Poincare Duality for Algebraic Varieties., by Ivan A. Panin and Serge A. Yagunov

Given an oriented theory on algebraic varieties we supply every smooth projective algebraic variety X with a fundamental class [X]. It is proven that the cap-product with the class [X] provides an isomorphism between the cohomology and the homology of the variety X.

The result holds for (co-)homology theories represented by oriented T-spectra. In particular, this class includes Motivic Cohomology represented by the Eilenberg-Mac Lane T-spectrum H and the algebraic cobordism represented by T-spectrum MGL.

In case the ground field is the complex numbers and the theory is singular (co-)homology (or more generally any theory represented by an oriented spectrum) the constructed isomorphism coincides with the classical Poincare Duality.

Ivan A. Panin <panin@pdmi.ras.ru>
Serge A. Yagunov <yagunov@pdmi.ras.ru>