Bivariant theory

In mathematics, a bivariant theory was introduced by Fulton and MacPherson (Fulton & MacPherson 1981), 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, specifically Borel-Moore homology, 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:

$A^{*}(X)\to \operatorname {H} ^{*}(X,\mathbb {Z} ).$

If X is smooth, such a map exists since $A^{*}(X)$ is the usual Chow ring of X. (Totaro 2014) 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 (1 June 2014). "Chow groups, Chow cohomology, and linear varieties". Forum of Mathematics, Sigma. 2 e17. doi:10.1017/fms.2014.15.
• Dan Edidin and Matthew Satriano, Towards an intersection Chow cohomology for GIT quotients
• Fulton, William (1998), Intersection Theory, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98549-7, MR 1644323
• Fulton, William; MacPherson, Robert (1981). Categorical Framework for the Study of Singular Spaces. American Mathematical Soc. ISBN 978-0-8218-2243-2.
• 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