniqueness in the universal properties.
So I left out a lot of details, but this sketch should give you a rough idea of what the proofs of 1)-3) look like, it's always a similar kind of thing : use the universal property to get maps, and check that these do the right thing by using the uniqueness part of the universal property.