Tohoku Mathematical Journal

A note on relative duality for Voevodsky motives

Luca Barbieri-Viale and Bruno Kahn

Full-text: Open access


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 $\operatorname{DM}_{\rm gm}(k)$.

Article information

Tohoku Math. J. (2), Volume 60, Number 3 (2008), 349-356.

First available in Project Euclid: 3 October 2008

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14C25: Algebraic cycles

Duality motives


Barbieri-Viale, Luca; Kahn, Bruno. A note on relative duality for Voevodsky motives. Tohoku Math. J. (2) 60 (2008), no. 3, 349--356. doi:10.2748/tmj/1223057732.

Export citation


  • L. Barbieri-Viale and V. Srinivas, Albanese and Picard 1-motives, Mémoire Soc. Math. France (N.S.) 87, Paris, 2001.
  • D. C. Cisinski and F. Déglise, Mixed Weil cohomologies, preprint, 2007.
  • F. Déglise, Interprétation motivique de la formule d'excès d'intersection, C. R. Math. Acad. Sci. Paris 338 (2004), 41--46.
  • F. Déglise, Motifs génériques, preprint, 2005.
  • F. Déglise, Around the Gysin triangle, preprint, 2005.
  • E. Friedlander and V. Voevodsky, Bivariant cycle cohomology, Ann. of Math. Stud. 143 (2000), 138--187.
  • A. Huber, Realization of Voevodsky's motives, J. Algebraic. Geom. 9 (2000), 755--799./Corrigendum, ibid. 13 (2004), 195--207.
  • F. Ivorra, Réalisation $\ell$-adique des motifs triangulés géométriques, I, Doc. Math. 12 (2007), 607--671.
  • M. Levine, Mixed motives, Math. Surveys Monogr. 57, American Mathematical Society, Providence, RI, 1998.
  • V. Voevodsky, Triangulated categories of motives over a field, Ann. of Math. Stud. 143 (2000), 188--238.