Mikhail Bondarko | On the motivic t-structure on DM(k,Q)
Channel: EIMI, PDMI RAS and Chebyshev Laboratory
Date: 2021-04-01

Seminar on A1-topology, motives and K-theory, April 01, 2021
Mikhail Bondarko (SPbU)
On the motivic t-structure on DM(k,Q)

I will describe my recent observations on certain candidates for the (conjectural) motivic t-structure and on objects that represent "classical" cohomology theories.

General (and recent) abstract nonsense easily implies that any homology theory H on the "big" category DM(k,Q) of motives over a field defines a t-structure t_H (that is, the "cohomological left hand side" of  t_H is characterized by the vanishing of H(M[i]) for i positive). In particular, this is the case when H is the Q_l-etale homology (clearly, one can also take etale cohomology here).

t_H restricts to compact motives whenever the weight structure w right adjacent to it "respects coproducts", and the converse implication follows from standard conjectures. Moreover, the "right hand side" of this t_H can be described "almost explicitly" if the object R that represents etale cohomology is connective (that is, there are no non-zero morphisms R→R[i] for i positive).

I have also proved that the so-called Chow t-structure splits this object R. Consequently, the weight filtration on the Q_l-etale cohomology of motives splits functorially; this is also true for any other cohomology that possesses similar properties. Moreover, I have some ideas how to prove that this splitting respects cup products and to verify the aforementioned connectivity.