Oriented cohomology theories of algebraic varieties II

Abstract

The concept of oriented cohomology theory is well-known in topology. Examples of these kinds of theories are complex cobordism, complex K-theory, usual singular cohomology, and Morava K-theories. A specific feature of these cohomology theories is the existence of trace operators (or Thom-Gysin operators, or push-forwards) for morphisms of compact complex manifolds. The main aim of the present article is to develop an algebraic version of the concept. Bijective correspondences between orientations, Chern structures, Thom structures and trace structures on a given ring cohomology theory are constructed. The theory is illustrated by singular cohomology, motivic cohomology, algebraic K-theory, the algebraic cobordism of Voevodsky and by other examples.

Torsion algebraic cycles and étale cobordism

“…Chern classes for E are then defined in the well known way. The existence and uniqueness of pushforward maps follows from [26].…”

Oriented cohomology, Borel-Moore homology, and algebraic cobordism

“…As our intention here is to add support conditions to Panin's theory, and this requires some additional commutativity conditions not imposed in [8], we will add the simplifying assumption that a ring cohomology theory will always be Z/2graded. We likewise require that the boundary maps in the underlying cohomology theory are of odd degree and that the pull-back maps preserve degree.…”

“…Thus ι ′ is subjected to ω, and hence ι = ι ′ by the uniqueness in theorem 1.10. Theorem 1.10 is proven by copying the construction in [8] of an integration subjected to a given orientation ω, making at each stage the extension to an integration with supports.…”

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

“…Roughly speaking, oriented cohomology theories (see [11,19,27]) are cohomology theories possessing push-forwards along arbitrary projective morphisms and satisfying certain natural properties. The theory in the oriented setting is quite well-developed: one may obtain a projective bundle theorem and introduce Chern classes of vector bundles [21,27], study morphisms between such theories and obtain Riemann-Roch type theorems [20,28], construct a universal oriented cohomology theory [11] that allows to perform computations in the universal setting, study the corresponding categories of motives and obtain various motivic decompositions [18], etc.…”

“…Using A. Nenashev's constructions (which follow in general the ones introduced in [21,27]) one can check that this definition does not depend on the choice of p and i and obtain the usual properties of push-forwards: functoriality, projection formula and compatibility with transversal base change. The paper is organized in the following way.…”