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In algebraic geometry, a correspondence between algebraic varieties V and W is a subset R of V×W, that is closed in the Zariski topology. 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.
A proof of the correspondence theorem can be found here. Similar results hold for rings, modules, vector spaces, and algebras. More generally an analogous result that concerns congruence relations instead of normal subgroups holds for any algebraic structure.
The correspondence has the following useful properties. It is inclusion-reversing. The inclusion of subgroups H 1 ⊆ H 2 holds if and only if the inclusion of fields E H 1 ⊇ E H 2 holds. Degrees of extensions are related to orders of groups, in a manner consistent with the inclusion-reversing property.
Suppose G is a closed subgroup of GL(n;C), and thus a Lie group, by the closed subgroups theorem.Then the Lie algebra of G may be computed as [2] [3] = {(;)}. For example, one can use the criterion to establish the correspondence for classical compact groups (cf. the table in "compact Lie groups" below.)
Correspondence (algebraic geometry), between two algebraic varieties; Corresponding sides and corresponding angles, between two polygons; Correspondence (category theory), the opposite of a profunctor; Correspondence (von Neumann algebra) or bimodule, a type of Hilbert space; Correspondence analysis, a multivariate statistical technique
A bijection, bijective function, or one-to-one correspondence between two mathematical sets is a function such that each element of the second set (the codomain) ...
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Generalizing this result, Hilbert's Nullstellensatz provides a fundamental correspondence between ideals of polynomial rings and algebraic sets. Using the Nullstellensatz and related results, mathematicians have established a strong correspondence between questions on algebraic sets and questions of ring theory. This correspondence is a ...