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# Homology, Homotopy and Applications

## Volume 11 (2009)

### Number 1

### Oriented cohomology theories of algebraic varieties II

Pages: 349 – 405

DOI: http://dx.doi.org/10.4310/HHA.2009.v11.n1.a14

#### Author

#### 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.

#### Keywords

motivic cohomology; algebraic cobordism; oriented cohomology theories

#### 2010 Mathematics Subject Classification

14F42, 14F43, 55N22