Oriented cohomology theories of algebraic varieties. II.

*(English)*Zbl 1169.14016The author continues his axiomatic study of oriented ring cohomology theories for algebraic varieties from I. Panin [K-Theory 30, No. 3, 265–314 (2003; Zbl 1047.19001)].

A ring cohomology theory \(A\) is called oriented if it is equipped with Thom isomorphism operators for vector bundles. In loc. cit., the author proved that an orientation is equivalent to a Chern struture or a Thom structure, i.e., existence of Chern classes or Thom classes for line bundles.

In the current paper, the author introduces another notion of trace structure, which is a rule assigning to each projective morphism of smooth varieties \(f:Y\to X\) a two-sided \(A(X)\) module operator \[ {\mathrm{tr}}_f:A(Y)\to A(X), \] satisfying certain properties like naturality, base change, normalization and localization. The map \({\mathrm{tr}}_f\) is called a push-forward, and other common names are Gysin homomorphism, transfer, etc. A trace structure \(f\mapsto {\mathrm{tr}}_f\) is called compatible with an orientation \(\omega\) or its associated Chern structure \(c\) if for a smooth divisor \(i:D\hookrightarrow X\) one has \[ {\mathrm{tr}}_i(1)=c(L(D)). \]

The main result of the current paper is that there is a bijection between orientations and trace structures. The difficult direction of this result is to construct a trace structure from an orientation. Since a projective morphism is the composition of a closed imbedding and a projection for projective spaces, the author deals with the two cases first and at the end proves that the result is independent of the decomposition. For closed imbeddings, deformation to the normal cone and the Thom isomorphism from the orientation are used in the construction. For projections of projective spaces, the author uses the formal group law associated to the orientation to study the image of a projective space \(\mathbb{P}^n\) in the coefficient ring by either applying Quillen’s result on complex cobordism or using residue analysis. Then the author carefully proves properties of such trace operators leading to the above-mentioned main result.

Examples of trace structures are also presented, including algebraic \(K\)-theory, étale cohomology, \(K\) cohomology, motivic cohomology and algebraic cobordism.

As a result, the author demonstrates a bijection between trace structures and local parameters of \(A^{\mathrm ev}(\mathbb{P}^\infty)\).

A ring cohomology theory \(A\) is called oriented if it is equipped with Thom isomorphism operators for vector bundles. In loc. cit., the author proved that an orientation is equivalent to a Chern struture or a Thom structure, i.e., existence of Chern classes or Thom classes for line bundles.

In the current paper, the author introduces another notion of trace structure, which is a rule assigning to each projective morphism of smooth varieties \(f:Y\to X\) a two-sided \(A(X)\) module operator \[ {\mathrm{tr}}_f:A(Y)\to A(X), \] satisfying certain properties like naturality, base change, normalization and localization. The map \({\mathrm{tr}}_f\) is called a push-forward, and other common names are Gysin homomorphism, transfer, etc. A trace structure \(f\mapsto {\mathrm{tr}}_f\) is called compatible with an orientation \(\omega\) or its associated Chern structure \(c\) if for a smooth divisor \(i:D\hookrightarrow X\) one has \[ {\mathrm{tr}}_i(1)=c(L(D)). \]

The main result of the current paper is that there is a bijection between orientations and trace structures. The difficult direction of this result is to construct a trace structure from an orientation. Since a projective morphism is the composition of a closed imbedding and a projection for projective spaces, the author deals with the two cases first and at the end proves that the result is independent of the decomposition. For closed imbeddings, deformation to the normal cone and the Thom isomorphism from the orientation are used in the construction. For projections of projective spaces, the author uses the formal group law associated to the orientation to study the image of a projective space \(\mathbb{P}^n\) in the coefficient ring by either applying Quillen’s result on complex cobordism or using residue analysis. Then the author carefully proves properties of such trace operators leading to the above-mentioned main result.

Examples of trace structures are also presented, including algebraic \(K\)-theory, étale cohomology, \(K\) cohomology, motivic cohomology and algebraic cobordism.

As a result, the author demonstrates a bijection between trace structures and local parameters of \(A^{\mathrm ev}(\mathbb{P}^\infty)\).

Reviewer: Zhaohu Nie (Altoona)

##### MSC:

14F43 | Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) |

55N22 | Bordism and cobordism theories and formal group laws in algebraic topology |

14F42 | Motivic cohomology; motivic homotopy theory |