In mathematics, given a category C, a quotient of an object X by an equivalence relation f : R → X × X {\displaystyle f:R\to X\times X} is a coequalizer for the pair of maps
R → f X × X → pr i X , i = 1 , 2 , {\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 f : R ( T ) = Mor ( T , R ) → X ( T ) × X ( T ) {\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 q : X → Q {\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 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 → Q {\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
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.
