Quotient by an equivalence relation

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 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 that is a closed scheme of a Hilbert scheme H. The quotient map can then be thought of as a relative version of the Abel map.