Quotient by an equivalence relation

<h2 id="examples">Examples</h2>
<ul><li>Let <i>X</i> be a set and consider some equivalence relation on it. Let <i>Q</i> be the set of all <a href="/facts/Equivalence_classes/1d2sh0TB">equivalence classes</a> in <i>X</i>. Then the map 
  
    
      
        q
        :
        X
        →
        Q
      
    
    {\displaystyle q:X\to Q}
  
 that sends an element <i>x</i> to the equivalence class to which <i>x</i> belongs is a quotient.</li>
<li>In the above example, <i>Q</i> is a <a href="/facts/Subset/SJGERmbA">subset</a> of the <a href="/facts/Power_set/FVuf8PDD">power set</a> <i>H</i> of <i>X</i>. In <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a>, one might replace <i>H</i> by a <a href="/facts/Hilbert_scheme/kJcbts4e">Hilbert scheme</a> or disjoint union of Hilbert schemes. In fact, Grothendieck constructed a relative <a href="/facts/Picard_scheme/lWH0CoWQ">Picard scheme</a> of a flat projective scheme <i>X</i><a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> as a quotient <i>Q</i> (of the scheme <i>Z</i> parametrizing <a href="/facts/Relative_effective_divisor/V0QDypeo">relative effective divisors</a> on <i>X</i>) that is a closed scheme of a Hilbert scheme <i>H</i>. The quotient map 
  
    
      
        q
        :
        Z
        →
        Q
      
    
    {\displaystyle q:Z\to Q}
  
 can then be thought of as a relative version of the <a href="/facts/Abel_map/W1f1S827">Abel map</a>.</li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Categorical_quotient/0KTb8W6M">Categorical quotient</a>, a special case</li></ul>
<h2 id="notes">Notes</h2>

<ul><li>Nitsure, N. <i>Construction of Hilbert and Quot schemes.</i> Fundamental algebraic geometry: Grothendieck’s FGA explained, Mathematical Surveys and Monographs 123, American Mathematical Society 2005, 105–137.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>One also needs to assume the geometric fibers are integral schemes; Mumford's example shows the "integral" cannot be omitted. <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
</ol>

Quotient by an equivalence relation open-in-new

Quotient by an equivalence relation