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Formally smooth map
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<h2 id="examples">Examples</h2> <h3>Smooth morphisms</h3> <p>All smooth morphisms f : X → S {\displaystyle f:X\to S} are equivalent to morphisms locally of finite presentation which are formally smooth. Hence formal smoothness is a slight generalization of smooth morphisms.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p> <h3>Non-example</h3><p> One method for detecting formal smoothness of a scheme is using infinitesimal lifting criterion. For example, using the truncation morphism k [ ε ] / ( ε 3 ) → k [ ε ] / ( ε 2 ) {\displaystyle k[\varepsilon ]/(\varepsilon ^{3})\to k[\varepsilon ]/(\varepsilon ^{2})} the infinitesimal lifting criterion can be described using the commutative square</p><blockquote><p> X ← Spec ( k [ ε ] ( ε 2 ) ) ↓ ↓ S ← Spec ( k [ ε ] ( ε 3 ) ) {\displaystyle {\begin{matrix}X&\leftarrow &{\text{Spec}}\left({\frac {k[\varepsilon ]}{(\varepsilon ^{2})}}\right)\\\downarrow &&\downarrow \\S&\leftarrow &{\text{Spec}}\left({\frac {k[\varepsilon ]}{(\varepsilon ^{3})}}\right)\end{matrix}}} </p></blockquote><p>where X , S ∈ S c h / S {\displaystyle X,S\in Sch/S} . For example, if</p><blockquote><p> X = Spec ( k [ x , y ] ( x y ) ) {\displaystyle X={\text{Spec}}\left({\frac {k[x,y]}{(xy)}}\right)} and Y = Spec ( k ) {\displaystyle Y={\text{Spec}}(k)} </p></blockquote><p>then consider the tangent vector at the origin ( 0 , 0 ) ∈ X ( k ) {\displaystyle (0,0)\in X(k)} given by the ring morphism</p><blockquote><p> k [ x , y ] ( x y ) → k [ ε ] ( ε 2 ) {\displaystyle {\frac {k[x,y]}{(xy)}}\to {\frac {k[\varepsilon ]}{(\varepsilon ^{2})}}} </p></blockquote><p>sending</p><blockquote><p> x ↦ ε y ↦ ε {\displaystyle {\begin{aligned}x&\mapsto \varepsilon \\y&\mapsto \varepsilon \end{aligned}}} </p></blockquote><p>Note because x y ↦ ε 2 = 0 {\displaystyle xy\mapsto \varepsilon ^{2}=0} , this is a valid morphism of commutative rings. Then, since a lifting of this morphism to</p><blockquote><p> Spec ( k [ ε ] ( ε 3 ) ) → X {\displaystyle {\text{Spec}}\left({\frac {k[\varepsilon ]}{(\varepsilon ^{3})}}\right)\to X} </p></blockquote><p>is of the form</p><blockquote><p> x ↦ ε + a ε 2 y ↦ ε + b ε 2 {\displaystyle {\begin{aligned}x&\mapsto \varepsilon +a\varepsilon ^{2}\\y&\mapsto \varepsilon +b\varepsilon ^{2}\end{aligned}}} </p></blockquote><p>and x y ↦ ε 2 + ( a + b ) ε 3 = ε 2 {\displaystyle xy\mapsto \varepsilon ^{2}+(a+b)\varepsilon ^{3}=\varepsilon ^{2}} , there cannot be an infinitesimal lift since this is non-zero, hence X ∈ S c h / k {\displaystyle X\in Sch/k} is not formally smooth. This also proves this morphism is not smooth from the equivalence between formally smooth morphisms locally of finite presentation and smooth morphisms. </p><h2 id="see-also">See also</h2> <ul><li><a href="/facts/Dual_number/DntuRu3F">Dual number</a></li> <li><a href="/facts/Smooth_morphism/GNxTBHTG">Smooth morphism</a></li> <li><a href="/facts/Deformation_theory/tVLS3kkG">Deformation theory</a></li></ul> <h2 id="external-links">External links</h2> <ul><li>Formally smooth with smooth fibers, but not smooth <a href="https://mathoverflow.net/q/333596">https://mathoverflow.net/q/333596</a></li> <li>Formally smooth but not smooth <a href="https://mathoverflow.net/q/195">https://mathoverflow.net/q/195</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Grothendieck, Alexandre; Dieudonné, Jean (1964). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Première partie". Publications Mathématiques de l'IHÉS. 20: 5–259. doi:10.1007/bf02684747. MR 0173675. <a href="/wiki/Alexander_Grothendieck" target="_blank">/wiki/Alexander_Grothendieck</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Grothendieck, Alexandre; Dieudonné, Jean (1967). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Quatrième partie". Publications Mathématiques de l'IHÉS. 32: 5–361. doi:10.1007/bf02732123. MR 0238860. <a href="/wiki/Alexander_Grothendieck" target="_blank">/wiki/Alexander_Grothendieck</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>"Lemma 37.11.7 (02H6): Infinitesimal lifting criterion—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-04-07. <a href="https://stacks.math.columbia.edu/tag/02H6" target="_blank">https://stacks.math.columbia.edu/tag/02H6</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> </ol>