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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0310REJJY8]] and let $A$ be a commutative [[$R$-algebra|DPBJ04JMH8]].\nThen $A$ is *étale* over $R$, and the structure map $R \\to A$ is an *étale morphism*, if both conditions hold:\n\n1. $A$ is a [[finitely presented|764GZHTGV8]] $R$-algebra.\n2. $A$ is formally étale over $R$, in the following sense.\n\nLet $C$ be a commutative $R$-algebra, let $J \\subseteq C$ be an [[ideal|096Z9MP2ME]] with $J^2 = 0$, and let $u \\colon A \\to C/J$ be an $R$-algebra [[homomorphism|0VTH8KPCG2]].\nThen there is exactly one $R$-algebra homomorphism $v \\colon A \\to C$ whose composite with $C \\to C/J$ is $u$.\n\n## Remarks\n\nCondition 2 says that the lift required by [[formal smoothness|8KPDNVKX7B]] exists and is unique.\nEquivalent forms: $A$ is [[smooth|286G0P1B69]] and [[unramified|GY3QA6NH0E]] over $R$; or $A$ is finitely presented, [[flat|73RYYEVDKA]] and unramified over $R$.\n\nExample: let $f \\in R[x]$ be monic and let $A = R[x]/(f)$; if the class of $f'$ is a unit in $A$, then $A$ is étale over $R$.\nFor instance, $R[x]/(x^2 - a)$ is étale when $2a$ is a unit of $R$, since $(2x)^2 = 4a$ in $A$.\nOver a field $k$, a finite [[separable extension|0WCX7NB4N7]] $L$ of $k$ is étale, and the étale $k$-algebras are exactly the finite products of such fields.\n\nNon-example: $A = k[x]$ over a field $k$ is smooth and finitely presented, but a map $A \\to C/J$ has many lifts: $x$ may go to any preimage of $u(x)$.\nSo uniqueness in clause 2 fails; accordingly $\\Omega_{A/k} = A \\, dx \\ne 0$, see [[Kähler differentials|7HCFARHG7D]].\n",
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