# Connectedness and full faithfulness *Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).* A morphism between two finite étale covers is determined by its graph, which is an open and closed subscheme of their fibre product. So the full faithfulness of \(Y\mapsto Y(\mathbf C)\) comes down to one statement: the open and closed subsets of a scheme \(X\) of finite type over \(\mathbf C\) are the same as those of \(X(\mathbf C)\). In other words, a connected scheme has a connected space of complex points. For a smooth projective variety this follows from the algebraic computation \(H^0(X,\mathcal O_X)=\mathbf C\) and GAGA. The general case is reduced to that one by resolution of singularities and smooth compactification. We use [Finite étale covers and their analytification](finite-etale-covers-and-their-analytification.md), [Principalization and resolution](course:resolution-of-singularities-in-characteristic-zero/principalization-and-resolution) from the course on resolution of singularities, [Serre's comparison theorems and Chow's theorem](course:AG-QC/serres-comparison-theorems-and-chows-theorem), and the Riemann extension theorem of [Holomorphic functions of several variables](course:complex-analytic-spaces-and-coherent-sheaves/holomorphic-functions-of-several-variables#4-the-riemann-extension-theorem). ## 1. Smooth points and divisors with normal crossings **Lemma 1.1.** Let \(X\) be a scheme smooth of dimension \(n\) over \(\mathbf C\) near a closed point \(x\), and let \(z_1,\ldots,z_n\) be regular functions near \(x\), vanishing at \(x\), whose differentials form a basis of \(\Omega_X\otimes\kappa(x)\). Then \((z_1,\ldots,z_n)\) maps an open neighbourhood of \(x\) in \(X(\mathbf C)\) homeomorphically onto an open subset of \(\mathbf C^n\). If \(X\) is a variety, these charts make \(X^{\rm sm}(\mathbf C)\) a complex manifold whose holomorphic functions are the sections of \(\mathcal O_{X^{\rm an}}\). If \(E\) is an snc divisor whose components through \(x\) are \(V(z_1),\ldots,V(z_p)\), then in such a chart \(E(\mathbf C)\) is the set \(\{z_1\cdots z_p=0\}\). **Proof.** On a neighbourhood \(U\) of \(x\) the differentials \(dz_i\) form a basis of \(\Omega_U\), so \(z:U\to\mathbf A^n\) is étale ([Smooth blow-ups and transforms of ideals, Section 1](course:resolution-of-singularities-in-characteristic-zero/smooth-blowups-and-transforms-of-ideals#1-coordinates-and-derivations)). By Lemma 2.2 of the previous lesson, \(U(\mathbf C)\to\mathbf C^n\) is a local homeomorphism. The argument in the proof of Proposition 3.1 there is local and uses only that the morphism is étale; it shows that \(z\) is a local isomorphism of analytic spaces, so the charts are holomorphically compatible and the holomorphic functions are those of \(X^{\rm an}\). The last statement holds because \(E=V(z_1\cdots z_p)\) near \(x\). \(\square\) **Lemma 1.2 (complements of divisors).** Let \(M\) be a connected complex manifold and \(A\subset M\) a closed subset such that every point of \(M\) has a connected chart neighbourhood \(W\) and a holomorphic function \(h\) on \(W\), not identically zero, with \(A\cap W\subset\{h=0\}\). Then \(M\setminus A\) is connected and dense in \(M\). **Proof.** In each such chart, \(W\setminus A\supset W\setminus\{h=0\}\), which is dense in \(W\) by [Holomorphic functions of several variables, Corollary 4.3](course:complex-analytic-spaces-and-coherent-sheaves/holomorphic-functions-of-several-variables#4-the-riemann-extension-theorem); so \(M\setminus A\) is dense. Suppose \(M\setminus A=P\sqcup Q\) with \(P,Q\) open and nonempty. The function \(f\) equal to \(1\) on \(P\) and \(0\) on \(Q\) is holomorphic and bounded on \(M\setminus A\). In each chart it is holomorphic on \(W\setminus\{h=0\}\) and bounded, so by the Riemann extension theorem ([Holomorphic functions of several variables, Theorem 4.2](course:complex-analytic-spaces-and-coherent-sheaves/holomorphic-functions-of-several-variables#4-the-riemann-extension-theorem)) it extends to a holomorphic function on \(W\); the extensions agree on overlaps, because they agree on a dense subset. The resulting continuous function on \(M\) takes the values \(0\) and \(1\) on a dense subset, hence everywhere, and both values occur. This decomposes \(M\) into two nonempty open sets, contradicting connectedness. \(\square\) ## 2. Open and closed subsets **Theorem 2.1.** Let \(X\) be a scheme locally of finite type over \(\mathbf C\). The map \(T\mapsto T(\mathbf C)\) is a bijection from the set of open and closed subsets of \(X\) to the set of open and closed subsets of \(X(\mathbf C)\). In particular, \(X\) is connected if and only if \(X(\mathbf C)\) is connected. **Proof.** *The map is defined and injective.* A Zariski closed subset is closed in the classical topology (in an affine chart it is a common zero set of polynomials), and so is the complement of a Zariski open subset; so \(T(\mathbf C)\) is open and closed. If \(T\ne T'\), their symmetric difference is a nonempty locally closed subset of \(X\), which contains a closed point; so \(T(\mathbf C)\ne T'(\mathbf C)\). *Reduction to connected schemes.* The scheme \(X\) is locally noetherian, so its connected components are open, and \(X(\mathbf C)\) is the disjoint union of their spaces of complex points. If each connected component \(X_\lambda\) has connected \(X_\lambda(\mathbf C)\), then every open and closed subset of \(X(\mathbf C)\) is a union of some \(X_\lambda(\mathbf C)\), hence of the form \(T(\mathbf C)\). So it suffices to prove: *if \(X\) is connected, then \(X(\mathbf C)\) is connected.* By Lemma 1.2(5) of the previous lesson we may assume \(X\) reduced. *Reduction to integral schemes.* Let \(X\) be connected and reduced, and assume the claim for integral schemes. The irreducible components \(X_i\) of \(X\), with their reduced structure, form a locally finite family, and each \(X_i(\mathbf C)\) is connected. If \(X_i\cap X_j\ne\emptyset\), this closed set contains a closed point, so \(X_i(\mathbf C)\cap X_j(\mathbf C)\ne\emptyset\). Let \(C\) be the connected component of \(X(\mathbf C)\) containing \(X_1(\mathbf C)\), and \(I\) the set of \(i\) with \(X_i(\mathbf C)\subset C\). If \(i\in I\) and \(X_j\) meets \(X_i\), then \(j\in I\). So \(\bigcup_{i\in I}X_i\) and \(\bigcup_{i\notin I}X_i\) are disjoint, and both are closed, being locally finite unions of closed sets. They cover the connected \(X\), so the second one is empty, and \(X(\mathbf C)=C\). *Reduction to affine integral schemes.* Let \(X\) be integral. Any two nonempty affine opens \(U,U'\) meet, since \(X\) is irreducible, and \(U\cap U'\) contains a closed point. If every \(U(\mathbf C)\) is connected, \(X(\mathbf C)\) is a union of pairwise intersecting connected subsets, hence connected. *The affine integral case.* Let \(X\) be affine and integral, and let \(\rho:R(X)\to X\) be its resolution ([Principalization and resolution, Theorem 4.8](course:resolution-of-singularities-in-characteristic-zero/principalization-and-resolution#4-resolution-of-singularities)). Since \(X\) has an embedding into an affine space, \(\rho\) is projective, so \(R(X)\) is a smooth quasi-projective variety. It is an isomorphism over the dense open subset \(X^{\rm sm}\), which is irreducible, so \(R(X)\) is irreducible, and \(\rho\) is surjective: its image is closed and dense. By [Principalization and resolution, Corollary 5.2](course:resolution-of-singularities-in-characteristic-zero/principalization-and-resolution#5-consequences-used-in-the-programme) there is a smooth projective variety \(R'\) containing \(R(X)\) as a dense open subset whose complement is the support of an snc divisor \(D\). As the closure of an irreducible set, \(R'\) is irreducible, hence integral. First, \(R'(\mathbf C)\) is connected. The proper integral scheme \(R'\) over the algebraically closed field \(\mathbf C\) has \(H^0(R',\mathcal O_{R'})=\mathbf C\) ([Stacks, Tag 0BUG](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/varieties.html#varieties-lemma-proper-geometrically-reduced-global-sections)). By the first comparison theorem ([Serre's comparison theorems and Chow's theorem, Theorem 3.1](course:AG-QC/serres-comparison-theorems-and-chows-theorem#3-the-first-comparison-theorem), in degree \(0\)), the holomorphic functions on \(R'(\mathbf C)\) are also only the constants. If \(R'(\mathbf C)=P\sqcup Q\) with \(P,Q\) open and nonempty, the function equal to \(1\) on \(P\) and \(0\) on \(Q\) would be a nonconstant holomorphic function. Second, \(R(X)(\mathbf C)=R'(\mathbf C)\setminus D(\mathbf C)\) is connected, by Lemma 1.2 applied to the connected complex manifold \(R'(\mathbf C)\) and \(A=D(\mathbf C)\): by Lemma 1.1, near each point of \(D(\mathbf C)\) there is a chart in which \(A\) is \(\{z_1\cdots z_p=0\}\). Finally, \(\rho\) is surjective, so every closed point of \(X\) is the image of a closed point of \(R(X)\) (the fibre is a nonempty scheme of finite type over \(\mathbf C\)). Hence \(X(\mathbf C)\) is the image of the connected space \(R(X)(\mathbf C)\) under the continuous map \(\rho(\mathbf C)\), and it is connected. \(\square\) ## 3. Full faithfulness **Corollary 3.1.** For every scheme \(X\) locally of finite type over \(\mathbf C\), the functor \(\Phi_X:\operatorname{F\acute Et}(X)\to\operatorname{Cov}(X(\mathbf C))\) is fully faithful. **Proof.** Let \(Y,Z\) be finite étale over \(X\). For an \(X\)-morphism \(f:Y\to Z\), the graph \(Y\to Y\times_XZ\) is the base change of the diagonal of \(Z\) over \(X\) along \(f\times\mathrm{id}_Z\). This diagonal is an open immersion, because \(Z\to X\) is unramified ([Stacks, Tag 02GE](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/morphisms.html#morphisms-lemma-diagonal-unramified-morphism)), and a closed immersion, because \(Z\to X\) is separated. So \(f\mapsto\Gamma_f\) identifies \(\operatorname{Hom}_X(Y,Z)\) with the set of open and closed subschemes \(\Gamma\subset Y\times_XZ\) for which the first projection \(\Gamma\to Y\) is an isomorphism; the inverse sends \(\Gamma\) to the second projection composed with the inverse of the first. In the same way, a continuous map \(g:Y(\mathbf C)\to Z(\mathbf C)\) over \(X(\mathbf C)\) is determined by its graph, a subset \(\Gamma'\) of \(Y(\mathbf C)\times_{X(\mathbf C)}Z(\mathbf C)\) on which the first projection is a homeomorphism onto \(Y(\mathbf C)\). The first projection of this fibre product is a finite covering map, and the graph is the image of a continuous section of it; such an image is open and closed, because over an open set \(W\) on which the covering is \(W\times F\), the section is given by a locally constant map \(W\to F\). By Lemma 1.2(6) of the previous lesson, \((Y\times_XZ)(\mathbf C)=Y(\mathbf C)\times_{X(\mathbf C)}Z(\mathbf C)\), and by Theorem 2.1 the open and closed subsets of the two sides correspond. An open and closed subscheme \(\Gamma\) of \(Y\times_XZ\) is finite étale over \(Y\). The morphism \(\Gamma\to Y\) is an isomorphism if and only if its degree is \(1\) at every point, since a finite locally free algebra of rank \(1\) is the structure sheaf itself. As the degree is locally constant and every nonempty open set contains a closed point, this holds if and only if every closed point of \(Y\) has exactly one preimage, that is, by Proposition 2.3 of the previous lesson, if and only if \(\Gamma(\mathbf C)\to Y(\mathbf C)\) is a covering of degree \(1\), a homeomorphism. So graphs correspond to graphs, and \(\operatorname{Hom}_X(Y,Z)\to\operatorname{Hom}_{X(\mathbf C)}(Y(\mathbf C),Z(\mathbf C))\) is bijective. \(\square\) In particular \(\Phi_X\) identifies automorphism groups: the group of automorphisms of a finite étale cover over \(X\) is the group of deck transformations of its space of complex points. **Proposition 3.2 (existence is local).** Let \(E\to X(\mathbf C)\) be a finite covering map and \(X=\bigcup_iU_i\) an open cover such that each restriction \(E|_{U_i(\mathbf C)}\) is isomorphic to \(Y_i(\mathbf C)\) for some \(Y_i\) finite étale over \(U_i\). Then \(E\cong Y(\mathbf C)\) for some \(Y\) finite étale over \(X\). **Proof.** Put \(U_{ij}=U_i\cap U_j\). The given isomorphisms induce isomorphisms \(Y_i(\mathbf C)|_{U_{ij}(\mathbf C)}\cong Y_j(\mathbf C)|_{U_{ij}(\mathbf C)}\), which by Corollary 3.1 for \(U_{ij}\) come from unique isomorphisms \(\varphi_{ij}:Y_i|_{U_{ij}}\to Y_j|_{U_{ij}}\). On triple intersections \(\varphi_{jk}\circ\varphi_{ij}\) and \(\varphi_{ik}\) induce the same map on complex points, so they are equal by faithfulness. The \(Y_i\) glue to a scheme \(Y\) finite étale over \(X\), and the given isomorphisms glue to \(E\cong Y(\mathbf C)\). \(\square\) **Corollary 3.3.** Let \(Y\) be finite étale over \(X\). Every open and closed subset of \(Y(\mathbf C)\), regarded as a finite covering of \(X(\mathbf C)\), is \(Y_1(\mathbf C)\) for a unique open and closed subscheme \(Y_1\subset Y\). In particular a covering that is a direct summand of one in the image of \(\Phi_X\) is itself in the image. **Proof.** Theorem 2.1 for \(Y\). \(\square\) ## 4. Examples **Example 4.1 (two lines).** Let \(X=V(xy)\subset\mathbf A^2\). Its components are the two axes, which meet at the origin, so \(X(\mathbf C)\) is connected, as Theorem 2.1 predicts; it is the union of two complex lines meeting in a point. The space \(X(\mathbf C)\) is contractible, since each line is contractible and they meet in a point, so every finite covering of \(X(\mathbf C)\) is trivial. By the Riemann existence theorem ([The Riemann existence theorem, Theorem 2.1](the-riemann-existence-theorem.md#2-the-theorem)), every finite étale cover of \(X\) is therefore trivial. **Example 4.2 (the role of the resolution).** For the cuspidal cubic \(X=V(y^2-x^3)\) the resolution is the normalization \(\mathbf A^1\to X\), \(t\mapsto(t^2,t^3)\), and the compactification of \(\mathbf A^1\) is \(\mathbf P^1\) with one boundary point. The proof of Theorem 2.1 obtains the connectedness of \(X(\mathbf C)\) from that of \(\mathbf P^1(\mathbf C)\), which follows from \(H^0(\mathbf P^1,\mathcal O)=\mathbf C\), then that of \(\mathbf C=\mathbf P^1(\mathbf C)\setminus\{\infty\}\), and finally that of its image \(X(\mathbf C)\). Here \(X(\mathbf C)\) is even homeomorphic to \(\mathbf C\), as shown in Section 6 of [Complex analytic spaces and analytification](course:AG-QC/complex-analytic-spaces-and-analytification). ## 5. Exercises **Exercise 5.1.** Show that the map of Theorem 2.1 is a bijection for \(X=\operatorname{Spec}\mathbf C[x,y]/(x^2,xy)\), and compute both sides. *Solution.* \(X_{\rm red}\) is the line \(V(x)\), connected, and \(X(\mathbf C)=\mathbf C\). Both sides consist of \(\emptyset\) and the whole space. **Exercise 5.2.** Prove directly, without Theorem 2.1, that \(\mathbf P^n(\mathbf C)\) is connected, and compare with the proof of Theorem 2.1 for \(R'=\mathbf P^n\). *Solution.* \(\mathbf P^n(\mathbf C)\) is the continuous image of the connected unit sphere in \(\mathbf C^{n+1}\). In the proof of Theorem 2.1 the same fact comes from \(H^0(\mathbf P^n,\mathcal O)=\mathbf C\) and the comparison theorem: a disconnection would give a nonconstant holomorphic function. **Exercise 5.3.** Let \(Y\to X\) be finite étale with \(X\) connected. Show that \(Y\) is connected if and only if \(Y(\mathbf C)\) is connected, and that a connected finite étale cover of degree \(d\) has an automorphism group of order at most \(d\), with equality if and only if the covering space \(Y(\mathbf C)\) is regular, that is, its deck group acts transitively on each fibre. *Solution.* The first statement is Theorem 2.1 for \(Y\). By Corollary 3.1, automorphisms of \(Y\) over \(X\) are deck transformations of the connected covering \(Y(\mathbf C)\). A deck transformation of a connected covering is determined by the image of one point, so there are at most \(d\) of them, and there are exactly \(d\) if and only if the deck group acts transitively on a fibre. ## References - [SGA 1] A. Grothendieck et al., *Revêtements étales et groupe fondamental (SGA 1)*, Exposé XII, Proposition 2.4 and Corollary 2.6 (connected components of \(X\) and of \(X^{\rm an}\)), and the first part of the proof of Theorem 5.1 (full faithfulness through graphs); free re-edition arXiv:math/0206203. The proof of Theorem 2.1 in this lesson uses resolution, smooth compactification and GAGA.