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<title>Smooth morphism</title>
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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Smooth morphism</span></span>
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<p>In <a href="Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, a <a href="Morphism_of_schemes" title="Morphism of schemes">morphism</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to S}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
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<annotation encoding="application/x-tex">{\displaystyle f:X\to S}</annotation>
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</math></span><img src="./a21a941b35354b73f5a62849067deac7ee9794dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.309ex; height:2.509ex;" alt="{\displaystyle f:X\to S}" loading="lazy"></span> between <a href="Scheme_(mathematics)" title="Scheme (mathematics)">schemes</a> is said to be <b>smooth</b> if
</p>
<ul><li>(i) it is <a href="Finitely_presented_scheme" class="mw-redirect" title="Finitely presented scheme">locally of finite presentation</a></li>
<li>(ii) it is <a href="Flat_morphism" title="Flat morphism">flat</a>, and</li>
<li>(iii) for every <a href="Geometric_point" class="mw-redirect" title="Geometric point">geometric point</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {s}}\to S}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>s</mi>
<mo accent="false">¯<!-- ¯ --></mo>
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<mo stretchy="false">→<!-- → --></mo>
<mi>S</mi>
</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle {\overline {s}}\to S}</annotation>
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</math></span><img src="./9810cdf5fd92f3d61be46ac1480eb022a960ba1f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.319ex; height:2.343ex;" alt="{\displaystyle {\overline {s}}\to S}" loading="lazy"></span> the <a href="Fiber_(mathematics)#In_algebraic_geometry" title="Fiber (mathematics)">fiber</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{\overline {s}}=X\times _{S}{\overline {s}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>X</mi>
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<annotation encoding="application/x-tex">{\displaystyle X_{\overline {s}}=X\times _{S}{\overline {s}}}</annotation>
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</math></span><img src="./2bed3af302f61e0a992b8ea262e7d1dbf04d922b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.438ex; height:2.843ex;" alt="{\displaystyle X_{\overline {s}}=X\times _{S}{\overline {s}}}" loading="lazy"></span> is regular.</li></ul>
<p>(iii) means that each geometric fiber of <i>f</i> is a <a href="Smooth_scheme" title="Smooth scheme">nonsingular variety</a> (if it is <a href="Separated_morphism" class="mw-redirect" title="Separated morphism">separated</a>). Thus, intuitively speaking, a smooth morphism gives a flat family of nonsingular varieties.
</p><p>If <i>S</i> is the <a href="Spectrum_of_a_ring" title="Spectrum of a ring">spectrum</a> of an <a href="Algebraically_closed" class="mw-redirect" title="Algebraically closed">algebraically closed</a> <a href="Field_(mathematics)" title="Field (mathematics)">field</a> and <i>f</i> is <a href="Morphism_of_finite_type" title="Morphism of finite type">of finite type</a>, then one recovers the definition of a nonsingular variety.
</p><p>A singular variety is called smoothable if it can be put in a flat family so that the nearby fibers are all smooth. Such a family is called a smoothning of the variety.
</p>
<meta property="mw:PageProp/toc">
<div class="mw-heading mw-heading2"><h2 id="Equivalent_definitions">Equivalent definitions</h2></div>
<p>There are many equivalent definitions of a smooth morphism. Let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to S}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>:</mo>
<mi>X</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>S</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f:X\to S}</annotation>
</semantics>
</math></span><img src="./a21a941b35354b73f5a62849067deac7ee9794dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.309ex; height:2.509ex;" alt="{\displaystyle f:X\to S}" loading="lazy"></span> be locally of finite presentation. Then the following are equivalent.
</p>
<ol><li><i>f</i> is smooth.</li>
<li><i>f</i> is formally smooth (see below).</li>
<li><i>f</i> is flat and the <a href="Cotangent_sheaf" title="Cotangent sheaf">sheaf of relative differentials</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{X/S}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
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<annotation encoding="application/x-tex">{\displaystyle \Omega _{X/S}}</annotation>
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</math></span><img src="./b403d4143c562a13c19a6f57a9870d8f9f12b8f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.193ex; height:3.009ex;" alt="{\displaystyle \Omega _{X/S}}" loading="lazy"></span> is locally free of rank equal to the <a href="Relative_dimension_(scheme_theory)" class="mw-redirect" title="Relative dimension (scheme theory)">relative dimension</a> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X/S}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
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<mi>S</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X/S}</annotation>
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</math></span><img src="./2fdd90e10ae0a9bed42cd09b98f9460e5c41d8b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.642ex; height:2.843ex;" alt="{\displaystyle X/S}" loading="lazy"></span>.</li>
<li>For any <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
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<annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation>
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</math></span><img src="./3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"></span>, there exists a neighborhood <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Spec} B}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Spec</mi>
<mo>⁡<!-- ⁡ --></mo>
<mi>B</mi>
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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Spec} B}</annotation>
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</math></span><img src="./624cf3b2f5b4c7735d78ebd18c53cacb9a991dc3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.801ex; height:2.509ex;" alt="{\displaystyle \operatorname {Spec} B}" loading="lazy"></span> of x and a neighborhood <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Spec} A}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Spec</mi>
<mo>⁡<!-- ⁡ --></mo>
<mi>A</mi>
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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Spec} A}</annotation>
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</math></span><img src="./7ad53059b6a9d6c309ab164aa4eaed021844b5ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.78ex; height:2.509ex;" alt="{\displaystyle \operatorname {Spec} A}" loading="lazy"></span> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>
</semantics>
</math></span><img src="./202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy"></span> such that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B=A[t_{1},\dots ,t_{n}]/(P_{1},\dots ,P_{m})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>B</mi>
<mo>=</mo>
<mi>A</mi>
<mo stretchy="false">[</mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">]</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mo stretchy="false">(</mo>
<msub>
<mi>P</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>P</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle B=A[t_{1},\dots ,t_{n}]/(P_{1},\dots ,P_{m})}</annotation>
</semantics>
</math></span><img src="./0bba842ca6591c40252716975ba6e6eaab353160.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.893ex; height:2.843ex;" alt="{\displaystyle B=A[t_{1},\dots ,t_{n}]/(P_{1},\dots ,P_{m})}" loading="lazy"></span> and the ideal generated by the <i>m</i>-by-<i>m</i> minors of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\partial P_{i}/\partial t_{j})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>P</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
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<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
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</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (\partial P_{i}/\partial t_{j})}</annotation>
</semantics>
</math></span><img src="./cda2995981cdf9581fb72d3e5fc9e306949c8a6b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.649ex; height:3.009ex;" alt="{\displaystyle (\partial P_{i}/\partial t_{j})}" loading="lazy"></span> is <i>B</i>.</li>
<li>Locally, <i>f</i> factors into <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X{\overset {g}{\to }}\mathbb {A} _{S}^{n}\to S}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mo stretchy="false">→<!-- → --></mo>
<mi>g</mi>
</mover>
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<msubsup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">A</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>S</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msubsup>
<mo stretchy="false">→<!-- → --></mo>
<mi>S</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X{\overset {g}{\to }}\mathbb {A} _{S}^{n}\to S}</annotation>
</semantics>
</math></span><img src="./05acd5a17bcffe5a6de96b8db12bf54ae65f5a39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.388ex; height:3.843ex;" alt="{\displaystyle X{\overset {g}{\to }}\mathbb {A} _{S}^{n}\to S}" loading="lazy"></span> where <i>g</i> is <a href="%C3%89tale_morphism" title="Étale morphism">étale</a>.</li></ol>
<p>A morphism of finite type is étale <a href="If_and_only_if" title="If and only if">if and only if</a> it is smooth and <a href="Quasi-finite_morphism" title="Quasi-finite morphism">quasi-finite</a>.
</p><p>A smooth morphism is stable under base change and composition.
</p><p>A smooth morphism is universally <a href="Locally_acyclic_morphism" title="Locally acyclic morphism">locally acyclic</a>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>
<p>Smooth morphisms are supposed to geometrically correspond to smooth <a href="Submersion_(mathematics)" title="Submersion (mathematics)">submersions</a> in <a href="Differential_geometry" title="Differential geometry">differential geometry</a>; that is, they are smooth locally trivial fibrations over some base space (by <a href="Ehresmann's_theorem" class="mw-redirect" title="Ehresmann's theorem">Ehresmann's theorem</a>).
</p>
<div class="mw-heading mw-heading3"><h3 id="Smooth_morphism_to_a_point">Smooth morphism to a point</h3></div>
<p>Let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> be the morphism of schemes
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Spec} _{\mathbb {C} }\left({\frac {\mathbb {C} [x,y]}{(f=y^{2}-x^{3}-x-1)}}\right)\to \operatorname {Spec} (\mathbb {C} )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>Spec</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">C</mi>
</mrow>
</mrow>
</msub>
<mo>⁡<!-- ⁡ --></mo>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">C</mi>
</mrow>
<mo stretchy="false">[</mo>
<mi>x</mi>
<mo>,</mo>
<mi>y</mi>
<mo stretchy="false">]</mo>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mi>f</mi>
<mo>=</mo>
<msup>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo stretchy="false">→<!-- → --></mo>
<mi>Spec</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">C</mi>
</mrow>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {Spec} _{\mathbb {C} }\left({\frac {\mathbb {C} [x,y]}{(f=y^{2}-x^{3}-x-1)}}\right)\to \operatorname {Spec} (\mathbb {C} )}</annotation>
</semantics>
</math></span><img src="./5e56e17a524cf3f057a18dd606fcb0360b26bd7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:43.876ex; height:6.509ex;" alt="{\displaystyle \operatorname {Spec} _{\mathbb {C} }\left({\frac {\mathbb {C} [x,y]}{(f=y^{2}-x^{3}-x-1)}}\right)\to \operatorname {Spec} (\mathbb {C} )}" loading="lazy"></span></dd></dl>
<p>It is smooth because of the Jacobian condition: the <a href="Jacobian_matrix" class="mw-redirect" title="Jacobian matrix">Jacobian matrix</a>
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [3x^{2}-1,y]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">[</mo>
<mn>3</mn>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo>,</mo>
<mi>y</mi>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle [3x^{2}-1,y]}</annotation>
</semantics>
</math></span><img src="./a3d90a6889cb2edd6eb90095f6808e1bd4c4fbc0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.032ex; height:3.176ex;" alt="{\displaystyle [3x^{2}-1,y]}" loading="lazy"></span></dd></dl>
<p>vanishes at the points <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1/{\sqrt {3}},0),(-1/{\sqrt {3}},0)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mn>1</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
<mo>,</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mo stretchy="false">(</mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
<mo>,</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (1/{\sqrt {3}},0),(-1/{\sqrt {3}},0)}</annotation>
</semantics>
</math></span><img src="./64d88b542d475c377efebb732e2c9b8077f69017.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.7ex; height:3.009ex;" alt="{\displaystyle (1/{\sqrt {3}},0),(-1/{\sqrt {3}},0)}" loading="lazy"></span> which has an empty intersection with the polynomial, since
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}f(1/{\sqrt {3}},0)&amp;=1-{\frac {1}{\sqrt {3}}}-{\frac {1}{3{\sqrt {3}}}}\\f(-1/{\sqrt {3}},0)&amp;={\frac {1}{\sqrt {3}}}+{\frac {1}{3{\sqrt {3}}}}-1\end{aligned}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
<mtr>
<mtd>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
<mo>,</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<msqrt>
<mn>3</mn>
</msqrt>
</mfrac>
</mrow>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mn>3</mn>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
<mo>,</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<msqrt>
<mn>3</mn>
</msqrt>
</mfrac>
</mrow>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mn>3</mn>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}f(1/{\sqrt {3}},0)&amp;=1-{\frac {1}{\sqrt {3}}}-{\frac {1}{3{\sqrt {3}}}}\\f(-1/{\sqrt {3}},0)&amp;={\frac {1}{\sqrt {3}}}+{\frac {1}{3{\sqrt {3}}}}-1\end{aligned}}}</annotation>
</semantics>
</math></span><img src="./cca6b7e1b85c017c63b4c0544d4bbba69f38e144.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:32.24ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}f(1/{\sqrt {3}},0)&amp;=1-{\frac {1}{\sqrt {3}}}-{\frac {1}{3{\sqrt {3}}}}\\f(-1/{\sqrt {3}},0)&amp;={\frac {1}{\sqrt {3}}}+{\frac {1}{3{\sqrt {3}}}}-1\end{aligned}}}" loading="lazy"></span></dd></dl>
<p>which are both non-zero.
</p>
<div class="mw-heading mw-heading3"><h3 id="Trivial_fibrations">Trivial fibrations</h3></div>
<p>Given a <a href="Smooth_scheme" title="Smooth scheme">smooth scheme</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>
</semantics>
</math></span><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span> the projection morphism
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y\times X\to X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Y</mi>
<mo>×<!-- × --></mo>
<mi>X</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y\times X\to X}</annotation>
</semantics>
</math></span><img src="./62cb7c50ae55dc3282690eaf2904205ad15bcf15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.188ex; height:2.176ex;" alt="{\displaystyle Y\times X\to X}" loading="lazy"></span></dd></dl>
<p>is smooth.
</p>
<div class="mw-heading mw-heading3"><h3 id="Vector_bundles">Vector bundles</h3></div>
<p>Every <a href="Vector_bundle" title="Vector bundle">vector bundle</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\to X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>E</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle E\to X}</annotation>
</semantics>
</math></span><img src="./c6e385963fc9a729e22866cd3c05d5002c361bbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.37ex; height:2.176ex;" alt="{\displaystyle E\to X}" loading="lazy"></span> over a scheme is a smooth morphism. For example, it can be shown that the associated vector bundle of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}(k)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">O</mi>
</mrow>
</mrow>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}(k)}</annotation>
</semantics>
</math></span><img src="./b0dc155fff30585091c53892f3c57f96c608fe7d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.87ex; height:2.843ex;" alt="{\displaystyle {\mathcal {O}}(k)}" loading="lazy"></span> over <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} ^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">P</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} ^{n}}</annotation>
</semantics>
</math></span><img src="./7e712abd7ff167cfb12ffa7c8c9bf7b0029cf192.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.639ex; height:2.343ex;" alt="{\displaystyle \mathbb {P} ^{n}}" loading="lazy"></span> is the <a href="Weighted_projective_space" title="Weighted projective space">weighted projective space</a> minus a point
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(k)=\mathbb {P} (1,\ldots ,1,k)-\{[0:\cdots :0:1]\}\to \mathbb {P} ^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>O</mi>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">P</mi>
</mrow>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<mn>1</mn>
<mo>,</mo>
<mi>k</mi>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<mo fence="false" stretchy="false">{</mo>
<mo stretchy="false">[</mo>
<mn>0</mn>
<mo>:</mo>
<mo>⋯<!-- ⋯ --></mo>
<mo>:</mo>
<mn>0</mn>
<mo>:</mo>
<mn>1</mn>
<mo stretchy="false">]</mo>
<mo fence="false" stretchy="false">}</mo>
<mo stretchy="false">→<!-- → --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">P</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle O(k)=\mathbb {P} (1,\ldots ,1,k)-\{[0:\cdots :0:1]\}\to \mathbb {P} ^{n}}</annotation>
</semantics>
</math></span><img src="./d7c83ffbdefab8e1c27598ebffbc872d53aae967.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.604ex; height:2.843ex;" alt="{\displaystyle O(k)=\mathbb {P} (1,\ldots ,1,k)-\{[0:\cdots :0:1]\}\to \mathbb {P} ^{n}}" loading="lazy"></span></dd></dl>
<p>sending
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x_{0}:\cdots :x_{n}:x_{n+1}]\to [x_{0}:\cdots :x_{n}]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">[</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>:</mo>
<mo>⋯<!-- ⋯ --></mo>
<mo>:</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo>:</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
<mo stretchy="false">]</mo>
<mo stretchy="false">→<!-- → --></mo>
<mo stretchy="false">[</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>:</mo>
<mo>⋯<!-- ⋯ --></mo>
<mo>:</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle [x_{0}:\cdots :x_{n}:x_{n+1}]\to [x_{0}:\cdots :x_{n}]}</annotation>
</semantics>
</math></span><img src="./e7797f1282119989bfc73904a99ab31cee66f5fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.847ex; height:2.843ex;" alt="{\displaystyle [x_{0}:\cdots :x_{n}:x_{n+1}]\to [x_{0}:\cdots :x_{n}]}" loading="lazy"></span></dd></dl>
<p>Notice that the direct sum bundles <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(k)\oplus O(l)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>O</mi>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo stretchy="false">)</mo>
<mo>⊕<!-- ⊕ --></mo>
<mi>O</mi>
<mo stretchy="false">(</mo>
<mi>l</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle O(k)\oplus O(l)}</annotation>
</semantics>
</math></span><img src="./cac4859b7391f93d53130825962c7a16f0af9dd7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.91ex; height:2.843ex;" alt="{\displaystyle O(k)\oplus O(l)}" loading="lazy"></span> can be constructed using the fiber product
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(k)\oplus O(l)=O(k)\times _{X}O(l)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>O</mi>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo stretchy="false">)</mo>
<mo>⊕<!-- ⊕ --></mo>
<mi>O</mi>
<mo stretchy="false">(</mo>
<mi>l</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>O</mi>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo stretchy="false">)</mo>
<msub>
<mo>×<!-- × --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>X</mi>
</mrow>
</msub>
<mi>O</mi>
<mo stretchy="false">(</mo>
<mi>l</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle O(k)\oplus O(l)=O(k)\times _{X}O(l)}</annotation>
</semantics>
</math></span><img src="./3cdaf8c48de9ef1f5b711075fcec8e7efe570d7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.551ex; height:2.843ex;" alt="{\displaystyle O(k)\oplus O(l)=O(k)\times _{X}O(l)}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading3"><h3 id="Separable_field_extensions">Separable field extensions</h3></div>
<p>Recall that a <a href="Field_extension" title="Field extension">field extension</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\to L}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>K</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>L</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle K\to L}</annotation>
</semantics>
</math></span><img src="./6ebdd971c4df7ed7e82835f5ed921cf5475d596d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.263ex; height:2.176ex;" alt="{\displaystyle K\to L}" loading="lazy"></span> is called <a href="Separable_field_extension" class="mw-redirect" title="Separable field extension">separable</a> iff given a presentation
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L={\frac {K[x]}{(f(x))}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>L</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>K</mi>
<mo stretchy="false">[</mo>
<mi>x</mi>
<mo stretchy="false">]</mo>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle L={\frac {K[x]}{(f(x))}}}</annotation>
</semantics>
</math></span><img src="./73819f6dc53c5ec57c924152260251d854844ea0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:11.744ex; height:6.509ex;" alt="{\displaystyle L={\frac {K[x]}{(f(x))}}}" loading="lazy"></span></dd></dl>
<p>we have that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle gcd(f(x),f'(x))=1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
<mi>c</mi>
<mi>d</mi>
<mo stretchy="false">(</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle gcd(f(x),f'(x))=1}</annotation>
</semantics>
</math></span><img src="./f956a68df6580ed98e7b19c7d5850b2036518c73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.005ex; height:3.009ex;" alt="{\displaystyle gcd(f(x),f'(x))=1}" loading="lazy"></span>. We can reinterpret this definition in terms of <a href="K%C3%A4hler_differential" title="Kähler differential">Kähler differentials</a> as follows: the field extension is separable iff
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{L/K}=0.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mi>K</mi>
</mrow>
</msub>
<mo>=</mo>
<mn>0.</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Omega _{L/K}=0.}</annotation>
</semantics>
</math></span><img src="./726d0abd5745e59aac81505ade6586d1a65d83a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.22ex; height:3.009ex;" alt="{\displaystyle \Omega _{L/K}=0.}" loading="lazy"></span></dd></dl>
<p>Notice that this includes every <a href="Perfect_field" title="Perfect field">perfect field</a>: <a href="Finite_field" title="Finite field">finite fields</a> and fields of <a href="Characteristic_(algebra)" title="Characteristic (algebra)">characteristic</a> 0.
</p>
<div class="mw-heading mw-heading2"><h2 id="Non-examples">Non-examples</h2></div>
<div class="mw-heading mw-heading3"><h3 id="Singular_varieties">Singular varieties</h3></div>
<p>If we consider <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Spec} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Spec</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {Spec} }</annotation>
</semantics>
</math></span><img src="./347e55d772e74c0d6934658fab909f65f6f5c960.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.65ex; height:2.509ex;" alt="{\displaystyle \operatorname {Spec} }" loading="lazy"></span> of the underlying algebra <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R}</annotation>
</semantics>
</math></span><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"></span> for a projective variety <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span>, called the affine cone of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span>, then the point at the origin is always singular. For example, consider the <b>affine cone</b> of a quintic <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>3</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 3}</annotation>
</semantics>
</math></span><img src="./991e33c6e207b12546f15bdfee8b5726eafbbb2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 3}" loading="lazy"></span>-fold given by
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}^{5}+x_{1}^{5}+x_{2}^{5}+x_{3}^{5}+x_{4}^{5}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>5</mn>
</mrow>
</msubsup>
<mo>+</mo>
<msubsup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>5</mn>
</mrow>
</msubsup>
<mo>+</mo>
<msubsup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>5</mn>
</mrow>
</msubsup>
<mo>+</mo>
<msubsup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>5</mn>
</mrow>
</msubsup>
<mo>+</mo>
<msubsup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>5</mn>
</mrow>
</msubsup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{0}^{5}+x_{1}^{5}+x_{2}^{5}+x_{3}^{5}+x_{4}^{5}}</annotation>
</semantics>
</math></span><img src="./57919e491fc90049cc49a291a5c91ccdd9406d4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.281ex; height:3.176ex;" alt="{\displaystyle x_{0}^{5}+x_{1}^{5}+x_{2}^{5}+x_{3}^{5}+x_{4}^{5}}" loading="lazy"></span></dd></dl>
<p>Then the Jacobian matrix is given by
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}5x_{0}^{4}&amp;5x_{1}^{4}&amp;5x_{2}^{4}&amp;5x_{3}^{4}&amp;5x_{4}^{4}\end{bmatrix}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>[</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>5</mn>
<msubsup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msubsup>
</mtd>
<mtd>
<mn>5</mn>
<msubsup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msubsup>
</mtd>
<mtd>
<mn>5</mn>
<msubsup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msubsup>
</mtd>
<mtd>
<mn>5</mn>
<msubsup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msubsup>
</mtd>
<mtd>
<mn>5</mn>
<msubsup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msubsup>
</mtd>
</mtr>
</mtable>
<mo>]</mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}5x_{0}^{4}&amp;5x_{1}^{4}&amp;5x_{2}^{4}&amp;5x_{3}^{4}&amp;5x_{4}^{4}\end{bmatrix}}}</annotation>
</semantics>
</math></span><img src="./7eda3a66662aa0bbaf0ebd1b5c923335b65338af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.713ex; height:3.176ex;" alt="{\displaystyle {\begin{bmatrix}5x_{0}^{4}&amp;5x_{1}^{4}&amp;5x_{2}^{4}&amp;5x_{3}^{4}&amp;5x_{4}^{4}\end{bmatrix}}}" loading="lazy"></span></dd></dl>
<p>which vanishes at the origin, hence the cone is singular. Affine hypersurfaces like these are popular in <a href="Singularity_theory" title="Singularity theory">singularity theory</a> because of their relatively simple algebra but rich underlying structures.
</p><p>Another example of a singular variety is the <b>projective cone</b> of a smooth variety: given a smooth projective variety <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\subset \mathbb {P} ^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo>⊂<!-- ⊂ --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">P</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X\subset \mathbb {P} ^{n}}</annotation>
</semantics>
</math></span><img src="./59ec74d93cc805a1c25dcee28747a3c4aa8fec7f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.717ex; height:2.343ex;" alt="{\displaystyle X\subset \mathbb {P} ^{n}}" loading="lazy"></span> its projective cone is the union of all lines in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} ^{n+1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">P</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} ^{n+1}}</annotation>
</semantics>
</math></span><img src="./8f3bf862e47a534f67ffe6435bfc6e29ee9aaeed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.739ex; height:2.676ex;" alt="{\displaystyle \mathbb {P} ^{n+1}}" loading="lazy"></span> intersecting <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span>. For example, the projective cone of the points
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Proj}}\left({\frac {\mathbb {C} [x,y]}{(x^{4}+y^{4})}}\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtext>Proj</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">C</mi>
</mrow>
<mo stretchy="false">[</mo>
<mi>x</mi>
<mo>,</mo>
<mi>y</mi>
<mo stretchy="false">]</mo>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msup>
<mo>+</mo>
<msup>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\text{Proj}}\left({\frac {\mathbb {C} [x,y]}{(x^{4}+y^{4})}}\right)}</annotation>
</semantics>
</math></span><img src="./b8c0db08521734a7a2fb470becb302d92fd17313.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:18.262ex; height:6.509ex;" alt="{\displaystyle {\text{Proj}}\left({\frac {\mathbb {C} [x,y]}{(x^{4}+y^{4})}}\right)}" loading="lazy"></span></dd></dl>
<p>is the scheme
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Proj}}\left({\frac {\mathbb {C} [x,y,z]}{(x^{4}+y^{4})}}\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtext>Proj</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">C</mi>
</mrow>
<mo stretchy="false">[</mo>
<mi>x</mi>
<mo>,</mo>
<mi>y</mi>
<mo>,</mo>
<mi>z</mi>
<mo stretchy="false">]</mo>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msup>
<mo>+</mo>
<msup>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\text{Proj}}\left({\frac {\mathbb {C} [x,y,z]}{(x^{4}+y^{4})}}\right)}</annotation>
</semantics>
</math></span><img src="./12ec315c09200e05932cb65c576508d5758fe1c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:18.262ex; height:6.509ex;" alt="{\displaystyle {\text{Proj}}\left({\frac {\mathbb {C} [x,y,z]}{(x^{4}+y^{4})}}\right)}" loading="lazy"></span></dd></dl>
<p>If we look in the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\neq 0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>z</mi>
<mo>≠<!-- ≠ --></mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle z\neq 0}</annotation>
</semantics>
</math></span><img src="./3b8b7eb2d2a30057811a7835502717d3d6ece962.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.349ex; height:2.676ex;" alt="{\displaystyle z\neq 0}" loading="lazy"></span> chart this is the scheme
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Spec} \left({\frac {\mathbb {C} [X,Y]}{(X^{4}+Y^{4})}}\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Spec</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">C</mi>
</mrow>
<mo stretchy="false">[</mo>
<mi>X</mi>
<mo>,</mo>
<mi>Y</mi>
<mo stretchy="false">]</mo>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msup>
<mo>+</mo>
<msup>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {Spec} \left({\frac {\mathbb {C} [X,Y]}{(X^{4}+Y^{4})}}\right)}</annotation>
</semantics>
</math></span><img src="./05e763a7dd2133516cff9101a307cb0de42a5e3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.562ex; height:6.509ex;" alt="{\displaystyle \operatorname {Spec} \left({\frac {\mathbb {C} [X,Y]}{(X^{4}+Y^{4})}}\right)}" loading="lazy"></span></dd></dl>
<p>and project it down to the affine line <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{Y}^{1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">A</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>Y</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msubsup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{Y}^{1}}</annotation>
</semantics>
</math></span><img src="./5e1211fbaad61936ce18ec735ebb446eee960117.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.164ex; height:3.176ex;" alt="{\displaystyle \mathbb {A} _{Y}^{1}}" loading="lazy"></span>, this is a family of four points degenerating at the origin. The non-singularity of this scheme can also be checked using the Jacobian condition.
</p>
<div class="mw-heading mw-heading3"><h3 id="Degenerating_families">Degenerating families</h3></div>
<p>Consider the flat family
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Spec} \left({\frac {\mathbb {C} [t,x,y]}{(xy-t)}}\right)\to \mathbb {A} _{t}^{1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Spec</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">C</mi>
</mrow>
<mo stretchy="false">[</mo>
<mi>t</mi>
<mo>,</mo>
<mi>x</mi>
<mo>,</mo>
<mi>y</mi>
<mo stretchy="false">]</mo>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mi>y</mi>
<mo>−<!-- − --></mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo stretchy="false">→<!-- → --></mo>
<msubsup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">A</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msubsup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {Spec} \left({\frac {\mathbb {C} [t,x,y]}{(xy-t)}}\right)\to \mathbb {A} _{t}^{1}}</annotation>
</semantics>
</math></span><img src="./99dca0c92c6b4a1c7cebc2db58fe62ecb959b6fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:23.618ex; height:6.509ex;" alt="{\displaystyle \operatorname {Spec} \left({\frac {\mathbb {C} [t,x,y]}{(xy-t)}}\right)\to \mathbb {A} _{t}^{1}}" loading="lazy"></span></dd></dl>
<p>Then the fibers are all smooth except for the point at the origin. Since smoothness is stable under base-change, this family is not smooth.
</p>
<div class="mw-heading mw-heading3"><h3 id="Non-separable_field_extensions">Non-separable field extensions</h3></div>
<p>For example, the field <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{p}(t^{p})\to \mathbb {F} _{p}(t)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">F</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>p</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<msup>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>p</mi>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo stretchy="false">→<!-- → --></mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">F</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>p</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{p}(t^{p})\to \mathbb {F} _{p}(t)}</annotation>
</semantics>
</math></span><img src="./5ab8158d4a96167b06f2b4e6a4209c5704020688.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.93ex; height:3.009ex;" alt="{\displaystyle \mathbb {F} _{p}(t^{p})\to \mathbb {F} _{p}(t)}" loading="lazy"></span> is non-separable, hence the associated morphism of schemes is not smooth. If we look at the minimal polynomial of the field extension,
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=x^{p}-t^{p}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>p</mi>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<msup>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>p</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(x)=x^{p}-t^{p}}</annotation>
</semantics>
</math></span><img src="./92502aaada8d48d5efce62a452c7cef5138627e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.644ex; height:2.843ex;" alt="{\displaystyle f(x)=x^{p}-t^{p}}" loading="lazy"></span></dd></dl>
<p>then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle df=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>d</mi>
<mi>f</mi>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle df=0}</annotation>
</semantics>
</math></span><img src="./4b7a58ad8f5eeb3da52247dc7e38b3f9f489dd2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.755ex; height:2.509ex;" alt="{\displaystyle df=0}" loading="lazy"></span>, hence the Kähler differentials will be non-zero.
</p>
<div class="mw-heading mw-heading2"><h2 id="Formally_smooth_morphism">Formally smooth morphism</h2></div>
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</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="Formally_smooth_map" title="Formally smooth map">Formally smooth map</a> and <a href="Geometrically_regular_ring" title="Geometrically regular ring">Geometrically regular ring</a></div>
<p>One can define smoothness without reference to geometry. We say that an <i>S</i>-scheme <i>X</i> is <b>formally smooth</b> if for any affine <i>S</i>-scheme <i>T</i> and a subscheme <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>T</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle T_{0}}</annotation>
</semantics>
</math></span><img src="./55b9e7d7b96196b5a6a26f4349caa3ac82fd67e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{0}}" loading="lazy"></span> of <i>T</i> given by a <a href="Nilpotent_ideal" title="Nilpotent ideal">nilpotent ideal</a>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X(T)\to X(T_{0})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo stretchy="false">(</mo>
<mi>T</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">→<!-- → --></mo>
<mi>X</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>T</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X(T)\to X(T_{0})}</annotation>
</semantics>
</math></span><img src="./55aea08c99a626989db94326714592b059c08056.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.241ex; height:2.843ex;" alt="{\displaystyle X(T)\to X(T_{0})}" loading="lazy"></span> is surjective where we wrote <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X(T)=\operatorname {Hom} _{S}(T,X)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo stretchy="false">(</mo>
<mi>T</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msub>
<mi>Hom</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>S</mi>
</mrow>
</msub>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>T</mi>
<mo>,</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X(T)=\operatorname {Hom} _{S}(T,X)}</annotation>
</semantics>
</math></span><img src="./6bf2e38d226b249805ff86b24629f9177fcafc9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.117ex; height:2.843ex;" alt="{\displaystyle X(T)=\operatorname {Hom} _{S}(T,X)}" loading="lazy"></span>. Then a morphism locally of finite presentation is smooth if and only if it is formally smooth.
</p><p>In the definition of "formally smooth", if we replace surjective by "bijective" (resp. "injective"), then we get the definition of <a href="Formally_%C3%A9tale_morphism" title="Formally étale morphism">formally étale</a> (resp. <b>formally unramified</b>).
</p>
<div class="mw-heading mw-heading2"><h2 id="Smooth_base_change">Smooth base change</h2></div>
<p>Let <i>S</i> be a scheme and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {char} (S)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>char</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>S</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {char} (S)}</annotation>
</semantics>
</math></span><img src="./c1724011b6a6ada6e501ca63ecef7688af71f999.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.708ex; height:2.843ex;" alt="{\displaystyle \operatorname {char} (S)}" loading="lazy"></span> denote the image of the structure map <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\to \operatorname {Spec} \mathbb {Z} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>S</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>Spec</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">Z</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S\to \operatorname {Spec} \mathbb {Z} }</annotation>
</semantics>
</math></span><img src="./3dab61b02665dac90b2cc38c8051a78b87650dce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.701ex; height:2.509ex;" alt="{\displaystyle S\to \operatorname {Spec} \mathbb {Z} }" loading="lazy"></span>. The <b>smooth base change theorem</b> states the following: let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to S}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>:</mo>
<mi>X</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>S</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f:X\to S}</annotation>
</semantics>
</math></span><img src="./a21a941b35354b73f5a62849067deac7ee9794dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.309ex; height:2.509ex;" alt="{\displaystyle f:X\to S}" loading="lazy"></span> be a <a href="Quasi-compact_morphism" title="Quasi-compact morphism">quasi-compact morphism</a>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g:S'\to S}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
<mo>:</mo>
<msup>
<mi>S</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">→<!-- → --></mo>
<mi>S</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g:S'\to S}</annotation>
</semantics>
</math></span><img src="./d88622e94e7f68928e8186c39931670369ed408c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.373ex; height:2.843ex;" alt="{\displaystyle g:S'\to S}" loading="lazy"></span> a smooth morphism and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">F</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation>
</semantics>
</math></span><img src="./205d4b91000d9dcf1a5bbabdfa6a8395fa60b676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}" loading="lazy"></span> a <a href="Torsion_sheaf" title="Torsion sheaf">torsion sheaf</a> on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{\text{et}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext>et</mtext>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X_{\text{et}}}</annotation>
</semantics>
</math></span><img src="./25f1bf7f69b2e2c5d2300e21b8e3fdf623f5f5dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.526ex; height:2.509ex;" alt="{\displaystyle X_{\text{et}}}" loading="lazy"></span>. If for every <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\neq p}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>0</mn>
<mo>≠<!-- ≠ --></mo>
<mi>p</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 0\neq p}</annotation>
</semantics>
</math></span><img src="./e4f534105c10360bd35c0093f90cb62b52b77545.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.43ex; height:2.676ex;" alt="{\displaystyle 0\neq p}" loading="lazy"></span> in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {char} (S)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>char</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>S</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {char} (S)}</annotation>
</semantics>
</math></span><img src="./c1724011b6a6ada6e501ca63ecef7688af71f999.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.708ex; height:2.843ex;" alt="{\displaystyle \operatorname {char} (S)}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p:{\mathcal {F}}\to {\mathcal {F}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
<mo>:</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">F</mi>
</mrow>
</mrow>
<mo stretchy="false">→<!-- → --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">F</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p:{\mathcal {F}}\to {\mathcal {F}}}</annotation>
</semantics>
</math></span><img src="./a6cd1cc5c88259e054772f428b6d70f1ecd582a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.663ex; height:2.509ex;" alt="{\displaystyle p:{\mathcal {F}}\to {\mathcal {F}}}" loading="lazy"></span> is injective, then the <a href="Base_change_morphism" class="mw-redirect" title="Base change morphism">base change morphism</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g^{*}(R^{i}f_{*}{\mathcal {F}})\to R^{i}f'_{*}(g'^{*}{\mathcal {F}})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>∗<!-- ∗ --></mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<msup>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msup>
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>∗<!-- ∗ --></mo>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">F</mi>
</mrow>
</mrow>
<mo stretchy="false">)</mo>
<mo stretchy="false">→<!-- → --></mo>
<msup>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msup>
<msubsup>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>∗<!-- ∗ --></mo>
</mrow>
<mo>′</mo>
</msubsup>
<mo stretchy="false">(</mo>
<msup>
<mi>g</mi>
<mrow>
<mo class="MJX-variant">′</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>∗<!-- ∗ --></mo>
</mrow>
</mrow>
</msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">F</mi>
</mrow>
</mrow>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g^{*}(R^{i}f_{*}{\mathcal {F}})\to R^{i}f'_{*}(g'^{*}{\mathcal {F}})}</annotation>
</semantics>
</math></span><img src="./febc3d5d1a9bbcb21cba5032bb04864822aa4f4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.398ex; height:3.176ex;" alt="{\displaystyle g^{*}(R^{i}f_{*}{\mathcal {F}})\to R^{i}f'_{*}(g'^{*}{\mathcal {F}})}" loading="lazy"></span> is an isomorphism.
</p>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Smooth_algebra" title="Smooth algebra">smooth algebra</a></li>
<li><a href="Regular_embedding" title="Regular embedding">regular embedding</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
<ul><li><a href="James_Milne_(mathematician)" title="James Milne (mathematician)">J. S. Milne</a> (2012). "<a rel="nofollow" class="external text" href="http://www.jmilne.org/math/CourseNotes/LEC.pdf">Lectures on Étale Cohomology</a>"</li>
<li>J. S. Milne. <i>Étale cohomology</i>, volume 33 of Princeton Mathematical Series. Princeton University Press, Princeton, N.J., 1980.</li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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