<p>We give a <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K">
<mml:semantics>
<mml:mi>K</mml:mi>
<mml:annotation encoding="application/x-tex">K</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>-theoretic criterion for a quasi-projective variety to be smooth. If <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper L">
<mml:semantics>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi mathvariant="double-struck">L</mml:mi>
</mml:mrow>
<mml:annotation encoding="application/x-tex">\mathbb {L}</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> is a line bundle corresponding to an ample invertible sheaf on <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X">
<mml:semantics>
<mml:mi>X</mml:mi>
<mml:annotation encoding="application/x-tex">X</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>, it suffices that <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K Subscript q Baseline left-parenthesis upper X right-parenthesis approximately-equals upper K Subscript q Baseline left-parenthesis double-struck upper L right-parenthesis">
<mml:semantics>
<mml:mrow>
<mml:msub>
<mml:mi>K</mml:mi>
<mml:mi>q</mml:mi>
</mml:msub>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>X</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo>≅<!-- ≅ --></mml:mo>
<mml:msub>
<mml:mi>K</mml:mi>
<mml:mi>q</mml:mi>
</mml:msub>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi mathvariant="double-struck">L</mml:mi>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:annotation encoding="application/x-tex">K_q(X)\cong K_q(\mathbb {L})</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> for all <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q less-than-or-equal-to dimension left-parenthesis upper X right-parenthesis plus 1">
<mml:semantics>
<mml:mrow>
<mml:mi>q</mml:mi>
<mml:mo>≤<!-- ≤ --></mml:mo>
<mml:mi>dim</mml:mi>
<mml:mo><!-- --></mml:mo>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>X</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo>+</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:annotation encoding="application/x-tex">q\le \dim (X)+1</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>.</p>
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