Atiyah–Hirzebruch for (possibly derived) qcqs schemes

Speaker: Shane Kelly (University of Tokyo)

Time: 2025-10-27 15:00-18:00
Location: 1110, SIMIS
Zoom Meeting ID:882 5464 8726 Passcode:SIMIS

Abstract:

The topological Atiyah–Hirzebruch spectral sequence expresses complex K-theory in terms of ordinary cohomology and underlies the index theorem. In the early 1980s, motivated by conjectures on special values of L-functions, Beilinson and Lichtenbaum proposed a universal cohomology theory with integral coefficients—motivic cohomology—whose associated Atiyah–Hirzebruch spectral sequence would compute algebraic K-theory. For smooth varieties over a field, this spectral sequence was subsequently constructed in multiple ways in work of Bloch, Grayson, Friedlander, Levine, Suslin, and Voevodsky.

In joint work with Shuji Saito, we extend this picture to general (possibly derived) qcqs schemes by proposing a definition of motivic cohomology that comes equipped with an Atiyah–Hirzebruch spectral sequence, agreeing with the classical one on smooth varieties. This framework may be viewed as a universal form of the Grothendieck–Riemann–Roch theorem. If time permits, I may also mention recent joint work with Shuji Saito and Georg Tamme.