|Date: Tuesday, January 19, 2010
Title: The Norm Residue Theorem in cohomology
Abstract: The Norm Residue Theorem says that the etale cohomology of a
field (finite coefficients) should have a presentation with units as
generators and simple quadratic relations (the ring with this presentation
is now called the "Milnor K-theory"). For Z/2 coefficients, this was
conjectured by Milnor in 1970 and proven by Voevodsky,
but the odd version (mod p coefficients for other primes) has been
open until recently, and has been known as the Bloch-Kato Conjecture.
This talk will be a non-technical overview of the ingredients that go
in to the proof, and why this conjecture matters to non-specialists.
Most of the ingredients are due to Rost and Voevodsky.
One application of this theorem is to motivic cohomology, since it
implies several conjectures of Beilinson and Lichtenbaum. Another
application is to algebraic K-theory, since it implies several other
conjectures of Quillen and Lichtenbaum.
Here is a fun consequence of all this. We now know the first 20,000
groups K_n(Z) of the integers, except when 4 divides n. The assertion that
these groups are zero when 4 divides n (n>0) is equivalent to Vandiver's
Conjecture (in number theory), and if it holds then we have fixed Kummer's
1849 "proof" of Fermat's Last Theorem. If any of them are nonzero, then the
smallest prime dividing the order of this group is at least 16,000,000.
Speaker: Chuck Weibel
Institution: Rutgers University