Comparison with the topological fundamental group over the complex numbers

Written by GPT-6.1 Sol (OpenAI) in Codex, Ultra setting, October 2026. Self-checked by the writing AI. Public domain (CC0).

An algebraic finite étale cover over the complex numbers can be read as a finite covering space in the classical topology. The comparison includes singular and nonreduced schemes. We first construct the analytic structure on a topological cover and prove the proper case using coherent algebras and GAGA. We then state Riemann existence in finite type, extend it by gluing, and prove the fundamental-group and curve computations.

Our algebraic fundamental groups, fibre functors and left-action conventions are those of Galois categories, Section 5.2 and The étale fundamental group, Section 1. A connected scheme is nonempty. A finite covering space is locally a disjoint union of finitely many copies of the base; the empty cover is allowed. Its degree is locally constant and need not be the same on different connected components.

1. Analytification and its precise inputs

For a scheme $X$ locally of finite type over $\mathbf C$, write $X(\mathbf C)$ for its complex points with the classical topology, and $X^{\mathrm{an}}$ for its analytification. An affine presentation $U=\operatorname{Spec}\bigl(\mathbf C[z_1,\ldots,z_r]/I\bigr)$ gives the common zero set of $I$ in $\mathbf C^r$, equipped with the analytic structure sheaf obtained by quotienting holomorphic functions by the ideal generated by $I$. Retain that ideal, including its nilpotent information. The construction is not just a space of points. Regular functions are holomorphic on their domains, so these charts glue, as do their maps. This constructs the functor $X\mapsto X^{\mathrm{an}}$ and the morphism of locally ringed spaces $a_X:X^{\mathrm{an}}\longrightarrow X. \tag{1.1}$ Analytification respects products, open subschemes and closed immersions: on charts the defining equations and their pullbacks are the same. In particular it respects fibre products.

For a nonseparated scheme we use the glued analytic locally ringed space, which can be non-Hausdorff. Statements about analytic spaces below permit this gluing convention. Each algebraic affine chart has a Hausdorff analytification, and all local constructions take place in such charts. Proper schemes are separated and have the usual Hausdorff analytification. No separatedness restriction is being inserted into the finite type comparison theorem.

Here are the local analytic prerequisites we use. The ring of convergent power series is Noetherian; its maximal-ideal completion is the formal power-series ring. Analytic structure sheaves and their finitely presented modules are coherent. These are the local analytic algebra and Oka coherence inputs. For every complex point $x$, put $R=\mathcal O_{X,x},\qquad S=\mathcal O_{X^{\mathrm{an}},x}.$ Then $R\to S$ is faithfully flat and induces an isomorphism on completions. The construction explains why this also holds without reducedness. After moving $x$ to the origin, both completions are $\mathbf C[[z_1,\ldots,z_r]]/ I\mathbf C[[z_1,\ldots,z_r]]. \tag{1.2}$ Completion of a Noetherian local ring is faithfully flat. Thus $\widehat S=\widehat R$ is flat over $R$ and faithfully flat over $S$; descent of flatness proves that $S$ is flat over $R$. The map is local with residue field $\mathbf C$, hence faithfully flat. This is the same local argument as the completed-local-ring comparison and faithful-flatness theorem, applied here to arbitrary defining ideals.

For an algebraic coherent module $\mathcal M$, analytification is pullback along (1.1), and $(\mathcal M^{\mathrm{an}})_x =\mathcal M_x\otimes_R S,\qquad (\mathcal M\otimes\mathcal N)^{\mathrm{an}} \simeq\mathcal M^{\mathrm{an}}\otimes\mathcal N^{\mathrm{an}}. \tag{1.3}$ It is exact by faithful flatness. These formulas also show compatibility with units, symmetry and associativity of tensor products. See the stalk formula and coherent-module comparison.

Analytic étaleness input. For a morphism $f:Y\to X$ of schemes of finite type over $\mathbf C$, $f\text{ is étale} \quad\Longleftrightarrow\quad f^{\mathrm{an}}\text{ is a local isomorphism of analytic spaces}. \tag{1.4}$ Stacks, Tag 0HFE

The same assertion for schemes locally of finite type follows by applying (1.4) to affine charts in source and target. Both properties are local on these charts. This input concerns analytic local isomorphisms, including the structure sheaf.

A local homeomorphism alone is insufficient. The reduction $\operatorname{Spec}\mathbf C\to\operatorname{Spec}\mathbf C[\epsilon]/(\epsilon^2)$ is a homeomorphism of the one-point classical spaces. It is not flat: tensor the inclusion $(\epsilon)\hookrightarrow\mathbf C[\epsilon]/(\epsilon^2)$ with $\mathbf C$. Its nonzero source maps to zero, so tensoring has destroyed injectivity. Its analytification is consequently not a local analytic isomorphism.

2. The analytic structure on a finite topological cover

Proposition 2.1. Let $p:E\to X(\mathbf C)$ be a finite topological covering space. There is a unique analytic-space structure on $E$, with the given topology, for which $p:E\to X^{\mathrm{an}}$ is a local analytic isomorphism. With this structure $p$ is finite étale analytically. This construction is an equivalence on categories, including every morphism over the base.

Proof. Choose an evenly covered open set $V$ and write $p^{-1}(V)=\coprod_{j=1}^{n}V_j,\qquad p|_{V_j}:V_j\xrightarrow{\sim}V. \tag{2.1}$ Transport the analytic structure of $V$ onto every $V_j$. On overlapping sheets the transition, when expressed on the base, is the identity on an open subset of $X^{\mathrm{an}}$. It is therefore analytic, and satisfies the cocycle condition. The transported sheaves glue. Equivalently the resulting structure sheaf is $p^{-1}\mathcal O_{X^{\mathrm{an}}}$.

Any structure making $p$ a local analytic isomorphism must give precisely these transported sheaves on every sheet. Their identifications with the base force the overlap identifications. This proves uniqueness as a locally ringed space over $X^{\mathrm{an}}$, not just uniqueness of the reduced structure.

A finite covering map is closed. This can be checked over each evenly covered open set, where the image of a closed subset is a finite union of closed subsets. Its fibres are finite. It is also proper: given an open cover of the preimage of a compact set, cover each finite fibre by finitely many members; closedness gives a base neighbourhood whose whole inverse image lies in those members. Compactness of the base set then gives a finite subcover. These assertions persist after base change. Thus the analytic local isomorphism is finite. Conversely, a finite analytic local isomorphism is a finite covering map. Around the finitely many points of a fibre, choose disjoint local-isomorphism neighbourhoods; closedness removes the image of their complement. Shrinking their common image gives an evenly covered neighbourhood. The assertions can be checked on Hausdorff base charts when the globally glued base is non-Hausdorff.

Finally let $h:E\to E'$ be a continuous map of finite covering spaces over the base. Near any point choose sheets for both covers. By continuity, after shrinking, $h$ maps the chosen source sheet into a chosen target sheet. In base coordinates it is the identity. It is therefore analytic with a unique map on structure sheaves. Conversely an analytic map is continuous. This proves full faithfulness and the category equivalence. $\square$

The sheaf $\mathcal E=p_*\mathcal O_E \tag{2.2}$ is a coherent, finite locally free $\mathcal O_{X^{\mathrm{an}}}$-algebra: over (2.1) it is $\mathcal O_V^n$, with coordinatewise multiplication. Its primitive factors recover the sheets of the cover. More explicitly an algebra homomorphism between two such product algebras sends the idempotents of the target product to orthogonal idempotents summing to one in the source product. On a connected small neighbourhood each source factor selects one target factor. This is exactly a map of sheets, in the contravariant direction. Hence analytic cover morphisms are precisely homomorphisms of these coherent algebras in the reverse direction.

3. The proper case from GAGA

We state the permitted GAGA input in the generality needed here.

Proper-scheme GAGA input. If $X$ is a proper scheme over $\mathbf C$, analytification induces an exact equivalence $\operatorname{Coh}(X)\simeq \operatorname{Coh}(X^{\mathrm{an}}) \tag{3.1}$ and isomorphisms $H^q(X,\mathcal M)\simeq H^q(X^{\mathrm{an}},\mathcal M^{\mathrm{an}})$ for coherent $\mathcal M$. See Hall, GAGA theorems, Theorem A and Example 9.4, pages 2 and 22. This includes nonreduced proper schemes. Serre's original Theorems 1–3, Section 3, no. 12, pages 19–20 and the frozen GAGA statements give the projective variety version; that version alone does not justify replacing “projective” by “proper.”

Theorem 3.1. For $X$ proper over $\mathbf C$, analytification is an equivalence from schemes finite étale over $X$ to finite topological covers of $X(\mathbf C)$, with all morphisms.

Proof. Take a topological cover $p:E\to X(\mathbf C)$, give it the structure of Proposition 2.1, and form the algebra $\mathcal E$ of (2.2). GAGA gives a coherent module $\mathcal B$ and an isomorphism $\mathcal B^{\mathrm{an}}\simeq\mathcal E$. Through (1.3), full faithfulness lifts the analytic maps $\mathcal E\otimes\mathcal E\longrightarrow\mathcal E, \qquad \mathcal O_{X^{\mathrm{an}}}\longrightarrow\mathcal E$ uniquely to $\mu:\mathcal B\otimes\mathcal B\longrightarrow\mathcal B, \qquad u:\mathcal O_X\longrightarrow\mathcal B. \tag{3.2}$ The associativity, commutativity and unit diagrams commute after analytification. Faithfulness makes them commute algebraically. Thus $\mathcal B$ is a coherent commutative unital algebra. The zero algebra on a component corresponds to the empty cover there.

We prove that $Y=\operatorname{Spec}_X\mathcal B\to X$ is finite étale. It is finite because $\mathcal B$ is coherent on the Noetherian scheme $X$. At a complex point $x$, (1.3) identifies $\mathcal B_x\otimes_R S\simeq S^n \quad\text{as algebras}. \tag{3.3}$ Faithfully flat descent of finite projectivity, proved in the module descent lesson, Section 3, implies that $\mathcal B_x$ is free of rank $n$ over the local ring $R$. The locally free locus of a coherent module is open. Its complement, if nonempty, would be a closed subset containing a closed point, since $X$ is of finite type over $\mathbf C$. Every closed point is a complex point, where (3.3) has proved freeness. Consequently $\mathcal B$ is finite locally free everywhere.

Moreover, differentials of a finitely presented algebra commute with extension of scalars, so $\Omega_{\mathcal B/\mathcal O_X,x}\otimes_R S \simeq\Omega_{S^n/S}=0. \tag{3.4}$ Faithful flatness kills the left algebraic module at every closed point. Its coherent support is therefore empty. Thus $\Omega_{\mathcal B/\mathcal O_X}=0$. A finite locally free algebra with vanishing relative differentials is étale, by the finite flat étale criterion. This proves the assertion about $Y$.

For clarity, we also verify that its analytification gives the original cover. A finite algebraic map becomes a proper map of classical spaces. Locally write $X=\operatorname{Spec}A$ and embed its finite affine inverse image in $X\times\mathbf A^r$ using algebra generators $b_i$. Each $b_i$ satisfies a monic polynomial with coefficients in $A$. Above a compact subset of $X(\mathbf C)$ those coefficients are bounded, so the bound $|z|\le1+\sum_{j<d}|a_j|$ for a root of $z^d+\sum_{j<d}a_jz^j$ bounds every $b_i$. For $|z|>1$, the equation gives $|z|^d\le(\sum|a_j|)|z|^{d-1}$, which proves that bound. The inverse image is a closed subset of the compact base set times a closed bounded polydisc, hence compact. This proves the local properness assertion; closedness is local on the base and follows as well. By (1.4), $q:Y^{\mathrm{an}}\to X^{\mathrm{an}}$ is a local isomorphism. Properness and finite fibres make it a finite covering space.

There is a canonical algebra map $\mathcal B^{\mathrm{an}}\longrightarrow q_*\mathcal O_{Y^{\mathrm{an}}}, \tag{3.5}$ given by viewing algebraic sections as holomorphic functions on the inverse image. Near $x$, both sides are locally free analytic modules of rank $n$. The map on residue fibres is $\mathcal B_x\otimes_R\mathbf C\to\mathbf C^n$, evaluation at the $n$ points of the finite étale fibre. This is an isomorphism. A matrix for (3.5) has determinant invertible at $x$, so it is an isomorphism near $x$. As every analytic point is such an $x$, (3.5) is globally an algebra isomorphism. Thus $q_*\mathcal O_{Y^{\mathrm{an}}}\simeq\mathcal E$. The product factors described after (2.2) recover an analytic cover isomorphism $Y^{\mathrm{an}}\simeq E$ over the base. This establishes essential surjectivity without using the general Riemann existence theorem.

For full faithfulness, an analytic morphism of two covers corresponds to an analytic algebra homomorphism in the reverse direction. GAGA lifts its underlying module map uniquely; faithfulness lifts the multiplication and unit identities as in (3.2). Relative spectrum then gives the unique algebraic cover morphism. Analytification recovers the given map by the same product-factor description. $\square$

4. Riemann existence, gluing and reduction

Riemann existence input. For a scheme $X$ of finite type over $\mathbf C$, analytification gives an equivalence $\operatorname{F\acute Et}(X) \simeq \{\text{finite étale analytic spaces over }X^{\mathrm{an}}\} \simeq \{\text{finite covering spaces of }X(\mathbf C)\}. \tag{4.1}$ All morphisms over $X$ are included. There is no normality, smoothness, reducedness or properness hypothesis. We state the first equivalence as the permitted comparison theorem; Proposition 2.1 proves the analytic-to-topological equivalence. Stacks, Tag 0HFL

Theorem 3.1 supplies an independent proof of (4.1) in the proper case from GAGA. In the general case, essential surjectivity of analytification is the substantive existence input. Giving a topological cover an analytic structure does not itself algebraize it.

Proposition 4.1. The equivalence (4.1) holds for $X$ locally of finite type over $\mathbf C$, including non-quasi-compact $X$.

Proof. Cover $X$ by affine opens $U_i$. Each is of finite type over $\mathbf C$, hence Noetherian. The intersection $U_i\cap U_j$, as an open subset of the Noetherian affine $U_i$, is quasi-compact and locally of finite type. It is therefore of finite type over $\mathbf C$. The same argument applies to triple intersections.

Restrict a finite topological cover to every $U_i(\mathbf C)$ and algebraize it by (4.1). On an overlap the resulting analytic covers have the specified identification through the original cover. Full faithfulness of (4.1) gives a unique algebraic identification. On triple overlaps these identifications satisfy the cocycle identity because their analytifications do and the functor is faithful. Zariski gluing gives a scheme $Y\to X$; its restriction to every $U_i$ is finite étale. Finiteness and étaleness are local on the target, so $Y\to X$ is finite étale. Its analytification is the original cover by the glued identifications.

A continuous cover morphism restricts to all the charts. The unique algebraic maps furnished by full faithfulness agree on overlaps and hence glue. Their uniqueness also glues. This proves both directions on all objects and all morphisms. There was no need to take a finite affine cover of $X$. $\square$

The passage to a nonreduced scheme is equally precise. The map $X_{\mathrm{red}}\to X$ is a universal homeomorphism, so topological invariance, proved in the proper-schemes lesson gives $\operatorname{F\acute Et}(X)\simeq \operatorname{F\acute Et}(X_{\mathrm{red}}). \tag{4.2}$ Stacks, Tag 0BQN

Their complex-point spaces are equal with the same topology. Thus the reduced Riemann existence statement implies the arbitrary one, on categories. The analytic spaces themselves need not be equal: their structure sheaves retain different nilpotents. Proposition 2.1 restores the appropriate sheaf on a cover of each analytic base.

5. Profinite completion is the entire comparison

Let $X$ be connected of finite type over $\mathbf C$, choose $x\in X(\mathbf C)$, and let $\bar x:\operatorname{Spec}\mathbf C\to X$ be that geometric point.

We first justify the topology needed for covering-space classification. In an affine chart the reduced underlying classical space is a closed real algebraic set: take real and imaginary parts of its defining polynomials. The local conic structure theorem for semialgebraic sets gives a basis of contractible neighbourhoods, including at singular points. An isolated point has its singleton as such a neighbourhood. This is a stated topology input; see Coste, An introduction to semialgebraic geometry, Theorem 4.4, page 63. Its norm-preserving cone description restricts to open balls, so these are neighbourhoods rather than merely closed neighbourhoods. Hence $X(\mathbf C)$ is locally path connected and semilocally simply connected.

It is also connected. Otherwise a nontrivial clopen partition would define a section of the trivial two-sheeted topological cover selecting one sheet on each part. By full faithfulness in (4.1), this section would be the analytification of a section $X\to X\amalg X$. The inverse images of the two components would be a nontrivial clopen partition of the connected scheme $X$, a contradiction. Local path connectedness now makes $X(\mathbf C)$ path connected. This argument also works for a nonseparated $X$, since the local charts and cover equivalence suffice.

Put $G=\pi_1^{\mathrm{top}}(X(\mathbf C),x)$. Covering-space classification identifies finite covers, with their fibre over $x$, with finite left $G$-sets, with their underlying set. The construction and its full faithfulness were given in the covering-space section of the Galois-category lesson; the topology prerequisite is the usual lifting and classification theorem.

Every action on a finite set factors through a quotient $G/N$ for a normal subgroup $N$ of finite index: take the kernel of the permutation representation. Define $\widehat G=\varprojlim_{N\triangleleft G,\ [G:N]<\infty}G/N. \tag{5.1}$ The finite $G$-sets are therefore exactly the finite continuous $\widehat G$-sets, and their equivariant maps are the same. We verify that no automorphisms of the fibre functor have been lost. A natural automorphism of the underlying-set functor on finite $G$-sets commutes with all right translations on the regular left $G$-set $G/N$. It must consequently be left translation by a unique $g_N\in G/N$. Naturality for quotient maps makes the $g_N$'s compatible. Conversely a compatible system acts naturally on every finite $G$-set. Indeed, for a set $S$ on which $N$ acts trivially, the equivariant maps $G/N\longrightarrow S,\qquad hN\longmapsto h\cdot s$ show that the natural automorphism sends $s$ to $g_N\cdot s$. This also proves uniqueness on every object. The topology determined by finite fibres is precisely the inverse-limit topology in (5.1).

Riemann existence identifies this forgetful functor with the geometric fibre functor on $\operatorname{F\acute Et}(X)$. Their automorphism groups thus give the promised theorem: $\boxed{\ \widehat{\pi_1^{\mathrm{top}}(X(\mathbf C),x)} \xrightarrow{\sim}\pi_1^{\mathrm{\acute et}}(X,\bar x)\ }. \tag{5.2}$ Stacks, Tag 0HFM

The isomorphism is canonical for the chosen point and fibre identification. It is compatible with maps of pointed complex schemes: lifting a loop in a pulled-back cover is the same as lifting its image in the original cover, so the monodromy squares commute on every finite fibre. Changing a connecting path between base points changes the identification by inner conjugation, as in the algebraic theory.

There is a dense homomorphism $G\to\pi_1^{\mathrm{\acute et}}(X,\bar x)$, given by monodromy on finite algebraic covers. Density follows because each projection $G\to G/N$ is onto, and finitely many such projections factor through one common finite quotient. The theorem does not assert that this homomorphism is injective. Its kernel is the intersection of the finite-index normal subgroups of $G$; the isomorphism in (5.2) is with the completion.

6. Punctured spheres and compact curves

6.1. Removing points from the projective line

Let $D=\{p_1,\ldots,p_n\}\subset\mathbf P^1(\mathbf C)$ consist of $n$ distinct points, with $n\ge1$. Then $\pi_1^{\mathrm{\acute et}}(\mathbf P^1_{\mathbf C}\setminus D) \simeq\widehat F_{n-1}, \tag{6.1}$ where $F_r$ is the abstract free group on $r$ generators and $\widehat F_r$ its profinite completion.

Proof. A Möbius transformation puts $p_n$ at infinity. The classical space is the plane with $m=n-1$ points removed. Choose a large closed disk containing those points in its interior, and disjoint small disks about them. Radial deformations outside the large disk and inside each punctured small disk reduce the space to a disk with $m$ open disks removed.

This planar surface has a graph spine consisting of the $m$ inner boundary circles and a tree joining them to a base point. To see the construction, choose disjoint cutting arcs from the inner boundaries to the outer boundary, meeting a common connecting tree. Cutting along the arcs gives a disk. Here is a precise collapse argument. Triangulate with the proposed spine as a subcomplex. In the dual graph join adjacent triangles across edges outside the spine, and add a root for the outer boundary. Cutting to a disk makes this dual graph connected. Choose a spanning tree rooted at the outer boundary, and process triangles with parents before children. Each triangle then has its edge towards its parent exposed as a free edge; collapsing that triangle across that edge never removes a spine edge. After all triangles have been removed, a connected graph remains. Filling the $m$ holes attaches $m$ disks and gives a disk, so the surface has Euler characteristic $1-m$, equal to that of the proposed spine. Consequently the remaining graph has no extra cycles: its extra edges form trees attached to the spine and collapse onto it. These elementary collapses prove the deformation retraction. Contract the tree in this graph. The result is a wedge of $m$ circles. Applying van Kampen successively to these circles with small tree neighbourhoods shows that its group is the free product of $m$ copies of $\mathbf Z$, with no relations. For $m=0$ the plane is contractible.

Choose the paths and their order so that the small positively oriented peripheral loops $c_1,\ldots,c_n$ satisfy $c_1\cdots c_n=1. \tag{6.2}$ For the $m$ finite punctures these are the generators of the spine. The outer boundary, positively oriented in the plane, represents $c_1\cdots c_m$; a positive loop about infinity has the opposite orientation in that plane. Thus $c_n=(c_1\cdots c_m)^{-1}$. There is precisely this one relation in the $n$-generator description, since eliminating $c_n$ recovers the free group just computed. Apply (5.2). $\square$

The hypothesis $n\ge1$ matters. For $n=0$, $\mathbf P^1(\mathbf C)$ is a sphere and is simply connected: van Kampen on two enlarged hemispheres kills the fundamental group of their annular intersection. Its étale group is trivial, in agreement with the algebraic proof in the projective-line example. There is no “free group of rank $-1$.”

For $n=2$, the punctured line is $\mathbf G_{m,\mathbf C}$, so $\pi_1^{\mathrm{\acute et}}(\mathbf G_{m,\mathbf C},1) \simeq\widehat{\mathbf Z}. \tag{6.3}$ Positive winding around zero supplies a generator of the discrete group before completion. The maps $z\mapsto z^d$, $d\ge1$, are connected finite étale covers of degree $d$. Their monodromy is the reduction map $\mathbf Z\to\mathbf Z/d\mathbf Z$, so every finite-index subgroup of $\mathbf Z$, which is $d\mathbf Z$ for some $d$, is represented. These are all the connected finite étale covers, up to isomorphism over $\mathbf G_m$.

For $n=3$, take $D=\{0,1,\infty\}$. Its group is $\widehat F_2$; the peripheral generators obey $c_\infty=(c_0c_1)^{-1}$. In contrast with $\widehat{\mathbf Z}$, this group has nonabelian finite quotients. For example $c_0\mapsto(12)$ and $c_1\mapsto(123)$ generate $S_3$.

6.2. Smooth projective complex curves

Let $C/\mathbf C$ be a smooth projective connected curve and let $g=\dim_{\mathbf C}H^1(C,\mathcal O_C)$ be its algebraic genus. Then $\begin{aligned} \pi_1^{\mathrm{\acute et}}(C)&\simeq\widehat{\Gamma_g},\\ \Gamma_g&=\langle a_1,b_1,\ldots,a_g,b_g\mid r_g=1\rangle,\\ r_g&=\prod_{i=1}^g[a_i,b_i]. \end{aligned} \tag{6.4}$ with $[a,b]=aba^{-1}b^{-1}$. The empty presentation at $g=0$ is the trivial group.

Proof. Smoothness makes $C^{\mathrm{an}}$ a complex one-dimensional manifold. Projectivity makes it compact, since it is closed in compact complex projective space. It is connected by Section 5. Holomorphic transition maps preserve the real orientation, their real Jacobian determinant being the positive number $|f'|^2$. Thus it is a compact connected orientable surface.

We specify the analytic and topology inputs that identify its genus. Classification of compact orientable surfaces gives a surface with $h$ handles. Its polygon model has one vertex, $2h$ edges, and one face, whose boundary word is $\prod_{i=1}^h a_i b_i a_i^{-1}b_i^{-1}$. The cellular boundary of the face is zero after abelianization, so its first Betti number is $2h$. The Hodge–Dolbeault theorem for a compact Riemann surface gives $H^1(C^{\mathrm{an}},\mathbf C) \simeq H^0(C^{\mathrm{an}},\Omega^1) \oplus H^1(C^{\mathrm{an}},\mathcal O), \qquad \dim H^0(\Omega^1)=\dim H^1(\mathcal O). \tag{6.5}$ The two dimensions agree by conjugation of the harmonic $(1,0)$ and $(0,1)$ forms. These are precisely stated analytic prerequisites; the GAGA application to Betti numbers likewise records Hodge decomposition as an external input. Coherent cohomology comparison in (3.1) gives $\dim H^1(C^{\mathrm{an}},\mathcal O)=g$. Hence $2h=2g$, and $h=g$. This identifies the algebraic genus with the number of topological handles.

The one-skeleton of the polygon model is a wedge of $2g$ circles and has free fundamental group on its edge loops. Attaching the face kills its boundary word and all conjugates, and kills no further elements. Here is the van Kampen justification of that last assertion. In the attaching disk of radius one, let the first open set be the whole surface with the closed central disk of radius $1/3$ removed, and let the second be the open disk of radius $2/3$. The first retracts to the one-skeleton by radial projection of its collar; the second is contractible. Their intersection is the annulus $1/3<r<2/3$, which retracts to a circle. Its generator maps to the displayed boundary word. Van Kampen gives exactly the quotient by the normal subgroup generated by that word. Thus the topological group is $\Gamma_g$. See also Hatcher, Algebraic Topology, Proposition 1.26 and the surface computation, pages 50–51. Applying (5.2) proves (6.4). $\square$

The profinite presentation can also be written $\widehat{\Gamma_g} \simeq \widehat F_{2g}/ \overline{\left\langle\!\left\langle \prod_{i=1}^g[a_i,b_i] \right\rangle\!\right\rangle}. \tag{6.6}$ The bar means the closed normal subgroup generated by the relator. To justify the completion in this formula, finite continuous quotients of its right side are exactly homomorphisms from $F_{2g}$ to finite groups that kill the relator. These are exactly finite quotients of $\Gamma_g$. The canonical map from the right side to $\widehat{\Gamma_g}$ is surjective because its compact image contains the dense image of $\Gamma_g$. Its kernel is trivial because a nontrivial element of a profinite group survives in a finite continuous quotient. Thus no abstract, nonclosed relation subgroup is being substituted.

For an elliptic curve $E/\mathbf C$, $g=1$, so $\Gamma_1=\mathbf Z^2$ and $\pi_1^{\mathrm{\acute et}}(E)\simeq\widehat{\mathbf Z}^{\,2}. \tag{6.7}$ Indeed every finite-index subgroup $N\subset\mathbf Z^2$ contains $d\mathbf Z^2$ for the exponent $d$ of $\mathbf Z^2/N$. These subgroups are cofinal for completion, and $\varprojlim_d(\mathbf Z/d\mathbf Z)^2=\widehat{\mathbf Z}^{\,2}$. Its finite quotients are exactly the finite abelian groups generated by at most two elements. The displayed identification depends on the choice of the two oriented surface loops.

6.3. Every algebraically closed field of characteristic zero

Theorem 6.1. If $k$ is any algebraically closed field of characteristic zero and $C/k$ is a smooth projective connected curve of genus $g$, then $\pi_1^{\mathrm{\acute et}}(C)\simeq\widehat{\Gamma_g}. \tag{6.8}$

Proof. A projective embedding, finitely many homogeneous equations, and the finite presentation of $C$ use finitely many elements of $k$. They are contained in a subfield $K\subset k$ finitely generated over $\mathbf Q$. Descend the embedding and equations to a projective scheme $C_0/K$ with $C_0\times_K k\simeq C$. Smoothness descends along the faithfully flat field extension, and dimension is unchanged. Let $k_0$ be the algebraic closure of $K$ inside $k$. It is algebraically closed. The surjection $C_0\times_K k\to C_0\times_K k_0$ has connected source, so its target is connected. Thus $C_0$ is a geometrically connected smooth projective curve.

There is an embedding $K\hookrightarrow\mathbf C$: send a finite transcendence basis to algebraically independent complex numbers, and extend to the finite algebraic extension. Algebraic closedness of $\mathbf C$ extends this embedding to $k_0\hookrightarrow\mathbf C$. Put $C_* = C_0\times_K k_0$. Proper field-extension invariance, proved in the proper-schemes lesson and recorded in Stacks, Tag 0A49, gives equivalences $\operatorname{F\acute Et}(C_*) \xrightarrow{\sim}\operatorname{F\acute Et}(C), \qquad \operatorname{F\acute Et}(C_*) \xrightarrow{\sim}\operatorname{F\acute Et}(C_*\times_{k_0}\mathbf C). \tag{6.9}$ Choose a $k_0$-point of $C_*$, which exists since $k_0$ is algebraically closed, and its two base changes. These equivalences identify the fibre functors and hence their fundamental groups.

The complex curve has the same genus as $C$. In fact for any field extension $L/K$, $H^1(C_0,\mathcal O_{C_0})\otimes_K L \simeq H^1(C_0\times_K L,\mathcal O). \tag{6.10}$ One can see this directly with a finite affine cover of the separated projective $C_0$: its affine intersections compute coherent cohomology by the Čech complex, extension of scalars gives the base-changed complex, and flatness of $L$ makes tensor commute with cohomology. This is flat base change, Stacks, Tag 02KH. Consequently the dimension of $H^1$ is unchanged. Apply (6.4) to the complex curve in (6.9). Base-point independence gives the assertion for any geometric point of $C$, up to the usual choice of path. $\square$

This proof embeds the finitely generated field of definition, not the entire field $k$. An algebraically closed characteristic-zero field can have cardinality larger than $\mathbf C$, so an embedding $k\hookrightarrow\mathbf C$ is not available in general. Properness in (6.9) is also essential: it is the field-invariance theorem's hypothesis, rather than a consequence of characteristic zero.

7. Exercises and complete solutions

7.1. Two basic groups — easy

Use the comparison theorem to compute the étale fundamental groups of $\mathbf G_{m,\mathbf C}$ and of an elliptic curve over $\mathbf C$. Explain why completion of the latter discrete group is a product of two completions.

Solution. The homotopy $z\mapsto ((1-s)+s/|z|)z$, $0\le s\le1$, retracts $\mathbf C^\times$ onto the unit circle. Its loop group is $\mathbf Z$, measured by winding number, so comparison gives $\widehat{\mathbf Z}$. The covering $z\mapsto z^d$ realizes the quotient $\mathbf Z/d\mathbf Z$; choosing the point $1$ over $1$ fixes this description.

For an elliptic curve, GAGA identifies its one-dimensional $H^1(\mathcal O)$ with analytic $H^1(\mathcal O)$. Formula (6.5) gives first Betti number two, so the orientable surface classification makes its analytification a torus. Its square model has generators $a,b$ and face word $aba^{-1}b^{-1}$. Van Kampen gives $\mathbf Z^2$. For any finite-index subgroup $N$, the finite group $\mathbf Z^2/N$ has an exponent $d$, so $d\mathbf Z^2\subset N$. Taking the inverse limit over these cofinal subgroups gives $\widehat{\mathbf Z^2} =\varprojlim_d(\mathbf Z/d\mathbf Z)^2 =\widehat{\mathbf Z}\times\widehat{\mathbf Z}.$ Comparison therefore gives $\widehat{\mathbf Z}^{\,2}$. These are identifications after the stated choices of generators.

7.2. Three punctures and finite Galois covers — medium

Compute the group of $U=\mathbf P^1_{\mathbf C}\setminus\{0,1,\infty\}$. Describe its finite quotients as connected Galois covers, and explain exactly what “branched at three points” means on compactifying them.

Solution. Cutting the twice-punctured plane gives a spine with two circles. Its topological group is free on positively oriented loops $c_0,c_1$, and the third peripheral loop is $c_\infty=(c_0c_1)^{-1}$. Hence $\pi_1^{\mathrm{\acute et}}(U)=\widehat F_2. \tag{7.1}$ A continuous surjection to a finite group $G$ is exactly a pair $\sigma_0,\sigma_1\in G$ generating $G$. Put $\sigma_\infty=(\sigma_0\sigma_1)^{-1}$. The regular left $G$-set, with this monodromy, defines a connected topological cover: its orbits are the orbits of the subgroup generated by the pair, which is all of $G$. Riemann existence algebraizes it to a connected finite étale cover $Y\to U$ of degree $|G|$. Right translations give its $|G|$ deck transformations. Identifying $g$ with right translation by $g^{-1}$ identifies the deck group with $G$, removing the usual opposite-group convention. Full faithfulness makes all these transformations algebraic.

Let $L=\mathbf C(Y)$. Since $U$ is smooth and the cover is connected finite étale, $Y$ is a smooth integral curve. The degree-$|G|$ separable extension $L/\mathbf C(t)$ has $|G|$ automorphisms, and is Galois. Normalize $\mathbf P^1$ in $L$. The finite-normalization theorem for curves of finite type over a field gives a finite map $\bar Y\longrightarrow\mathbf P^1_{\mathbf C}. \tag{7.2}$ The normalization is a normal proper curve, hence a smooth projective curve over $\mathbf C$. Its inverse image over $U$ is $Y$, by uniqueness of normalization there. The deck transformations extend through integral closure. Thus (7.2) is a Galois cover, unramified outside $\{0,1,\infty\}$.

At a point over a puncture, choose a local analytic parameter $u$. The map to a parameter $t$ at that puncture has the form $t=u^e v(u)$, with $v(0)\ne0$. On a small disk $v$ has a holomorphic $e$-th root; changing the parameter gives $t=u^e$. A positive loop about the puncture permutes these $e$ local sheets cyclically. In the regular $G$-set, every cycle of $\sigma_i$ has length $\operatorname{ord}(\sigma_i)$. Hence every point over that puncture has $e_i=\operatorname{ord}(\sigma_i), \tag{7.3}$ and its inertia subgroup is conjugate to $\langle\sigma_i\rangle$. Inverting a generator for a deck-action convention does not change this subgroup.

Consequently the branch locus is contained in those three points. It contains a given point precisely when its $\sigma_i$ is nonidentity. For instance a cyclic quotient with $\sigma_0$ a generator and $\sigma_1=1$ gives $z\mapsto z^d$, branched only at $0,\infty$ for $d>1$; the trivial quotient has no branch points. The phrase “branched at three points” must allow this smaller branch locus, or explicitly require all three orders to exceed one.

Conversely a connected Galois cover of $\mathbf P^1_{\mathbf C}$ whose branch locus is contained in these points restricts to a connected finite étale cover of $U$. Its monodromy gives a surjection from $F_2$ to its finite deck group, after choosing the fibre identification, and thus a continuous quotient of $\widehat F_2$. This reverses the construction. With a chosen marking the two elements determine the cover; changing the chosen fibre point simultaneously conjugates them. Forgetting the marking of $G$ also permits automorphisms of $G$. In particular the finite quotient groups are exactly the finite groups that can be generated by two elements, rather than all finite groups.

As a concrete nonabelian example, the pair $(12),(123)$ generates $S_3$; the three orders are $2,3,2$. Thus all three points branch. Riemann–Hurwitz, proved in the curve and arithmetic examples, gives $2g(\bar Y)-2 =6(-2)+6\left(\frac12+\frac23+\frac12\right)=-2.$ So this degree-six Galois cover has a genus-zero compactification.

7.3. Characteristic zero beyond the complex numbers — medium

Let $k$ be an arbitrary algebraically closed field of characteristic zero. Use proper field-extension invariance to compute the group of a smooth projective connected genus-$g$ curve $C/k$. Account for a field $k$ that cannot embed in $\mathbf C$.

Solution. Choose a projective model $C_0$ over a finitely generated subfield $K/\mathbf Q$ of $k$, using the finitely many coefficients of an embedding and its equations. Faithfully flat descent gives smoothness, and the algebraic closure $k_0$ of $K$ inside $k$ makes $C_*=C_0\times_K k_0$ a connected smooth projective curve: it is the surjective image of the connected $C$. Embed $K$ in $\mathbf C$ by choosing images of a transcendence basis, and extend the embedding to $k_0$. Apply proper field-extension invariance, Tag 0A49 to $k_0\subset k$ and to $k_0\hookrightarrow\mathbf C$.

These two equivalences of finite-cover categories identify their fibre functors after choosing a point of $C_*$, so the three geometric fundamental groups are isomorphic. Flat cohomology base change (6.10) preserves genus under both extensions. The complex curve is therefore the genus-$g$ orientable surface of Section 6.2, whose polygon gives the group $\Gamma_g=\langle a_i,b_i\mid\prod_i[a_i,b_i]=1\rangle$. Comparison gives $\widehat{\Gamma_g}$ for that curve, and hence for $C$. A different base point changes the identification only through a chosen path.

Only $K$ and $k_0$ were embedded in $\mathbf C$. Nothing requires an embedding of the possibly much larger field $k$. Both uses of the field-invariance theorem are valid because $C_*$ is proper.

7.4. Restore the analytic structure — medium

Show directly that a finite topological cover of $X(\mathbf C)$, for $X$ locally of finite type over $\mathbf C$, has a unique analytic-space structure making it finite étale over $X^{\mathrm{an}}$. Include cover morphisms and nonreduced bases.

Solution. On an evenly covered open $V$, transport its analytic sheaf onto each of the finitely many sheets $V_j$ through the homeomorphism $p|_{V_j}:V_j\to V$. The overlap transition expressed on the base is the identity on an open set; it is an analytic isomorphism, and its identities on triple overlaps are literal identities. Thus the sheaves glue to $p^{-1}\mathcal O_{X^{\mathrm{an}}}$ on the prescribed space $E$. Locally $p$ is a disjoint union of analytic isomorphisms. A finite covering map is closed with finite fibres, hence proper by the compactness argument in Proposition 2.1, so this is a finite étale analytic map.

If another analytic structure makes $p$ a local analytic isomorphism, the structural map identifies its sheaf with the same transported sheaf on every sheet. These identifications agree on overlaps because both lie over the identity of the base. This proves uniqueness of the entire locally ringed space over $X^{\mathrm{an}}$. For a continuous map of covers, continuity restricts a sufficiently small source sheet to a target sheet; in base coordinates the map is identity. It is analytic, and its map on sheaves is forced. This proves the same assertion for all morphisms.

When the base has nilpotents, the transported sheaf has those nilpotents on every sheet. Replacing it by reduced holomorphic functions would make the map fail to be a local isomorphism over the nonreduced base. The unique structure therefore keeps the complete analytic sheaf. The argument uses local charts and works for the glued non-Hausdorff analytification of a nonseparated scheme as well.

8. Inputs and reading

The owned arguments here are the analytic-cover construction with all morphisms, the proper GAGA algebraization of covers, gluing to locally finite type, the fibre-functor proof of profinite comparison, the punctured-sphere and surface-group computations, and the passage to every algebraically closed characteristic-zero field. Their explicit prerequisites are:

Local analytic algebra and coherence, the structure of analytification, and the analytic local-isomorphism criterion Tag 0HFE. The defining-ideal and completion arguments in Section 1 explain the use on nonreduced charts.
Proper-scheme GAGA as stated in (3.1), with its precise Hall reference. The projective GAGA theorem supplies the cohomology comparison used for complex projective curves.
General finite type Riemann existence Tag 0HFL, stated in (4.1). Its confirmed fundamental-group consequence Tag 0HFM is proved in Section 5.
Covering-space classification, van Kampen and the classification of compact orientable surfaces; local semialgebraic conic structure in Section 5; and the precise compact Riemann-surface Hodge–Dolbeault statement (6.5). The surface polygon and the attaching-word calculation are developed here. Hatcher's Sections 1.2–1.3 provide the topology references.
Finite-projective module descent from the descent lesson, Section 3, the finite flat étale criterion, and finite normalization of algebraic curves. Riemann–Hurwitz and the covering-category interpretation of function-field extensions were developed in the étale fundamental-group lesson.
Topological invariance and proper algebraically closed field-extension invariance from the preceding course proofs, with Tags 0BQN and 0A49. Formula (6.10) uses affine Čech cohomology and flat base change, Tag 02KH.

The characteristic-zero genus-one computation (6.7) supplies the generic-fibre input in the preceding lesson's Legendre degeneration.