Bivariant theory
In mathematics, a bivariant theory was introduced by Fulton and MacPherson (Fulton 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 [math]\displaystyle{ f : X \to Y }[/math] be a map. For such a map, we can consider the fiber square
- [math]\displaystyle{ \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]\displaystyle{ f }[/math].
Now, a birational class of [math]\displaystyle{ f }[/math] is a family of group homomorphisms indexed by the fiber squares:
- [math]\displaystyle{ A_k Y' \to A_{k-p} X' }[/math]
satisfying the certain compatibility conditions.
Operational Chow ring
The basic question was whether there is a cycle map:
- [math]\displaystyle{ A^*(X) \to \operatorname{H}^*(X, \mathbb{Z}). }[/math]
If X is smooth, such a map exists since [math]\displaystyle{ A^*(X) }[/math] 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
- Fulton, William; MacPherson, Robert (1981) (in en). Categorical Framework for the Study of Singular Spaces. American Mathematical Soc.. ISBN 978-0-8218-2243-2. https://books.google.com/books?id=pR7UCQAAQBAJ.
- The last two lectures of Vakil, Math 245A Topics in algebraic geometry: Introduction to intersection theory in algebraic geometry
External links
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