Around the Gysin triangle. II.

Abstract

The notions of orientation and duality are well understood in algebraic topology in the framework of the stable homotopy category. In this work, we follow these lines in algebraic geometry, in the framework of motivic stable homotopy, introduced by F. Morel and V. Voevodsky. We use an axiomatic treatment which allows us to consider both mixed motives and oriented spectra over an arbitrary base scheme. In this context, we introduce the Gysin triangle and prove several formulas extending the traditional panoply of results on algebraic cycles modulo rational equivalence. We also obtain the Gysin morphism of a projective morphism and prove a duality theorem in the (relative) pure case. These constructions involve certain characteristic classes (Chern classes, fundamental classes, cobordism classes) together with their usual properties. They imply statements in motivic cohomology, algebraic K-theory (assuming the base is regular) and “abstract” algebraic cobordism as well as the dual statements in the corresponding homology theories. They apply also to ordinary cohomology theories in algebraic geometry through the notion of a mixed Weil cohomology theory, introduced by D.-C. Cisinski and the author in [CD06], notably rigid cohomology.