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  "body": "\n## Definition\n\nLet $X$ be a connected [[scheme|5A35TYZD02]] and let $\\bar{x} \\colon \\operatorname{Spec} \\Omega \\to X$ be a geometric point, with $\\Omega$ an algebraically closed field.\nA *finite étale cover* of $X$ is a morphism $Y \\to X$ that is [[finite|N9C4X7XKE4]] and [[étale|H84VHC8PX2]].\nLet $\\mathrm{FEt}_X$ be the category of finite étale covers of $X$, with morphisms over $X$.\nThe *fiber functor* $F_{\\bar{x}} \\colon \\mathrm{FEt}_X \\to \\mathrm{Set}$ sends $Y$ to the finite set of geometric points of $Y$ over $\\bar{x}$, that is, of morphisms $\\operatorname{Spec} \\Omega \\to Y$ that lift $\\bar{x}$.\nThe *étale fundamental group* is the automorphism group of this functor:\n$$\\pi_1^{\\text{ét}}(X, \\bar{x}) = \\Aut(F_{\\bar{x}}).$$\n\n## Remarks\n\nThe group $\\pi_1^{\\text{ét}}(X, \\bar{x})$ is a [[profinite group|VABPHNZAVV]], as a closed subgroup of $\\prod_Y \\operatorname{Sym}(F_{\\bar{x}}(Y))$.\nBy [[Grothendieck's Galois theory|218WAD58W3]], $F_{\\bar{x}}$ is an equivalence between $\\mathrm{FEt}_X$ and the category of finite sets with a continuous action of $\\pi_1^{\\text{ét}}(X, \\bar{x})$.\nExample: for $X = \\operatorname{Spec} k$ with $k$ a field, the finite étale covers are finite products of finite separable extensions of $k$.\nChoosing $\\bar{x}$ amounts to choosing a separable closure $k^{s}$, and $\\pi_1^{\\text{ét}}(X, \\bar{x}) \\cong \\Gal(k^{s}/k)$, the [[absolute Galois group|DPFMXS71D1]].\nNon-example: the exponential map $\\C \\to \\C^{*}$ is a covering space with infinitely many sheets, so it is not a finite étale cover and does not enter $\\mathrm{FEt}_X$.\nTherefore $\\pi_1^{\\text{ét}}$ of the complex punctured line is $\\widehat{\\Z}$, the profinite completion of the topological fundamental group $\\Z$, and not $\\Z$ itself.\nFor a variety $X$ over $\\C$, the Riemann existence theorem gives $\\pi_1^{\\text{ét}}(X) \\cong \\widehat{\\pi_1(X(\\C))}$.\nIn characteristic $p \u003e 0$ the affine line has nontrivial finite étale covers, such as the Artin–Schreier covers $y^p - y = x$.\n",
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