bivariant theory

In mathematics, a bivariant theory was introduced by Fulton and MacPherson {{harv|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 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. {{harv|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

{{cite journal |last1=Totaro |first1=Burt |title=Chow groups, Chow cohomology, and linear varieties |journal=Forum of Mathematics, Sigma |date=1 June 2014 |volume=2 |pages=e17 |doi=10.1017/fms.2014.15|doi-access=free }}
Dan Edidin and Matthew Satriano, [https://faculty.missouri.edu/~edidind/Papers/ChowTheorems-07-26.pdf Towards an intersection Chow cohomology for GIT quotients]
{{Citation | last1=Fulton | first1=William | author1-link=William Fulton (mathematician) | title=Intersection Theory | publisher=Springer-Verlag | location=Berlin, New York | isbn=978-0-387-98549-7 | mr=1644323 | year=1998 }}
{{cite book |last1=Fulton |first1=William |last2=MacPherson |first2=Robert |title=Categorical Framework for the Study of Singular Spaces |date=1981 |publisher=American Mathematical Soc. |isbn=978-0-8218-2243-2 |url=https://books.google.com/books?id=pR7UCQAAQBAJ |language=en}}
The last two lectures of Vakil, [https://math.stanford.edu/~vakil/245/ Math 245A Topics in algebraic geometry: Introduction to intersection theory in algebraic geometry]

External links

[https://ncatlab.org/nlab/show/bivariant+cohomology+theory nLab- bivariant cohomology theory]

Category:Abelian group theory

Category:Algebraic geometry

Category:Cohomology theories

Category:Functors

Category:Homology theory