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Mathematics Colloquium


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


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