Английская Википедия:Bivariant theory

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In mathematics, a bivariant theory was introduced by Fulton and MacPherson Шаблон:Harv, in order to put a ring structure on the Chow group of a singular variety, the resulting ring called an operational Chow ring.

On technical levels, a bivariant theory is a mix of a homology theory and a cohomology theory. In general, a homology theory is a covariant functor from the category of spaces to the category of abelian groups, while a cohomology theory is a contravariant functor from the category of (nice) spaces to the category of rings. A bivariant theory is a functor both covariant and contravariant; hence, the name “bivariant”.

Definition

Unlike a homology theory or a cohomology theory, a bivariant class is defined for a map not a space.

Let <math>f : X \to Y</math> be a map. For such a map, we can consider the fiber square

<math>

\begin{matrix} X' & \to & Y' \\ \downarrow & & \downarrow \\ X & \to & Y \end{matrix} </math> (for example, a blow-up.) Intuitively, the consideration of all the fiber squares like the above can be thought of as an approximation of the map <math>f</math>.

Now, a birational class of <math>f</math> is a family of group homomorphisms indexed by the fiber squares:

<math>A_k Y' \to A_{k-p} X'</math>

satisfying the certain compatibility conditions.

Operational Chow ring

Шаблон:Expand section The basic question was whether there is a cycle map:

<math>A^*(X) \to \operatorname{H}^*(X, \mathbb{Z}).</math>

If X is smooth, such a map exists since <math>A^*(X)</math> is the usual Chow ring of X. Шаблон:Harv has shown that rationally there is no such a map with good properties even if X is a linear variety, roughly a variety admitting a cell decomposition. He also notes that Voevodsky's motivic cohomology ring is "probably more useful " than the operational Chow ring for a singular scheme (§ 8 of loc. cit.)

References

• Шаблон:Cite journal
• Dan Edidin and Matthew Satriano, Towards an intersection Chow cohomology for GIT quotients
• Шаблон:Citation
• Шаблон:Cite book
• The last two lectures of Vakil, Math 245A Topics in algebraic geometry: Introduction to intersection theory in algebraic geometry

External links

• nLab- bivariant cohomology theory