## Homology, Homotopy and Applications

- Homology Homotopy Appl.
- Volume 11, Number 1 (2009), 349-405.

### 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.

#### Article information

**Source**

Homology Homotopy Appl., Volume 11, Number 1 (2009), 349-405.

**Dates**

First available in Project Euclid: 1 September 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.hha/1251832570

**Mathematical Reviews number (MathSciNet)**

MR2529164

**Zentralblatt MATH identifier**

1169.14016

**Subjects**

Primary: 14F42: Motivic cohomology; motivic homotopy theory [See also 19E15] 14F43: Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) 55N22: Bordism and cobordism theories, formal group laws [See also 14L05, 19L41, 57R75, 57R77, 57R85, 57R90]

**Keywords**

Motivic cohomology algebraic cobordism oriented cohomology theories

#### Citation

Panin, Ivan. Oriented cohomology theories of algebraic varieties II. Homology Homotopy Appl. 11 (2009), no. 1, 349--405. https://projecteuclid.org/euclid.hha/1251832570