In
algebraic geometry, a correspondence between
algebraic varieties
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
''V'' and ''W'' is a subset ''R'' of ''V''×''W'', that is closed in the
Zariski topology
In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is n ...
. In set theory, a subset of a Cartesian product of two sets is called a
binary relation or correspondence; thus, a correspondence here is a relation that is defined by algebraic equations. There are some important examples, even when ''V'' and ''W'' are
algebraic curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane ...
s: for example the
Hecke operator In mathematics, in particular in the theory of modular forms, a Hecke operator, studied by , is a certain kind of "averaging" operator that plays a significant role in the structure of vector spaces of modular forms and more general automorphic repr ...
s of
modular form theory may be considered as correspondences of
modular curve
In number theory and algebraic geometry, a modular curve ''Y''(Γ) is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex upper half-plane H by the action of a congruence subgroup Γ of the modular ...
s.
However, the definition of a correspondence in algebraic geometry is not completely standard. For instance, Fulton, in his book on
intersection theory
In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. The theory for varieties is older, with roots in Bézout's theorem o ...
, uses the definition above. In literature, however, a correspondence from a variety ''X'' to a variety ''Y'' is often taken to be a subset ''Z'' of ''X''×''Y'' such that ''Z'' is finite and surjective over each component of ''X''. Note the asymmetry in this latter definition; which talks about a correspondence from ''X'' to ''Y'' rather than a correspondence between ''X'' and ''Y''. The typical example of the latter kind of correspondence is the graph of a function ''f'':''X''→''Y''. Correspondences also play an important role in the construction of
motives (cf.
presheaf with transfers In algebraic geometry, a presheaf with transfers is, roughly, a presheaf that, like cohomology theory, comes with pushforwards, “transfer” maps. Precisely, it is, by definition, a contravariant additive functor from the category of finite corr ...
).
See also
*
Adequate equivalence relation
References
{{algebraic-geometry-stub
Algebraic geometry