Trying to understand answer to this question.
What is the (Beilinson) higher regulator of a number field?
$\begingroup$
$\endgroup$
asked Oct 23, 2009 at 17:35
Add a comment
|
2 Answers
$\begingroup$
$\endgroup$
5
Here is an attempt to answer, but I hope that someone else can give a better explanation.
As Rob H. pointed out in his answer to the previous question, the survey of Nekovar is very nice, and it is also available online here.
About your question: The Beilinson regulator can be defined for number fields but also for varieties over number fields. It is a map from motivic cohomology (or algebraic K-theory) with rational coefficients, to Deligne-Beilinson cohomology, and can be thought of as a kind of Chern character.
The precise definition of the regulator is quite nontrivial, and there are several equivalent ways of defining it. Philosophically, it should be a map between certain Ext groups, induced from a "Hodge realization" of motives. Sorry for not explaining this well - it belongs to your other question about the yoga of motives.
Forgetting about philosophy, there are several ways of actually constructing the regulator. One approach is to use the general framework of characteristic classes developed by Gillet. Nekovar has a "direct" construction in his paper. For another construction in terms of explicit complexes, see the recent thesis of Elisenda Feliu, available on her webpage. Another reference is the Bourbaki talk by Soule, available here.
If you are only interested in number fields, there is another construction of the regulator, named after Borel. An excellent online reference for this, and its relation to the Beilinson regulator, is the book of Burgos, available here. The regulator generalizes the Dirichlet regulator which is covered in most introductory books on algebraic number theory. For a computational approach to Borel's regulator, see recent papers on the arXiv by Choo, Mannan, Sánchez-Garcia and Snaith.
answered Oct 23, 2009 at 21:04
-
$\begingroup$ You're intriguing me with "Philosophically, it should be a map between certain Ext groups, induced from a "Hodge realization" of motives." I wonder if there is a "middle ground", the way to grok what's the motivation without understanding the whole philosophy (which may be beyond my abilitiess :) ) $\endgroup$ Commented Oct 23, 2009 at 21:07
-
$\begingroup$ Like, "kind of Chern character." maybe there's a good way to explain that? $\endgroup$ Commented Oct 23, 2009 at 21:08
-
$\begingroup$ I could probably try to give an answer, but it would be very long... and it wouldn't be much more informative than the introductions contained in the various references above. Sorry... I might give it a try though, but not right now. $\endgroup$ Commented Oct 23, 2009 at 21:19
-
$\begingroup$ Ok, so I tried to give a better "middle ground" answer about the Beilinson regulator, over at your yoga question, although I did not talk about the Chern character interpretation: mathoverflow.net/questions/2146/whats-the-yoga-of-motives $\endgroup$ Commented Oct 23, 2009 at 22:49
-
$\begingroup$ Georg Tamme, Karoubi’s relative Chern character and Beilinson’s regulator, Ann. Sci. Ec. Norm. Super. (4) 45 (2012), no. 4, 601-636, seems to define the Beilinson regulator in terms of Chern character valued in real Deligne cohomology. $\endgroup$– Z. MCommented Oct 13 at 20:58
$\begingroup$
$\endgroup$
5
Many, many years ago I happened to be in the room when somebody asked exactly the same question --- "What is the Beilinson regulator?" --- of Marc Levine. Marc went straight to the blackboard and drew the key diagram that makes everything clear. I knew it had to be preserved for posterity, so I ran across the street to buy a disposable camera. I am delighted to be able to share this:
(Click picture for larger version.)
answered Oct 24, 2014 at 19:00
-
2$\begingroup$ I am so glad I finally understand this! $\endgroup$ Commented Oct 24, 2014 at 20:06
-
$\begingroup$ It is especially fascinating how the word "regulator" acquires new depths of meaning in light of this $\endgroup$ Commented Nov 13, 2017 at 18:38
-
$\begingroup$ Dear @StevenGubkin, just in case you are not ironic here, mind you share a little bit of your understanding ? The picture and its story are pretty, but im not sure how much farther it gets me to understanding regulators. I can see the homology of some classifying spaces used in K-theory, i can see Kähler differentials, some exact sequences, Deligne-Beilinson cohomology groups,... but i'm not sure what the flow is between them.. Anyway, congratulations to Steven Landsburg for his composure and celerity in reacting to this scene. $\endgroup$– plmCommented Jul 15, 2023 at 23:15
-
2$\begingroup$ @plm My comment was nearly a decade ago. I think I intended it as a joke. $\endgroup$ Commented Jul 16, 2023 at 17:28
-
1$\begingroup$ @plm I am not capable to understand this photo, but the exponential exact sequence is used to define the first Chern class, and then use the projective bundle formula to define usual Chern classes. The appearance of $\operatorname{BGL}$ should be comparable to Weibel's K-book, Chap 5, section 11. $\endgroup$– Z. MCommented Oct 14 at 8:36