Two new motivic complexes for non-smooth schemes
Organizers
Speaker
Shane Kelly
Time
Thursday, October 17, 2024 10:00 AM - 11:00 AM
Venue
A6-101
Online
Zoom 638 227 8222
(BIMSA)
Abstract
The Atiyah-Hirzebruch spectral sequence calculating topological K-theory from singular cohomology has been available since around 1960. The analogous spectral sequence for algebraic K-theory has been more elusive. According to conjectures of Beilinson and Lichtenbaum from the 80s, the E2-page of such a spectral sequence is made of motivic cohomology groups. These are generalisations of Chow groups and are intimately related to a large swath of the open conjectures in algebraic and arithmetic geometry.
For smooth schemes over a field, these were constructed by Bloch in the 80s under the guise of higher Chow groups and later by Voevodsky around 2000 inside a very general and robust theory of A1-homotopy theory. Atiyah-Hirzebruch style spectral sequences have been developed by Friedlander-Suslin and Levine.
Recently, two new generalisations of this motivic cohomology to non-smooth schemes have independently appeared, which turn out to be equivalent. One constructed by Elmanto-Morrow and the other by the speaker in joint work with Shuji Saito. We discuss these two new cohomologies and their comparison.
For smooth schemes over a field, these were constructed by Bloch in the 80s under the guise of higher Chow groups and later by Voevodsky around 2000 inside a very general and robust theory of A1-homotopy theory. Atiyah-Hirzebruch style spectral sequences have been developed by Friedlander-Suslin and Levine.
Recently, two new generalisations of this motivic cohomology to non-smooth schemes have independently appeared, which turn out to be equivalent. One constructed by Elmanto-Morrow and the other by the speaker in joint work with Shuji Saito. We discuss these two new cohomologies and their comparison.