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I’ve created a page for the Witt vectors. It seems that even with all that I wrote here (don’t worry I had a set of about 10 blog entries I wrote a few months ago that I just condensed, so I didn’t write this whole thing tonight) there are all sorts of things still missing here. The Witt functor is mention at Lambda-ring and there seems to be connections to the field with one element (?!). I just needed to refer to Witt vectors in the next few pages I want to make, so I decided this had to come first. Dieudonne module will need it and obviously Witt cohomology will need it.
Oh. Also, there seems to be a strange thing going on in the first paragraph of the section “Operations on the Witt vectors”. It moved the projection map up to the top.
Once Dmitri Kaledin gets out of Harkov and filming Dau, we may hope for his next paper in which he has beautiful idea on de Rham-Witt cohomology in the noncommutative world, which he gave some gist at the conference in Split few days ago. In particular he has a relative version of Witt vectors, which is an elementary construction which is quite nice (my notes are not with me, so I can not report any details now). See also the abstract for a similar talk he gave elsewhere: pdf.
Nice material, hilbertthm90! I will edit it a bit once back to work (I am there now but leaving soon). Maybe word Verschiebung could be mentioned.
I have tried to clean-up the structure of the entry Witt vector a bit. More could be done.
Mostly, I have
polished the Idea-section such as to make the central claim there more transparent, that the (big) Witt vector ring is the co-free Lambda-ring.
in the definition section I have taken apart the “Definition and Theorem”-environment into a definition and a theorem. The theorem is now under “Universal characterization”.
But this needs attention. The whole discussion in the entry wants to prove that $W(-)$ is right adjoint to the forgetful functor from $Lambda$-rings to commutative rings, but never quite comes around to doing so.
From what I see, this is indeed true for the “big” Witt ring, but care needs to be exercised, and the literature on this seems to tend to be sub-optimal on this point. If somebody has the time and energy to put a definite statement (and maybe proof) of this point into the entry, that would be nice.
(I don’t feel like I have the leisure to dwell on this point further right now, but I would enjoy to be able to securely cite it…)
I guess the statement is in Hazewinkel, Witt vectors on p. 87 and the proof is on p. 97. I put that into the entry Witt vector now, but would still be happy if somebody could look over this.
So I gather from Borger08, prop. 1.10 that also the p-typical Witt vectors and all the other flavors are all right adjoint… to the corresponding p-typical etc flavors of Lambda rings. The flavor is parameterized by the set of maximal ideals for which one requires lifts of the Frobenius map.
Maybe I’ll add that in later when I am properly online again.
So I suppose there must then be some deep relationship between Borger’s absolute geometry and chromatic homotopy theory, via Lubin-Tate rings. This must have been mentioned somewhere. Where?
If someone has some idea of what this deep relationship is, I would really like to know about it!
It’s a koan:
$\array{Spec(\mathbb{Z}) & Spec(Spectra) \\ Spec(\mathbb{F}_1) & (?)}$
The other day someone showed me they could make quite a loud clap with just one hand, so I’m in need of a new koan.
So that $Spec(Spectra)$ is determined by the thick subcategory theorem. Now we need some mythical symmetric monoidal stable (∞,1)-category, whose $Spec$ sits under it? And Lubin-Tate appears as a deformation theory to allow the construction of Morava E-theories which stitch together the Morava K-theories in $Spec(Spectra)$ in the chromatic direction.
@David: yes, so this was a late night thought that came to mind. Charles is in way better position than I am to say anything here that goes beyond just free association. But it seems an interesting question. Maybe we should bounce it off the MO community.
The prime spectrum of a symmetric monoidal stable (∞,1)-category construction just talks about a “collection” of thick subcategories. Does this collection have a structure? Can you tensor different such collections, like happens as in the ordinary case, e.g., when looking for $Spec Z \times_{Spec F_1} Spec Z$?
At the beginning of the idea-section of ring of Witt vectors I have added a comment on how they are arithmetic analogs of rings of power series. This needs to be expanded on, for the moment I just included a pointer to arithmetic jet space (which also needs to be expanded on, though)
Maybe we should bounce it off the MO community.
Perhaps things have moved on since this question from November 09.
I am going to start a page differential cohesion and idelic structure with some comments on how in a context of differential cohesion, cohesive over infinitesimal cohesion, at least the key ingredients in the number theoretic and geometric Langlands correspondence all naturally appear structurally, such as the adeles, the ideles, the idele class group, automorphy, moduli of bundles (of course), their relation to the idele class group. (I don’t see the Langlands correspondence itself drop out yet, but various ideas here suggest themselves).
Here “structurally” means that I presently don’t have a model of differential cohesion in which the above would give the actual adeles, ideles, etc., just that it gives objects which conceptually behave just as the adeles, ideles etc. do in arithmetic geometry.
However, as mentioned elsewhere, it seems striking that in Borger’s absolute geometry the direct image $Et(Spec(\mathbb{Z})) \to Et(Spec(\mathbb{F}_1))$ is, being the “arithmetic jet space construction”, conceptually just like the direct image from differential cohesion to the base infinitesimal cohesion. That makes me think that maybe with a little massaging (factoring?) Borger’s geometry could yield differential cohesive structure on $Et(Spec(\mathbb{Z}))$ that would be such that unwinding the “structural” adeles, ideles etc. in this context yields the classical such concepts.
It appears there is An alternative to Witt vectors.
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