View source for Etale morphism

A smooth morphism of algebraic varieties or schemes of relative dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036290/e0362901.png" />. An étale morphism of schemes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036290/e0362902.png" /> can be defined equivalently as a locally finitely-presentable [[Flat morphism|flat morphism]] such that for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036290/e0362903.png" /> the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036290/e0362904.png" />-scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036290/e0362905.png" /> is finite and separable. An étale morphism has the lifting property for infinitesimal deformations: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036290/e0362906.png" /> is an étale morphism, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036290/e0362907.png" /> is an affine <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036290/e0362908.png" />-scheme and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036290/e0362909.png" /> is a closed subscheme of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036290/e03629010.png" /> given by a nilpotent sheaf of ideals, then the natural mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036290/e03629011.png" /> is bijective. This property characterizes the étale morphisms. Finally, an étale morphism can be defined as being flat and unramified. (A locally finitely-presentable morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036290/e03629012.png" /> is unramified if the diagonal imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036290/e03629013.png" /> is a local isomorphism.)

Being étale (like being smooth and being unramified) is preserved under composition of morphism and under base change. An open imbedding is an étale morphism. Any morphism between étale <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036290/e03629014.png" />-schemes is étale. For smooth varieties the fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036290/e03629015.png" /> is étale means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036290/e03629016.png" /> induces an isomorphism of the tangent spaces. Locally, an étale morphism is given by a polynomial with non-zero derivative.

Etale morphisms play an important role in étale cohomology theory (cf. [[Etale cohomology|Etale cohomology]]) in the definitions of the fundamental group of a scheme, of an [[Algebraic space|algebraic space]] and of a [[Hensel ring|Hensel ring]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Grothendieck,   J. Dieudonné,   "Eléments de géometrie algébrique"  ''Publ. Math. IHES'' , '''32'''  (1967)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Grothendieck (ed.)  et al. (ed.) , ''Revêtements étales et groupe fondamental. SGA 1'' , ''Lect. notes in math.'' , '''224''' , Springer  (1971)</TD></TR></table>



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Grothendieck,   J. Dieudonné,   "Eléments de géometrie algébrique: Etude locale des schémas et de morphismes de schémas"  ''Publ. Math. IHES'' , '''4'''  (1965)  pp. Part 4, Sect. 17.6</TD></TR></table>