Quotient by an equivalence relation

In mathematics, given a category C, a quotient of an object X by an equivalence relation ${\displaystyle f:R\to X\times X}$ is a coequalizer for the pair of maps

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

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 ${\displaystyle f:R(T)=\operatorname {Mor} (T,R)\to X(T)\times X(T)}$ 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 ${\displaystyle q:X\to Q}$ 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[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 ${\displaystyle q:Z\to Q}$ 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.