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Function field (scheme theory)
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<p>The sheaf of rational functions <i>KX</i> of a <a href="/facts/Scheme_(mathematics)/c7uSqGqL">scheme</a> <i>X</i> is the generalization to <a href="/facts/Scheme_theory/c7uSqGqL">scheme theory</a> of the notion of <a href="/facts/Function_field_of_an_algebraic_variety/54cc62wP">function field of an algebraic variety</a> in classical <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a>. In the case of <a href="/facts/Algebraic_varieties/Vk7f97vn">algebraic varieties</a>, such a sheaf associates to each open set <i>U</i> the <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a> of all <a href="/facts/Rational_function/2nJGu0Ko">rational functions</a> on that open set; in other words, <i>KX</i>(<i>U</i>) is the set of fractions of <a href="/facts/Regular_function/5jBzIgps">regular functions</a> on <i>U</i>. Despite its name, <i>KX</i> does not always give a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> for a general scheme <i>X</i>. </p>