In algebraic geometry, a branch of mathematics, an adequate equivalence relation is an equivalence relation on

algebraic cycles In mathematics, an algebraic cycle on an algebraic variety ''V'' is a formal linear combination of subvarieties of ''V''. These are the part of the algebraic topology of ''V'' that is directly accessible by algebraic methods. Understanding the alg ...

of smooth

projective varieties In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables wi ...

used to obtain a well-working theory of such cycles, and in particular, well-defined intersection products. Pierre Samuel formalized the concept of an adequate equivalence relation in 1958. Since then it has become central to theory of motives. For every adequate equivalence relation, one may define the

category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...

of pure motives with respect to that relation. Possible (and useful) adequate equivalence relations include ''rational'', ''algebraic'', ''homological'' and ''numerical equivalence''. They are called "adequate" because dividing out by the equivalence relation is

functorial 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 m ...

, i.e. push-forward (with change of codimension) and pull-back of cycles is well-defined. Codimension 1 cycles modulo rational equivalence form the classical

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...

s modulo linear equivalence. All cycles modulo rational equivalence form the

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 (s ...

Definition

Let ''Z''^*(''X'') := Z 'X''be the free abelian group on the algebraic cycles of ''X''. Then an adequate equivalence relation is a family of equivalence relations, ''∼_X'' on ''Z''^*(''X''), one for each smooth projective variety ''X'', satisfying the following three conditions: # (Linearity) The equivalence relation is compatible with addition of cycles. # ( Moving lemma) If

\alpha, \beta \in Z^(X)

are cycles on ''X'', then there exists a cycle

\alpha' \in Z^(X)

such that

\alpha

''~_X''

\alpha'

and

\alpha'

intersects

\beta

properly. # (Push-forwards) Let

\alpha \in Z^(X)

and

\beta \in Z^(X \times Y)

be cycles such that

\beta

intersects

\alpha \times Y

properly. If

\alpha

''~_X'' 0, then

(\pi_Y)_(\beta \cdot (\alpha \times Y))

''~_Y'' 0, where

\pi_Y : X \times Y \to Y

is the projection. The push-forward cycle in the last axiom is often denoted :

\beta(\alpha) := (\pi_Y)_(\beta \cdot (\alpha \times Y))

\beta

is the

graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...

of a

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...

, then this reduces to the push-forward of the function. The generalizations of functions from ''X'' to ''Y'' to cycles on ''X × Y'' are known as

correspondences Correspondence may refer to: *In general usage, non-concurrent, remote communication between people, including letters, email, newsgroups, Internet forums, blogs. Science * Correspondence principle (physics): quantum physics theories must agree ...

. The last axiom allows us to push forward cycles by a correspondence.

Examples of equivalence relations

The most common equivalence relations, listed from strongest to weakest, are gathered in the following table.

Notes

References

* * {{DEFAULTSORT:Adequate Equivalence Relation Algebraic geometry Equivalence (mathematics)