Fun with F_un
Amidst all LHC-noise,
Yuri I. Manin arXived today an interesting paper
Cyclotomy and analytic geometry over 
.
The paper gives a nice survey of the existent literature and focusses on the crucial role of roots of unity in the algebraic geometry over the non-existent field with one element (in French called ‘F-un’). I have tried to do a couple of
posts on F-un some time ago but now realize, reading Manin’s paper, I may have given up way too soon…
At several places in the paper, Manin hints at a possible noncommutative geometry over :
This is the appropriate place to stress that in a wider context of Toen-Vaqui ‘Au-dessous de Spec Z’, or eventually in noncommutative
Soule’s algebrasare a very important element of the structure, in particular, because they form a bridge to Arakelov geometry. Soule uses concrete choices of them in order to produce ‘just right’ supply of morphisms, without a priori constraining these choices formally. In this work, we use these algebras and their version also to pave a way to the analytic (and possibly non-commutative) geometry over
Back when I was writing the first batch of F-un posts, I briefly contemplated the possibility of a noncommutative geometry over , but quickly forgot about it because I thought it would be forced to reduce to commutative geometry.
Here is the quick argument : noncommutative geometry is really the study of coalgebras (see for example
my paper or if you prefer more trustworthy sources the
Kontsevich-Soibelman paper). Now, unless I made a mistake, I think all coalgebras over must be co-commutative (even group-like), so reducing to commutative geometry.
Surely, I’m missing something…
arxiv, coalgebras, geometry, Kontsevich, Manin, non-commutative, noncommutative
10 comments
Posted in geometry
Written on Wed, 10 September 2008 at 6:26 pm
Tags: arxiv, coalgebras, geometry, Kontsevich, Manin, non-commutative, noncommutative
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September 10th, 2008 at 8:03 pm
I assume your reassoning relies on the interpretation of groups as Hopf algebras over
, right?
If this is the case, same argument would allow to state that all algebras over the field with one element are division rings, and I don’t think that is what these people have in mind.
On the other hand, if you think about “going to
” as “forgetting addition”, an algebra should be something like a monoid (or a monad), and the right dual would be a comonad.
Still I am myself not sure this is the right approach…
September 11th, 2008 at 2:47 am
Hmmmm. Ross Street likes to say that quantum sets are coalgebras, and Hopf algebras come along with this philosophy. Co-commutative but non commutative algebras already greatly generalise ordinary ‘commutative’ groups (as sets with only multiplication). This would tie in with the ‘GL(n) is a braid group’ idea of F1 maths. Must check out the paper … now that the LHC news has died down.
September 11th, 2008 at 11:15 pm
I’ve kept thinking about this, and convinced myself about what I said above. Reasoning goes as follows: over an arbitrary field
, an algebra 
defines a monad 
in the category of 
-vector spaces. Conversely, given any monad 
on the category of 
-vector spaces gives rise to an ordinary algebra 
by evaluation on the unit object of the category. Thus algebras over a field are the same thing as monads on vector spaces over the same field. Since we have a pretty well defined category of vector spaces over 
, which is the category of sets, an 
-algebra should be a monad on the category of sets. Commutativity of an algebra can be given in monadic terms by means of a distributive law (which is the fanshy-wanshi way of calling a twisting map) between two copies of the monad, but this is by no means necessary.
Same argument goes for coalgebras and comonads.
September 12th, 2008 at 7:32 am
@javier : no i was just using definitions. if there is only 1 in the field then this determines the counit and then from coalgebra-defns one gets that all elements must be grouplike and so the coalgebra is cocommutative. of course i know the weak spot in my ‘argument’. the interpretation of coalgebras as noncommutative geometry is based on the Kostant duality between algebras and coalgebras and this duality breaks down over F-un… also, i think to understand what Manin has in mind when he calls for a noncomm geometry over F-un. probably he means the gadget bit (the algebra A_X) in Soule’s framework.
September 12th, 2008 at 9:21 am
I posted a comment yesterday evening digging deeper on the categorical approach that I hinted above, but since it was tex-intensive, it possibly got lost in the moderation queue. I am not convinced of deducing grouplikeness just from the counit, if you are not allowed to work with linear combinations (because you have no addition).
September 12th, 2008 at 9:35 am
what about
and the other way around, given the fact that you know already that 
for all 
?
tex doesnt look too well on black. so i meant
what about id = (\epsilon \otimes id) \circ \Delta and the other way around, given the fact that you know already that \epsilon(g)=1 for all g?
September 12th, 2008 at 12:31 pm
Yeah, that seems right… what I an not convinced on is the existence of maps landing on
, not as normal set-theoretical maps anyway (since the field with one element is not really defined as a set). Sort of the same way as you cannot explicitly take a ring map from 
to 
that would make the integers into a 
algebra. In Soulé’s paper, the existence of such hypothetical map is ‘assumed’ to build the notion of scheme, but there is no attempt of providing a realization of it.
My impression was that one should always avoid hitting
directly, and always try to use things defined over 
as workarounds, that is why I thouhgt of defining algebras and coalgebras as functors on the corresponding categories of vector spaces.
But it is very likely that I got the wrong intuition here, so nevermind…
September 15th, 2008 at 5:33 pm
Looks like Connes and Consani have also a new paper out on Geometry over F_1, in fact posted just one day before Manin’s http://www.alainconnes.org/docs/ak.pdf
September 17th, 2008 at 3:58 pm
@leon: Thanks for the reference! One more paper to read on…
@lieven: Some days ago I wrote another comment on the notion of coalgebra over F1 that got swallowed. I think your spam filter hates me… Talking different things, I am planning an informal study seminar around F1 here at MPI, starting at the end of the month. If you are interested on participating and sharing your thoughts about it, let me know! Bonn is pretty close to Antwerp, so I am sure we can easily arrange something.
September 17th, 2008 at 4:44 pm
Javier, next time one of your comments doesnt appear at once send me an email and ill fish it out of the Akismet spam-filter…
Fun seminar : well, lets wait for one more week. next week courses start here and apart from the 240hrs i already have to teach anyway this semester (yeah, good news being my 2nd semester is semi-free) i also run a master-course ’seminar noncommutative geometry’. our regular antwerp-seminar starts mid october and i promised to give some talks (combining them with the seminar-course). depending on whether or not students show up for the seminar-course (i rate as highly unlikely, but if they do ill talk about less virtual stuff) i can do some talks on Fun (btw. Raf is also interested in this for entirely different reasons…).
So, maybe we can use this ending-blog as a vehicle to have some interaction between the two seminars?