On the motivic t-structure on DM(k,Q)

by Mikhail Bondarko (SPbU)

Thursday 1 Apr 2021, 13:00 → 15:00 Europe/Moscow

818-1526-4739 (Zoom)

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 $i>0$). 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 $R\to R[i]$ for $i>0$).

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.
 

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