# Quotient by an equivalence relation

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__: Generalization of equivalence classes to scheme theory__

**Short description**In mathematics, given a category *C*, a **quotient** of an object *X* **by an equivalence relation** [math]\displaystyle{ f: R \to X \times X }[/math] is a coequalizer for the pair of maps

- [math]\displaystyle{ R \ \overset{f}{\to}\ X \times X \ \overset{\operatorname{pr}_i}{\to}\ X,\ \ i = 1,2, }[/math]

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 [math]\displaystyle{ f: R(T) = \operatorname{Mor}(T, R) \to X(T) \times X(T) }[/math] 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 [math]\displaystyle{ q: X \to Q }[/math] 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*^{[1]}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 [math]\displaystyle{ q: Z \to Q }[/math] can then be thought of as a relative version of the Abel map.

## See also

- Categorical quotient, a special case

## Notes

- ↑ One also needs to assume the geometric fibers are integral schemes; Mumford's example shows the "integral" cannot be omitted.

## 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.

Original source: https://en.wikipedia.org/wiki/Quotient by an equivalence relation.
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