Kapranov-Smirnov on F_un

Posted by on Oct 15, 2008 in quotesNo comments

One of the true treasures of our F_un library is the unpublished paper by M. Kapranov and A. Smirnov “Cohomology determinants and reciprocity laws: number field case”.

It is far from easy to read (at least for me), but I’ve found every effort trying to make sense of even a single paragraph time well invested. Here are the opening lines :

“Analogies between number fields and function fields have been a long-time source of inspiration in arithmetic. However, one of the most intruiging problems in this approach, namely the problem of the absolute point, is still far from being satisfactorily understood.

The scheme \wis{spec}(\Z), the final object in the category of schemes, has dimension 1 with respect to the Zariski topology and at least 3 with respect to the etale topology.

This generated a long-standing desire to introduce a more mythical object P, the “absolute point” with a natural morphism

\pi_X~:~X \rightarrow P

given for any arithmetic scheme X so that global invariants of X have an interpretation in terms of a version of direct image with respect to \pi_X.”

And a few paragraphs further they issue a firm warning to all who think they can attack this problem without proper preparation.

“First of all, it is an old idea to interpret combinatorics of finite sets as the q \rightarrow 1 limit of linear algebra over the finite fields \mathbb{F}_q. This had lead to frequent consideration of the folklore object \mathbb{F}_1, the “field with one element”.

One can postulate, of course, that \wis{spec}(\mathbb{F}_1) is the absolute point, but the real problem is to develop non-trivial consequences from this point of view.

Perhaps, the \mathbb{F}_1-idea might have led to a real breakthrough if it had landed around 1930 in Germany. Whether it will do the same today, remains to be seen…

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