## Tohoku Mathematical Journal 2008 September SECOND SERIES VOL. 60, NO. 3

 Tohoku Math. J. 60 (2008), 349-356

Title A NOTE ON RELATIVE DUALITY FOR VOEVODSKY MOTIVES

Author Luca Barbieri-Viale and Bruno Kahn

(Received April 27, 2007, revised January 15, 2008)
Abstract. Let $k$ be a perfect field which admits resolution of singularities in the sense of Friedlander and Voevodsky (for example, $k$ of characteristic $0$). Let $X$ be a smooth proper $k$-variety of pure dimension $n$ and $Y,Z$ two disjoint closed subsets of $X$. We prove an isomorphism $M(X-Z,Y)\simeq M(X-Y,Z)^*(n)[2n],$ where $M(X-Z,Y)$ and $M(X-Y,Z)$ are relative Voevodsky motives, defined in his triangulated category $\operatorname{DM}_{\rm gm}(k)$.

2000 Mathematics Subject Classification. Primary 14C25.

Key words and phrases. Duality, motives.