Quotient by an equivalence relation - Alchetron, the free social encyclopedia

In mathematics, given a category C, a quotient of an object X by an equivalence relation f : R → X × X is a coequalizer for the pair of maps

R → f X × X → pr i 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 ) 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 → 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 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 can then be thought of as a relative version of the Abel map.

References

Quotient by an equivalence relation Wikipedia

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