In mathematics, a bivariant theory was introduced by

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and MacPherson , in order to put a ring structure on the

Chow group In algebraic geometry, the Chow groups (named after Wei-Liang Chow by ) of an algebraic variety over any field are algebro-geometric analogs of the homology of a topological space. The elements of the Chow group are formed out of subvarieties ( ...

of a

singular variety In the mathematical field of algebraic geometry, a singular point of an algebraic variety is a point that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In ca ...

, the resulting ring called an operational Chow ring. On technical levels, a bivariant theory is a mix of a

homology Homology may refer to: Sciences Biology *Homology (biology), any characteristic of biological organisms that is derived from a common ancestor * Sequence homology, biological homology between DNA, RNA, or protein sequences *Homologous chrom ...

theory and a

cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...

theory. In general, a homology theory is a covariant

functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...

from the category of spaces to the category of

abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...

s, while a cohomology theory is a

contravariant functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ma ...

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
X' & \to & Y' \\
\downarrow & & \downarrow \\
X & \to & Y
\end

(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_ X'

satisfying the certain compatibility conditions.

Operational Chow ring

The basic question was whether there is a

cycle map Cycle, cycles, or cyclic may refer to: Anthropology and social sciences * Cyclic history, a theory of history * Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr. * Social cycle, various cycles in s ...

: :

A^*(X) \to \operatorname^*(X, \mathbb).

If ''X'' is smooth, such a map exists since

A^*(X)

is the usual

Chow ring In algebraic geometry, the Chow groups (named after Wei-Liang Chow by ) of an algebraic variety over any field are algebro-geometric analogs of the homology of a topological space. The elements of the Chow group are formed out of subvarieties (so-c ...

of ''X''. has shown that rationally there is no such a map with good properties even if ''X'' is a

linear variety In geometry, a flat or Euclidean subspace is a subset of a Euclidean space that is itself a Euclidean space (of lower dimension). The flats in two-dimensional space are points and lines, and the flats in three-dimensional space are points, lin ...

, roughly a variety admitting a cell decomposition. He also notes that Voevodsky's

motivic cohomology ring Motivic cohomology is an invariant of algebraic varieties and of more general schemes. It is a type of cohomology related to motives and includes the Chow ring of algebraic cycles as a special case. Some of the deepest problems in algebraic geome ...

is "probably more useful " than the operational Chow ring for a singular scheme (§ 8 of loc. cit.)

References

* * Dan Edidin and Matthew Satriano
''Towards an intersection Chow cohomology for GIT quotients''
* * * 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
Algebraic geometry {{Improve categories, date=December 2019 Homology theory Cohomology theories Functors Abelian group theory