Bivariant theory explained

In mathematics, a bivariant theory was introduced by Fulton and MacPherson, 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

f : X \to Y be a map. For such a map, we can consider the fiber square\begin{matrix} X' & \to & Y' \\ \downarrow & & \downarrow \\ X & \to & Y \end{matrix} (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 f .

Now, a birational class of

f is a family of group homomorphisms indexed by the fiber squares:A _{k} Y' \to A _{k-p} X' satisfying the certain compatibility conditions.

Operational Chow ring

The basic question was whether there is a cycle map:

A ^{*(X)} \to \operatorname{H} ^{*(X,} Z ). If X is smooth, such a map exists since A ^{*(X)} is the usual Chow ring of X. 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

• Totaro . Burt . Chow groups, Chow cohomology, and linear varieties . Forum of Mathematics, Sigma . 1 June 2014 . 2 . e17 . 10.1017/fms.2014.15. free .
• Dan Edidin and Matthew Satriano, Towards an intersection Chow cohomology for GIT quotients
  • Book: Fulton . William . MacPherson . Robert . Categorical Framework for the Study of Singular Spaces . 1981 . American Mathematical Soc. . 978-0-8218-2243-2 . en.
• 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