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Over at Lambda-ring there appears to be a glitch:
Corollary 1. There is an equivalence between the category of torsion-free -rings and the category of torsion-free commutative rings.
Could the author fix this to make it say what he meant to say?
This was introduced in revision 19 by Stephan.
It seems to be something like a copy-and-paste error, since it replaces the full statement that was introduced in revision 6 by John Baez, where it says (and does so up to and including revision 18):
Thus we have an explicit equivalence between the category of torsion-free λ-rings and the category of torsion-free commutative rings equipped with commuting Frobenius lifts.
Clearly something needs to be rolled back. I won’t do it now, though.
Thanks for finding this, I corrected it now. I rearranged the article that time and inserted theorem environments, …
The corollary is to Wilkerson’s theorem and the theorem below it. The Frobenius lifts are crucial since they form the “Adam’s operations” occurring in the characterization of morphisms of lambda rings (theorem 2).
Adam’s operations
You mean Adams or Adams’ after Frank Adams, I guess ? Strangely enough, Lab has a confusing, hard to comprehend, person entry Adam about hard to identify Lab visitor and graduate student, and also a page called John Adams, though John Frank Adams was in math community almost always referred to as Frank Adams and almost never John Adams without Frank.
Tim, could you comment ?
nLab has a confusing, hard to comprehend, person entry Adam about hard to identify nLab visitor and graduate student,
Indeed. Can we fix that? So sombody claimed two years ago to be “Adam, then a grad student at UC Berkeley”, Toby indicated that there were least four people who this could refer to, by pointing to a now defunct (?) page, and then some John Cartmell meant to drop that Adam a message.
This should be cleaned up. Anyone an idea who that Adam might be?
I’m pretty sure Sridhar Ramesh knows exactly which Adam is meant. Although I don’t know his last name, he is uniquely specified as the UC student Adam whose avatar at MO was once a picture of Skolem. Possibly he doesn’t want to be identified by last name; we should find out what’s to be done.
You mean Adams or Adams’ after Frank Adams, I guess ?
Yes, sorry. … I would prefer “Adams“ in case someone feels like creating Adams operation.
As it happens, I do know the relevant Adam, though I haven’t seen him in a while. For what it’s worth, I suspect he enjoys the relative anonymity.
@Zoran I have tidied up and extended a smidgin the entry on Frank Adams.
Sridhar,
he can remain anonymous all he wants, but the page titled Adam should not be weird. (Also, it’s not quite clear to me what the purpose of a page is if by design it is meant not to give away information. But okay.) I have now tidied up the page Adam. If he cares, maybe you can tell him when you see him next. Also that some John Cartmell once tried to get into contact with him.
Thanks, Tim.
I have tried to clean up the entry Lambda-ring a bit, but more clean-up is necessary here. Among other things:
in the Idea-section I moved the expositional introduction before the paragraph that alludes to universal algebra;
in that exposition I reduced the tone of excitement a bit. There was a sequence of four consecutive sentences that ended with an exclamation mark. I edited that a bit for soberness.
added to the Properties-section the statement that the co-free -ring is the ring of Witt vectors.
More on that in the Witt-vector thread…
added at Lambda-ring under Propositions the following statement:
+– {: .num_prop}
The forgetful functor from -rings to commutative rings has
a left adjoint, given by forming the ring of symmetric functions;
a right adjoint given by forming the ring of Witt vectors .
Hence
rings of Witt-vectors are the _co-free Lambda-rings;
rings of symmetric functions are the free Lambda-rings.
=–
This statement appears in (Hazewinkel 08, p. 87, p. 97, 98).
There was a sequence of four consecutive sentences that ended with an exclamation mark. I edited that a bit for soberness.
I see; the previous version did not have enough points.
BTW, the page with the four “Adam”s can be found on the Web Archive, if anybody really wants it.
Wherever you see a liberal sprinkling of exclamation points in an nLab or Café article, it is likely the handiwork of one John C. Baez.
I have tracked the exclamation marks to this Café comment (but it was David Corfield who copied them to the Lab).
In my defence, it was quite early in the life of the nLab, and we needed to flll it with material.
Hey, i know where this stuff comes from and how and why. No problem! But i think editing it now and announcing it here is worthwhile.
remember the adjoint triple in #13 above.
If I understand well (still need to look at some details) then upto generalizing the concept of Lambda-ring a little, it is this adjoint triple which Borger in his absolute geometry applies the sheaf construction to, to obtain the essential geometric morphism .
I have added in the entry (here) a remark to that effect.
Let’s see, on the risk of mixing up my variances:
from that adjoint triple
we get an adjoint quintuple on presheaves
Borger’s adjoint triple on sheaves is the restriction of the middle three of these. For the bottom one to extend to sheaves one needs to think, but the top one always will, by restriction and postcomposing with sheafification. So there should actually an adjoint quadruple
Ah, maybe Borger’s is not “sheaves on Lambda-rings” but “sheaves with Lambda-structure on rings” and maybe that makes a difference. Need to check.
The latter, no?
On another matter, I’d forgotten we had Borger and Morava at the Cafe in discussion on a raft of Lambda-related things, culminating in this exchange.
Yeah, I suppose the latter. But why? Is this set in stone?
Meanwhile, I was looking in the entry Lambda-ring at the section Heterodox definition. We were not being attentive here: this was written way back in revision 19 but it (still) contains plenty of issues. (Whatch out with that contributor.)
It used to start with a nonsense definition of the Frobenius map and then became weirder still by not actually saying what a -ring in the “heterodox” view actually is, but using the term.
I just spent some time editing, but I only fixed small parts. For one, I have added the actual “heterodox” definition (here):
+– {: .num_defn}
For a prime number, then a -typical -ring is
equipped with an endomorphism
such that under tensor product with it becomes the Frobenius morphism, def.\ref{FrobeniusMorphism}:
A big -ring is a commutative ring equipped with commuting endomorphisms, one for each prime number , such that each of them makes the ring -typical, respectively.as above.
=–
This is def. 1.7 in (Borger 08), formulated for the special case of example 1.15 there (which is stated in terms of Witt vectors) and translated to -rings in view of prop. 1.10 c) (the adjunction) there.
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