On the push-forwards for motivic cohomology theories with invertible stable Hopf element

TL;DR

This paper develops a geometric method for constructing push-forward maps in motivic cohomology theories with an inverted stable Hopf element, extending the framework for cohomology computations in algebraic geometry.

Contribution

It introduces a new geometric construction of push-forward maps in motivic cohomology theories with inverted Hopf elements, generalizing previous approaches for derived Witt groups.

Findings

Constructed push-forward maps along projective morphisms

Defined twisted cohomology groups via Thom spaces

Computed twisted cohomology groups of projective spaces

Abstract

We present a geometric construction of push-forward maps along projective morphisms for cohomology theories representable in the stable motivic homotopy category assuming that the element corresponding to the stable Hopf map is inverted in the coefficient ring of the theory. The construction is parallel to the one given by A. Nenashev for derived Witt groups. Along the way we introduce cohomology groups twisted by a formal difference of vector bundles as cohomology groups of a certain Thom space and compute twisted cohomology groups of projective spaces.