Torsion Motives

Abstract

Abstract In this paper, we study Chow motives whose identity map is killed by a natural number. Examples of such objects were constructed by Gorchinskiy and Orlov [ 10]. We introduce various invariants of torsion motives, in particular, the $p$-level. We show that this invariant bounds from below the dimension of the variety a torsion motive $M$ is a direct summand of and imposes restrictions on motivic and singular cohomology of $M$. We study in more details the $p$-torsion motives of surfaces, in particular, the Godeaux torsion motive. We show that such motives are in $1$-to-$1$ correspondence with certain Rost cycle submodules of free modules over $H^*_{et}$. This description is parallel to that of mod-$p$ reduced motives of curves.