Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
I’ve constructed the page p-divisible group since I need it for my height of a variety page. I have to admit that I’m incredibly embarrassed that no matter how many times I look up the words “directed” “inductive” “projective” “limit” “colimit” etc I never seem to use them correctly. All of the systems are as I showed $G_{\nu}\to G_{\nu +1}$ I thought this corresponded to directed, inductive, or colimit, but when I looked up inductive limit in the nlab it seemed to be indicating the opposite, so maybe some of the uses are wrong.
Harrison, you’re right: that page on inductive limit got it backwards. So over at p-divisible group, it should read “colimit”, not “limit”. I’m going to go ahead and change both.
Edit: Okay, fixed. The words “directed” and “filtered” are trickier, and there may be some variance between authors. Could be we’ve had a discussion about that here at the Forum. Personally, I say “directed colimit” and “filtered colimit”, perhaps out of force of habit. I think I say “cofiltered limit”. I don’t think I would ever say “codirected”.
I would probably say “codirected limit” if for some reason I wanted to talk about a cofiltered limit over a poset. It is a bit unfortunate that “directed” and “filtered” get “co”d oppositely to “limit”, but if I heard “directed limit” or “filtered limit” I would probably guess that what was meant was the same meaning as codirected/cofiltered; there doesn’t see mto be much point in taking a limit over a filtered (as opposed to cofiltered) category.
Wow, both inductive limit and projective limit were entirely backwards! I’ve fixed both of these (and there was more to fix than Todd fixed).
As for terminology, nobody uses the terms “codirected colimit” or “cofiltered colimit”; if it’s really a colimit, then it can only be directed or filtered. People do use the terms “directed limit” and “filtered limit”; these are synonyms for “codirected limit” and “cofiltered limit” (respectively). So it looks like both Todd and Mike have usages that I would accept (although I use Mike’s myself).
Whoops, I called hilbertthm90 “Harrison” (thinking of Harrison Brown), but that apparently was an error. I’m sorry!
I’m sort of relieved those were backwards in the nLab. I thought I had finally gotten these terms down, and then when I looked them up I felt like I was going crazy.
This appeared at almost the same time as divisible group, which also defines ‘$p$-divisible group’. I think that there is a connection, but for now, I’ve just put notes on each page mentioning the other as if it were completely unrelated.
Somebody sent me an email saying
Your contribution to the nlab article on p-divisible group contains the sentence ’It can be checked that a p-divisible group over R is a p-torsion commutative formal group G for which p:G→G is an isogeny’.
This is only true for connected p-divisible groups.
This sentence has been in the entry form first version/revision 0 on. I have now changed it, but people who made substantial contributions to the entry might want to have a look.
I’m not sure I understand which part requires connected. Maybe they are thinking of a slightly different definition? When I wrote this I was looking at the (unnumbered) first “Proposition” in section 6 of the Shatz article from the book Arithmetic Geometry. I just re-looked at it and there is no connected hypothesis.
Unless I’m going crazy the proof is basically immediate by the definition given in the article. (Shatz provides more details than this.) By definition $p: G\to G$ is surjective (that is the “p-divisible” property) and also by the definition we require the kernel to be exactly $G_1$ which is a finite group. I think this is the standard definition of isogeny: surjective with finite kernel.
The same type of reasoning shows that it is $p$-torsion (any element must exist at some finite stage and hence is in the kernel of multiplication by some power of $p$).
Did they provide a candidate counterexample so we can see where our definitions get mixed up?
They might be thinking of a related Theorem due to Tate which says that the connected $p$-divisible groups are exactly the ones of the form $\Gamma [p^\infty]$ for some divisible, commutative formal Lie group $\Gamma/R$.
Thanks. I’ll get back to you on this. I hope the person who made the comment will come by here and discuss this directly.
I am not getting a reply from the person who emailed me now.
So for the moment I have rolled back my latest change. Sorry, I should have checked first.
But maybe you’d enjoy adding a comment on that relation to the theorem of Tate? Or else, maybe my correspondent will come by here later and explain.
The theorem is there already as the third bullet under “Examples” even though it isn’t set aside formally as a theorem.
Okay, thanks. I gave them formal Examples-environments.
(I wish people would read/take serious the request on the nLab home page to send comments not by email to suspected (and often wrongly identified) authors, but directly to the nForum here. What might we do to make this clearer on the home page (or elsewhere)?)
I think this is just the general issue that “people do not read”. I have the same problem all the time in other contexts. Let me know if you find a solution. (-:
1 to 14 of 14