Around the Gysin triangle II

TL;DR

This paper develops an axiomatic framework for the Gysin triangle in triangulated motives, enabling new constructions of Gysin morphisms, duality results, and applications to rigid cohomology, generalizing classical theorems.

Contribution

It introduces a unified axiomatic approach to the Gysin triangle applicable to motives and MGl-modules, extending duality and cobordism concepts.

Findings

Constructed Gysin morphisms for smooth schemes over an arbitrary base.

Proved duality for projective smooth schemes within this framework.

Generalized the Myschenko theorem and connected it to classical stable homotopy.

Abstract

We study the construction and properties of the Gysin triangle in an axiomatic framework which covers triangulated mixed motives and MGl-modules over an arbitrary base S. This allows to define the Gysin morphism associated to a projective morphism between smooth S-schemes and prove duality for projective smooth S-schemes. As part of the construction, cobordism classes are considered and we give a proof of the Myschenko theorem generalized in our context - this in fact gives another proof of the latter theorem in classical stable homotopy through complex realization. Finally, these constructions apply to rigid cohomology through the notion of a mixed Weil theory introduced by D.-C. Cisinski and the author in another work.