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

$R \overset{f}\to X \times X \overset{\operatorname{pr}_i}\to X, \, i = 1,2,$

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 $f: R(T) = \operatorname{Mor}(T, R) \to X(T) \times X(T)$ is an equivalence relation; that is, $(x, y)$ is in it if and only if $(y, x)$ is in it, etc.

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 an equivalence class to which x belong 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 $q: Z \to Q$ can then be thought of as a relative version of the Abel map.

See also

• categorical quotient, a special case

Notes

1. ↑ 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.