Quotient by an equivalence relation

In mathematics, given a category ''C'', a quotient of an object ''X'' by an equivalence relation $f: R \to X \times X$ is a coequalizer for the pair of maps
:$R \ \overset\ X \times X \ \overset\ X,\ \ i = 1,2,$
where ''R'' is an object in ''C'' and "''f'' is an equivalence relation" means that, for any object ''T'' in ''C'', the image (which is a set) of $f: R(T) = \operatorname(T, R) \to X(T) \times X(T)$ is an equivalence relation; that is, a reflexive, symmetric and transitive relation.

The basic case in practice is when ''C'' is the category of all schemes over some scheme ''S''. But the notion is flexible and one can also take ''C'' to be the category of sheaves.

Examples

*Let ''X'' be a set and consider some equivalence relation on it. Let ''Q'' be the set of all equivalence classes in ''X''. Then the map $q: X \to Q$ that sends an element ''x'' to the equivalence class to which ''x'' belongs is a quotient.
*In the above example, ''Q'' is a subset of the power set ''H'' of ''X''. In algebraic geometry, one might replace ''H'' by a Hilbert scheme or disjoint union of Hilbert schemes. In fact, Grothendieck constructed a relative Picard scheme of a flat projective scheme ''X''One also needs to assume the geometric fibers are integral schemes; Mumford's example shows the "integral" cannot be omitted. as a quotient ''Q'' (of the scheme ''Z'' parametrizing relative effective divisors on ''X'') that is a closed scheme of a Hilbert scheme ''H''. The quotient map $q: Z \to Q$ can then be thought of as a relative version of the Abel map.

See also

*Categorical quotient, a special case

Notes

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References

*Nitsure, N. ''Construction of Hilbert and Quot schemes.'' Fundamental algebraic geometry: Grothendieck’s FGA explained, Mathematical Surveys and Monographs 123, American Mathematical Society 2005, 105–137.

Binary relations
Scheme theory