Loops on schemes and the algebraic fundamental group

Abstract

In this note we give a re-interpretation of the algebraic fundamental group for proper schemes that is rather close to the original definition of the fundamental group for topological spaces. The idea is to replace the standard interval from topology by what we call interval schemes. This leads to an algebraic version of continuous loops, and the homotopy relation is defined in terms of the monodromy action. Our main results hinge on Macaulayfication for proper schemes and Lefschetz type results.

1 Introduction

The fundamental group $\pi _1(X,x_0)$ of a topological space X with respect to a base point $x_0$ is an invariant of great significance, even more so as its definition is elementary and intuitive: the elements are loops up to homotopy, where a loop is a continuous morphism $I\rightarrow X$ from the standard interval $I=[0,1]$, such that the end points are mapped to the base point $x_0$. Roughly speaking, two loops are homotopic if one can be deformed to the other, respecting the base point. For connected and locally simply-connected spaces X, one may interpret the fundamental group also as the group of deck transformations of the universal covering ${\tilde{X}}\rightarrow X$.

In the realm of algebraic topology, where one works with schemes rather than topological spaces, the first construction above makes little sense. However, Grothendieck [9] realized that the second description has an analog in algebraic geometry. He introduced the notion of a Galois category $\mathcal {C}$, the objects of which should be considered as abstract finite coverings, which admits a fiber functor $\Phi :\mathcal {C}\rightarrow ({{\,\textrm{FinSet}\,}})$ to the category of finite sets satisfying certain properties. These properties ensure that the automorphism group of $\Phi$ is equal to the opposite group of the automorphisms of an abstract pro-finite universal covering. The algebraic fundamental group $\pi _1^\textrm{alg}(X,x_0)$ of a connected scheme X with respect to a geometric point $x_0$ is then defined by applying this general construction to the category $({{\,\textrm{FinEt}\,}}/X)$ of finite étale coverings of X, with fiber functor given by base-changing along $x_0$.

If X is of finite type over the complex numbers, the group $\pi _1^\textrm{alg}(X,x_0)$ equals the pro-finite completion of the classical fundamental group of $X(\mathbb {C})$, endowed with the classical topology. If $X={\text {Spec}}(K)$ is the spectrum of a field and $x_0$ is given by some separable closure, $\pi _1^\textrm{alg}(X,x_0)$ gives back the corresponding Galois group.

In this note we observe that the original construction of the fundamental group using loops has a meaningful analogy for schemes, once the notions of intervals and loops are interpreted in an algebraic manner. More precisely, the crucial properties of the interval $I=[0,1]$ in the construction above are: I is connected, quasi-compact, one dimensional, with no non-trivial coverings, and is endowed with two distinguished points. Translating these properties to algebraic geometry we define in Sect. 3interval schemes as reduced, connected, affine, one-dimensional schemes I, which have no non-trivial finite étale coverings and contain two distinguished closed points with separably closed residue fields. The latter are called end points. An algebraic loop on a scheme X based at a geometric point $x_0$ is a morphism of schemes $I\rightarrow X$ mapping the end points to the base point.

Algebraic loops define monodromy transformations. We call two algebraic loops homotopic if the resulting monodromies agree. The algebraic loop group $\pi _0\Omega ^\textrm{alg}(X,x_0)$ is defined as the set of homotopy classes of algebraic loops; the group structure is induced by concatenating algebraic loops, see Sect. 4.

Interval schemes are very often non-noetherian. One example of an interval scheme is the universal Galois covering (introduced by Grothendieck as a pro-object) of a noetherian, connected, affine, reduced, and one dimensional scheme. Such universal Galois coverings were systematically studied in [24], where Vakil and Wickelgren define the fundamental group scheme using universal coverings, which are certain pro-finite étale maps. The notion of interval scheme introduced above is also inspired by their work. But there are many other examples of interval schemes, which are more direct to obtain. For example, if R is an integral noetherian one-dimensional ring and A is its integral closure in the separable closure of ${\text {Frac}}(R)$, then the choice of two closed geometric points in ${\text {Spec}}(A)$ turns it into an interval scheme.

By construction, the monodromy induces an injective homomorphism

$$\begin{aligned} \pi _0\Omega ^\textrm{alg}(X,x_0)^{\textrm{op}}\longrightarrow \pi _1^\textrm{alg}(X,x_0), \end{aligned}$$

(1)

of groups, where “$\textrm{op}$” refers to the opposite group structure. The main result of this note is the following, see Theorem 4.4:

Theorem

Let X be a connected scheme that is separated and of finite type over a ground field k, endowed with a geometric point $x_0:{\text {Spec}}(k^{{\text {sep}}})\rightarrow X$. Then the injection (1) has dense image. It is actually bijective, provided that X is proper.

The main step in the proof of the above theorem for proper X is a Lefschetz type result saying that for a proper and connected k-scheme X we find a closed connected curve $C \subset X$ such that the algebraic fundamental group of C surjects to the one of X, see Proposition 5.5. This is well-known in the case where X is Cohen–Macaulay and projective over a field, see [10], Exposé XII. We reduce the general situation to this using a van-Kampen-like argument and Macaulayfication, which was in a special case constructed by Faltings [4] and in full generality by Kawasaki [18]. For further results on Macaulayfication, see the recent work of Česnavičius [2]. The proof of Theorem 4.4 is given in Sect. 7. We do not expect the map (1) to be an isomorphism for affine schemes of finite type over a field in general.

2 Monodromy

Let Y be a scheme, and write $({{\,\textrm{FinEt}\,}}/Y)$ for the category of Y-schemes X whose structure morphism $f:X\rightarrow Y$ is finite and étale. Note that such an f is proper, affine, flat, of finite presentation, and for each point $b\in Y$ the fiber $f^{-1}(b)$ is the spectrum of some étale algebra over the residue field $k=\kappa (b)$. For the following result, see for example [20], Theorem 5.10 and Exercise 5.21.

Proposition 1.1

Suppose Y is connected, and $f:X\rightarrow Y$ finite and étale. Then there is a surjective finite étale morphism $Y'\rightarrow Y$ such that $X'=X\times _Y Y'$ is isomorphic over $Y'$ to the disjoint union $\coprod _{i=1}^r Y'$ for some integer $r\geqslant 0$.

This has an immediate consequence:

Corollary 1.2

Suppose Y is connected, and $f:X\rightarrow Y$ finite and étale. Then X has only finitely many connected components $U\subset X$, each of which is open-and-closed. Moreover, the induced morphism $U\rightarrow Y$ is finite and étale.

Proof

Take $Y'\rightarrow Y$ as in Proposition 1.1. Since the projection $X'\rightarrow X$ is surjective, the connected components of X are images of the connected components of $X'$, hence there are only finitely many. This implies that the connected components U of X are open and closed and hence the composition $U\hookrightarrow X\rightarrow Y$ is étale and finite. $\square$

The proposition tells us that the Y-scheme X is a twisted form of the disjoint union $\coprod _{i=1}^r Y$, with respect to the étale topology. It thus corresponds to a class in the non-abelian cohomology set $H^1(Y,S_r)$, with coefficients in the symmetric group $S_r$ on $r\geqslant 0$ letters ([8], Chapter III, Section 2.3). To summarize:

Proposition 1.3

If Y is connected, the following are equivalent:

• (i) Every finite étale Y-scheme is isomorphic to some $\coprod _{i=1}^r Y$, $r\geqslant 0$.

• (ii) We have $H^1(Y,S_r)=\{*\}$ for all integers $r\geqslant 0$.

• (iii) Each finite étale morphism $X\rightarrow Y$ from a non-empty connected scheme X is an isomorphism.

Definition 1.4

We say a connected scheme Y is simply connected, if it satisfies the equivalent conditions of Proposition 1.3.

Let X be a Y-scheme, with structure morphism $f:X\rightarrow Y$, and $a:A\rightarrow Y$ be some other morphism. To simplify notation, we write

$$\begin{aligned} X(A)={\text {Hom}}_Y(A,X)=\{a':A\longrightarrow X\mid f\circ a'=a\} \end{aligned}$$

of liftings of $a:A\rightarrow Y$ with respect to $f:X\rightarrow Y$.

Proposition 1.5

Suppose that Y is connected, with $H^1(Y,S_r)=\{*\}$ for all $r\geqslant 0$, and $f:X\rightarrow Y$ is finite and étale. Let $a:A\rightarrow Y$ and $b:B\rightarrow Y$ be morphisms with connected and non-empty domains. Then the sets X(A) and X(B) are finite, and for each $a'\in X(A)$ there is a unique $b'\in X(B)$ such that the images $a'(A)$ and $b'(B)$ lie in the same connected component of X.

Proof

By Proposition 1.3, we may assume $X=\coprod _{i=1}^r Y$, for some $r\geqslant 0$. The set X(A) is in bijection with the set of sections of $\coprod _{i=1}^r A= X\times _Y A\rightarrow A$, whence is finite, and we see that every $a'\in X(A)$ corresponds uniquely to one of the maps

given by including A into one of the r summands. This implies the statement. $\square$

1.6. In the situation of Proposition 1.5 we obtain a mapping

$$\begin{aligned} \mu _X:X(A)\longrightarrow X(B),\quad a'\longmapsto b', \end{aligned}$$

which is called the monodromy, and will play a crucial role throughout. We regard it as a natural transformation between $X\mapsto X(A)$ and $X\mapsto X(B)$, viewed as functors $({{\,\textrm{FinEt}\,}}/Y)\rightarrow ({{\,\textrm{FinSet}\,}})$. By Proposition 1.5, the monodromy $\mu _X$ is a natural isomorphism, and is given as the composition of the following natural bijections

$$\begin{aligned} \mu _X: X(A)\longrightarrow \pi _0(X\times _Y A)\longrightarrow \pi _0(X)\longrightarrow \pi _0(X\times _Y B)\longrightarrow X(B), \end{aligned}$$

where $\pi _0(X)$ denotes the set of connected components of X.

3 Galois categories

In this section we recall the notion of Galois categories, which were introduced by Grothendieck to unify Galois theory from algebra and covering space theory from topology ([9], Exposé V).

2.1. Recall that a category $\mathcal {C}$ is called a Galois category if there exists a functor $\Phi :\mathcal {C}\rightarrow ({{\,\textrm{FinSet}\,}})$ such that the following six axioms hold:

• (G1) Fiber products and final objects exist in $\mathcal {C}$.

• (G2) Finite sums and quotients by finite group actions exist in $\mathcal {C}$.

• (G3) Every morphism $X'\rightarrow X$ in $\mathcal {C}$ factors into a strict epimorphism $X'\rightarrow U$ and the inclusion of a direct summand $U\subset X$.

• (G4) The functor $\Phi$ commutes with fiber products and final objects.

• (G5) It also commutes with finite direct sums and forming quotients by finite group actions, and transforms strict epimorphisms into surjections.

• (G6) If for a morphism $u:X'\rightarrow X$ in $\mathcal {C}$ the resulting map $\Phi (u)$ is bijective, then u is an isomorphism.

One calls $\Phi$ a fundamental functor or fiber functor for the Galois category $\mathcal {C}$, and denotes by $\pi ={\text {Aut}}(\Phi )$ the group of natural isomorphisms of the fundamental functor to itself. In turn, we have an inclusion

$$\begin{aligned} \pi \subset \prod _{X\in \mathcal {C}} S_{\Phi (X)} \end{aligned}$$

inside a product of symmetric groups ${\text {Aut}}(\Phi (X))=S_{\Phi (X)}$. These groups are finite. We endow them with the discrete topology, and the product with the product topology. The latter becomes a topological group that is compact and totally disconnected. Such topological groups are also called pro-finite groups. One easily checks that the subgroup $\pi$ is closed, and thus inherits the structure of a pro-finite group. Every fiber functor on $\mathcal {C}$ is (non-canonically) isomorphic to $\Phi$ and hence, up to a uncanonical isomorphism, the pro-finite group $\pi$ depends only on the Galois category $\mathcal {C}$, and not on the choice of fiber functor $\Phi$.

Now write $(\pi \text {-FinSet})$ for the category of finite sets F endowed with a $\pi$-action from the left, where the kernel of the canonical homomorphism $\pi \rightarrow S_F$ is closed. In other words, the action $\pi \times F\rightarrow F$ is continuous, when the finite set F is endowed with the discrete topology. With respect to the natural $\pi$-action on the $\Phi (X)$, $X\in \mathcal {C}$, the fundamental functor becomes a functor

$$\begin{aligned} \Phi :\mathcal {C}\longrightarrow (\pi \text {-FinSet}), \end{aligned}$$

and Grothendieck deduced from the axioms (G1)–(G6) that the above is an equivalence of categories. Conversely, if G is a pro-finite group, the category $(G\text {-FinSet})$ is a Galois category: The functor $\Phi$ that forgets the G-action is a fundamental functor, and the resulting $\pi ={\text {Aut}}(\Phi )$ becomes identified with G.

One should see $\pi ={\text {Aut}}(\Phi )$ as a common generalization of the opposite Galois group ${\text {Gal}}(F^{{\text {sep}}}/F)^\textrm{op}$ for fields F, and the pro-finite completion ${\widehat{\pi }}_1(Y,y_0)$ of the fundamental group, say for connected and locally simply-connected topological spaces Y.

2.2. Let $H:\mathcal {C}\rightarrow \mathcal {C}'$ be an exact covariant functor between Galois categories. By [9, Exposé V, Proposition 6.1] the exactness is equivalent to the statement, that the composition $\Phi '\circ H$ is a fiber functor for $\mathcal {C}$, whenever $\Phi '$ is a fiber functor for $\mathcal {C}'$. An exact functor H induces a morphism $h:\pi '\rightarrow \pi$ between the corresponding fundamental groups with reversed direction, which is well-defined up to conjugation. The following statements are equivalent (see [9, Exposé V, Proposition 6.9])

• (1) $H: \mathcal {C}\rightarrow \mathcal {C}'$ is fully faithful.

• (2) $h:\pi '\rightarrow \pi$ is surjective.

• (3) For each connected object $X\in \mathcal {C}$, the object H(X) is connected.

2.3. Let Y be a connected scheme. Then $\mathcal {C}=({{\,\textrm{FinEt}\,}}/Y)$ becomes a Galois category: For each morphism $y_0:{\text {Spec}}(K)\rightarrow Y$, where K is a separably closed field, we get a fiber functor

$$\begin{aligned} \Phi _{y_0}:({{\,\textrm{FinEt}\,}}/Y)\longrightarrow ({{\,\textrm{FinSet}\,}}),\quad X\longmapsto X(K) \end{aligned}$$

given by the set of morphisms $b':{\text {Spec}}(K)\rightarrow X$ lifting the given $b:{\text {Spec}}(K)\rightarrow Y$. The resulting group

$$\begin{aligned} \pi _1^\textrm{alg}(Y,y_0) = \pi = {\text {Aut}}(\Phi _{y_0}) \end{aligned}$$

is called the algebraic fundamental group of the connected scheme Y with respect to $y_0$. As in topology, the latter is called base point.

4 Interval schemes

In algebraic topology, the standard interval $I=[0,1]$ plays a central role. From our perspective, the following are the crucial properties:

• (i) The topological space I is connected and quasi-compact.

• (ii) There are two distinguished points $0,1\in I$.

• (iii) The universal covering ${\tilde{I}}\rightarrow I$ is a homeomorphism.

• (iv) The interval I is one-dimensional.

In this section we introduce a class of schemes with analogous properties. Fix a separably closed field K.

Definition 3.1

An interval scheme with K-valued endpoints is a triple $(I,a_0,a_1)$, where I is a reduced, connected, simply connected, affine, and one-dimensional scheme and $a_i:{\text {Spec}}(K)\rightarrow X$ are two closed embeddings.

The point $a_0$ is called the left endpoint, whereas $a_1$ is the right endpoint. By abuse of notation, we usually write I for the interval scheme $(I,a_0,a_1)$. The simplest example of an interval scheme is $I=\mathbb {A}^1_k$ where k is an algebraically closed field of characteristic zero, and the end points are the rational points $a_0=0$ and $a_1=1$. However, if k is not algebraically closed or of positive characteristic, this will not be an interval scheme. In fact, interval schemes are very often non-noetherian. The following gives the most basic class of interval schemes:

Proposition 3.2

Let A be a one-dimensional integral ring that is normal and whose field of fractions $F={\text {Frac}}(A)$ is separably closed, and suppose there are two surjections $\varphi _i:A\rightarrow K$. Then $I={\text {Spec}}(A)$ becomes an interval scheme, where the endpoints $a_i$ correspond to the homomorphisms $\varphi _i$.

Proof

Let $X\rightarrow I$ be a connected finite étale covering. It follows that X is an affine connected normal scheme, and thus is integral, see [14], Proposition (17.5.7). In particular the function field L of X is a finite separable field extension $L/{\text {Frac}}(A)$. Since ${\text {Frac}}(A)$ is separably closed we find $L\cong {\text {Frac}}(A)$. Hence $X\rightarrow I$ is an isomorphism. Thus I has no non-trivial finite étale covering and therefore defines an interval scheme. $\square$

Example 3.3

Rings as in Proposition 3.2 easily occur as follows: Suppose that R is a one-dimensional noetherian ring, endowed with two integral homomorphisms $\psi _i:R\rightarrow K$. The latter means that each $\lambda \in K$ is the root of a monic polynomial with coefficients from R. Choose a separable closure $F^{{\text {sep}}}$ for the field of fractions $F={\text {Frac}}(R)$, and write $A=R^{{\text {sep}}}$ for the integral closure of $R\subset F^{{\text {sep}}}$. By construction A is integral and normal, with field of fractions $F^{{\text {sep}}}$, and the ring extension $R\subset A$ is integral. According to the Going-Up Theorem, the map ${\text {Spec}}(A)\rightarrow {\text {Spec}}(R)$ is surjective, thus the $\psi _i:R\rightarrow K$ extend to some homomorphisms $\varphi _i:R^{{\text {sep}}}\rightarrow K$. Proposition 3.2 yields that the scheme $I={\text {Spec}}(R^{{\text {sep}}})$ in an interval scheme, where the endpoints $a_i$ correspond to the homomorphisms $\varphi _i$.

Recall that a local ring R is called strictly henselian if each factorization $P\equiv P_1P_2$ into coprime polynomials over the residue field $k=R/\mathfrak {m}_R$ of a monic polynomial $P\in R[T]$ is induced by a factorization over R, and moreover the residue field is separably closed. See [14], Proposition (18.8.1) for the next example of an interval scheme, for which the image points of the two end points agree.

Proposition 3.4

Let A be a one-dimensional local ring that is strictly henselian, and whose residue field is isomorphic to K, and let $\varphi _i:A/\mathfrak {m}_A\rightarrow K$ be two isomorphisms. Then $I={\text {Spec}}(A)$ becomes an interval scheme, where the endpoints $a_i$ correspond to the homomorphisms $\varphi _i$.

3.5. Let I, J be two interval schemes, with K-valued endpoints $a_0,a_1$ and $b_0,b_1$, respectively. We write $I* J$ for the concatenation of I and J with respect to the right endpoint on I and the left endpoint on J. In other words, we have a cocartesian square in the category of schemes

(3.1)

Note that the cocartesian square above exits in the category of schemes by [6], Théorème 5.4, and is in fact also cartesian. The scheme $I*J$ comes with closed embeddings of I and J, and we take $a_0\in I\subset I\star J$ as new left endpoint, and $b_1\in J\subset I* J$ as new right endpoint.

Lemma 3.6

In the above situation, the concatenation $I*J$ is an interval scheme, with endpoints $a_0$ and $b_1$.

Proof

Write $I={\text {Spec}}(A)$ and $J={\text {Spec}}(B)$. Then $I* J={\text {Spec}}(A\times _ {K} B)$ is affine and reduced. Here the fiber product is formed with respect to the homomorphisms $A\rightarrow K\leftarrow B$ corresponding to the morphisms $a_1$ and $b_0$. By construction we have $I*J= I\cup J$ and $I\cap J={\text {Spec}}(K)$, where the latter can be viewed as the image of both $a_1$ and $b_0$. Hence $I*J$ is also one-dimensional and connected. Let $X\rightarrow I*J$ be a finite étale covering. Set $X(Z)={\text {Hom}}_{I*J}(Z,X)$, where Z is an $I*J$-scheme. By [14], Corollaire (17.9.4), the set $X(I*J)$ is in bijection with the connected components of X. By construction of $I*J$ the set $X(I*J)$ is the pushout of the maps $X(I)\leftarrow X(I\cap J)\rightarrow X(J)$ induced by the morphisms $a_1$ and $b_0$. Since I, J, and ${\text {Spec}}(K)$ are simply connected the monodromy (see 1.6) induces bijections

$$\begin{aligned} X(I)\cong X(a_1)\cong X(I\cap J)\cong X(b_0)\cong X(J). \end{aligned}$$

Hence $|X(I*J)|=|X(I\cap J)|=\deg (X_{I\cap J}\rightarrow I\cap J)=\deg (X\rightarrow I*J)$, where $X_{I\cap J}\rightarrow I\cap J$ denotes the base change of $X\rightarrow I*J$ along $I\cap J\hookrightarrow I*J$.

Thus any finite étale covering of $I*J$ is trivial and hence $I*J$ is an interval scheme. $\square$

5 The algebraic loop group

Fix some separably closed field K and let Y be a scheme, endowed with two K-valued points $y_i:{\text {Spec}}(K)\rightarrow Y$. We may regard this as an object in the category $(K^2/\textrm{Sch})$ of schemes endowed with two K-valued points.

Definition 4.1

An algebraic path in Y starting at $y_0$ and ending in $y_1$ is an interval scheme $(I,a_0,a_1)$ with K-valued endpoints, together with a morphism

$$\begin{aligned} w:(I,a_0,a_1)\longrightarrow (Y,y_0,y_1) \end{aligned}$$

of schemes endowed with two K-valued points. An algebraic path w is called algebraic loop if $y_0=y_1$.

By abuse of notation, we often write $w:I\rightarrow Y$ for the algebraic path $w:(I,a_0, a_1)\rightarrow (Y, b_0, b_1)$. For each finite étale map $X\rightarrow Y$, the base change induces a finite étale map $X\times _Y I\rightarrow I$, which takes the form $\coprod _{i=1}^r I$, for some $r\geqslant 0$, and we have an identification $X(y_i)= (X\times _Y I)(a_i)$ of fiber sets. In turn, the monodromy gives a transformation

$$\begin{aligned} \mu _w:X(y_0)=(X\times _Y I)(a_0)\longrightarrow (X\times _YI)(a_1)= X(y_1) \end{aligned}$$

that is bijective, and natural in the objects $X\in ({{\,\textrm{FinEt}\,}}/Y)$. In other words, the monodromy $\mu _w$ attached to the path $w:I\rightarrow Y$ from $y_0$ to $y_1$ is a bijective natural transformation between fiber functors

$$\begin{aligned} \Phi _{y_0}, \Phi _{y_1}:({{\,\textrm{FinEt}\,}}/Y)\longrightarrow ({{\,\textrm{FinSet}\,}}). \end{aligned}$$

We now use this monodromy to give an algebraic version of homotopy:

Definition 4.2

We say that two algebraic paths $w:I\rightarrow Y$ and $v:J\rightarrow Y$ from $y_0$ to $y_1$ are homotopic if $\mu _w=\mu _v$ as natural transformations from $\Phi _{y_0}$ to $\Phi _{y_1}$.

4.3. We denote the class of algebraic loops in Y based at the geometric point $y_0$ by

$$\begin{aligned} \Omega ^\textrm{alg}(Y,y_0)=\left\{ w:(I,a_0,a_1)\longrightarrow (Y,y_0,y_0) \mid (I,a_0,a_1) \text { interval scheme }\right\} . \end{aligned}$$

Let $w:(I, a_0,a_1)\rightarrow (Y, y_0,y_0)$ and $v:(J, b_0,b_1)\rightarrow (Y,y_0,y_0)$ be two loops based at $y_0$. It follows from the pushout diagram (3.1) and Lemma 3.6 that we can concatenate these loops to get a new loop

$$\begin{aligned} w*v: (I*J, a_0, b_1)\longrightarrow (Y,y_0,y_0). \end{aligned}$$

The monodromy transformation corresponding to $w*v$ can be factored, for $X\in ({{\,\textrm{FinEt}\,}}/Y)$ as

where we use the notation $X_Z=X\times _Y Z$, the vertical maps in the upper square are induced by base change of $w*v$, and the isomorphisms $X_I(a_i)\cong X_{I*J}(a_i)$ and $X_J(b_i)\cong X_{I*J}(b_i)$, for $i=0,1$, are induced by the natural closed immersions $I\hookrightarrow I*J$ and $J\hookrightarrow I*J$. The upper square commutes by the definition of $\mu _{w*v}$, the lower square commutes by the construction of the isomorphisms, see the proof of Proposition 1.5. Thus the whole square commutes and by definition of the monodromy we obtain the equality

$$\begin{aligned} {\mu _{w*v}=\mu _{v}\circ \mu _w} \end{aligned}$$

(4.1)

of automorphisms of the fiber functor $\Phi _{y_0}:({{\,\textrm{FinEt}\,}}/Y)\rightarrow ({{\,\textrm{FinSet}\,}})$. Furthermore if $u: (I,a_0, a_1)\rightarrow (Y, y_0, y_0)$, $v: (J,b_0, b_1)\rightarrow (Y, y_0, y_0)$, and $w: (L,c_0, c_1)\rightarrow (Y, y_0, y_0)$ are three loops based at $y_0$, then the universal property of pushout diagrams yields a canonical isomorphism $\tau : I*(J*L)\rightarrow (I*J)*L$ and an equality

$$\begin{aligned} {(u*v)*w\circ \tau = u*(v*w)\quad \text {in } (\textrm{Sch}/Y).} \end{aligned}$$

(4.2)

Clearly, homotopy between paths defines an equivalence relation $w\sim v$ and we denote by

$$\begin{aligned} \pi _0\Omega ^\textrm{alg}(Y,y_0) \end{aligned}$$

the set of homotopy classes of algebraic loops based at $y_0$. We denote by [w] the homotopy class of a loop $w:I\rightarrow Y$ at $y_0$. According to (4.1) we obtain a well defined operation

$$\begin{aligned} *: \pi _0\Omega ^\textrm{alg}(Y,y_0)\times \pi _0\Omega ^\textrm{alg}(Y,y_0)\longrightarrow \pi _0\Omega ^\textrm{alg}(Y,y_0),\quad ([w], [v])\longmapsto [w*v]. \end{aligned}$$

This operation is associative by (4.2) and clearly any constant loop $I\rightarrow y_0\rightarrow Y$ has the same homotopy class denoted by e, which defines a neutral element for $*$. Moreover, for a loop $w: (I, a_0, a_1)\rightarrow (Y, y_0, y_0)$ we define the loop $w': (I, a_1, a_0)\rightarrow (Y,y_0,y_0)$ by switching the end points of I. Clearly $[w]*[w']=e$. Hence concatenation of algebraic loops defines a group structure on $\pi _0\Omega ^\textrm{alg}(Y,y_0)$, which we therefore call the algebraic loop group.

By definition of the algebraic fundamental group, our homotopy relation, and the relation (4.1) the algebraic loop space $\Omega ^\textrm{alg}(Y,y_0)$ induces an injective homomorphism

$$\begin{aligned} { \pi _0\Omega ^\textrm{alg}(Y,y_0)^{\textrm{op}}\longrightarrow \pi _1^\textrm{alg}(Y,y_0),\quad w\longmapsto \mu _w } \end{aligned}$$

(4.3)

of groups, where we use the opposite group structure on the left hand side. We regard this as an inclusion of groups.

The following is the main result of this note.

Theorem 4.4

Let X be a connected scheme that is separated and of finite type over a field k, endowed with a geometric point $x_0:{\text {Spec}}(k^{{\text {sep}}})\rightarrow X$. Then the canonical injection (4.3) has dense image. It is actually bijective, provided X is proper.

The proof of Theorem 4.4 requires some preparations and will be given in Sect. 7. We remark that we do not expect (4.3) to be an isomorphism for non-proper schemes.

6 A Lefschetz type theorem

The Lefschetz Hyperplane Theorem gives a strong relation between the homology of a projective complex manifold X of dimension $n\geqslant 2$ and the homology of an ample divisor $D\subset X$. The original arguments appear in [19], Chapter V, Section III. Analogous statements for fundamental groups were first obtained by Bott [1]: The induced map $\pi _1(D,x_0)\rightarrow \pi _1(X,x_0)$ is bijective provided $n\geqslant 3$, and at least surjective if $n\geqslant 2$.

The latter statement extends to projective schemes X over arbitrary ground fields k: According to [10], Exposé XII, Corollary 3.5 the map $\pi ^\textrm{alg}_1(D,x_0)\rightarrow \pi ^\textrm{alg}_1(X,x_0)$ is surjective provided that ${\text {depth}}(\mathscr {O}_{X,a})\geqslant 2$ for each closed point $a\in X$. If X is additionally Cohen–Macaulay, i.e., at every point $a\in X$ the equality $\dim (\mathscr {O}_{X,a})={\text {depth}}(\mathscr {O}_{X,a})$ holds, the above can be iterated and one finds a connected curve $C\subset X$ such that $\pi ^\textrm{alg}_1(C, x_0)\rightarrow \pi ^\textrm{alg}_1(X, x_0)$ is surjective.

In this section we generalize the latter statement to arbitrary proper k-schemes.

5.1. Let X be a non-empty connected noetherian scheme. We consider the following property:

• (C) For each closed subscheme $Z\subset X$ with $\dim (Z)\leqslant 0$, there is a connected closed subscheme $C\subset X$ with $0\leqslant \dim (C)\leqslant 1$ and $Z\subset C$ such that, for each finite étale covering $U\rightarrow X$ with connected total space, the restriction $C_U=C\times _X U$ remains connected.

Remark 5.2

We remark that property (C) does not hold for affine schemes in general, as the following simple example shows (confer Lemma 5.4 in [3]): Let k be a ground field of characteristic $p>0$, and C be an connected affine plane curve inside $\mathbb {A}^2={\text {Spec}}(R)$, defined by some non-constant polynomial $f=f(x,y)$ inside the ring $R=k[x,y]$. Then there exists a connected finite étale covering $X\rightarrow \mathbb {A}^2$ whose restriction to C becomes trivial. To see this, take any $h\in (f)$ not of the form $g^p-g$ with $g\in R$. Via the identification

$$\begin{aligned} H^1_{{\textrm{et}}}(\mathbb {A}^2_k, \mathbb {Z}/p)=R/\{ g^p-g\mid g\in R\} \end{aligned}$$

coming from the Artin–Schreier sequence, the polynomial h corresponds to a non-trivial $\mathbb {Z}/p$-torsor $X\rightarrow \mathbb {A}^2_k$, in particular it is a connected finite étale covering of $\mathbb {A}^2_k$. On the other hand h maps to zero in ${\bar{R}}/\{a^p-a\mid a\in {\bar{R}}\}$, where ${\bar{R}}=R/(f)$. In other words, the restriction of X to C is trivial.

Lemma 5.3

Let X be a non-empty connected noetherian scheme, $f:X'\rightarrow X$ be a proper and surjective morphism, and $X'_v\subset X'$ be the connected components. If property (C) holds for all $X'_v$, then it also holds for X.

The proof of this lemma is inspired by the Seifert–van Kampen Theorem from [23], but is more elementary. Note that even if one wants to use property (C) on X with $Z=\varnothing$, the proof for the lemma relies in an essential way on property (C) on $X'_v$ with non-empty $Z'$. We first gather some basic material on graphs.

5.4. Let $f:X'\rightarrow X$ be as in the statement of Lemma 5.3. Set $X''=X'\times _XX'$ and write ${\text {pr}}_1,{\text {pr}}_2:X''\rightarrow X'$ for the two projections. The schemes $X'$ and $X''$ are noetherian, hence the sets of connected components $\pi _0(X')$ and $\pi _0(X'')$ are finite. Consider the induced maps

$$\begin{aligned} {\text {pr}}_1\times {\text {pr}}_2:\pi _0(X'')\longrightarrow \pi _0(X')\times \pi _0(X'). \end{aligned}$$

This defines an oriented graph $\Gamma =(E,V,{\text {pr}}_1\times {\text {pr}}_2)$ in the sense of Serre [21], Section 2.1: the set of vertices is $V=\pi _0(X')$, and the set of oriented edges is $E=\pi _0(X'')$. The endpoints of an edge $e\in E$ are the images $v_i={\text {pr}}_i(e)$. The orientation is given by declaring $v_1$ as the initial vertex, and $v_2$ as the terminal vertex. We usually write $X'_v\subset X'$ and $X''_e\subset X''$ for the connected components corresponding to a vertex v and an edge e.

Note that edges could have the same initial and terminal vertices, and several edges could share their initial and terminal vertices. By abuse of notation, we also write $v\in \Gamma$ and $e\in \Gamma$ to denote vertices and edges of the graph, if there is no risk of confusion. A morphism $f:\Gamma \rightarrow \Gamma '$ between oriented graphs comprises compatible maps $V\rightarrow V'$ and $E\rightarrow E'$. We simply say that f is a map of oriented graphs. Also note that the graph $\Gamma$ constructed above is connected, since the scheme X is connected.

Proof of Lemma 5.3

Since $f:X'\rightarrow X$ is proper, the image f(Z) of a closed subscheme $Z\subset X'$ is closed and satisfies $\dim f(Z)\leqslant \dim (Z)$. In particular, closed points are mapped to closed points. The assertion is trivial for $\dim (X)\leqslant 1$. We now assume $\dim X\geqslant 2$. We use the notation from 5.4. Let $Z\subset X$ be a zero-dimensional closed subscheme or the empty set. For each edge $e\in \Gamma$ choose a closed point $x_e\in X''_e$. Set

$$\begin{aligned} x_{e,i}:= {\text {pr}}_i(x_{e})\in X'_{v_i},\quad \text {where } v_i={\text {pr}}_i(e)\in \Gamma , \, i=1,2. \end{aligned}$$

Since $\Gamma$ is finite we find for each vertex $v\in \Gamma$ a 0-dimensional closed subset $Z'_v\subset X'_v$, such that

• (a) $Z\cap f(X_v')\subset f(Z'_v)$ and

• (b) $x_{e,i}\in Z'_v$, for all edges $e\in \Gamma$ with $v={\text {pr}}_i(e)$ for $i=1$ or 2.

Condition (b) is immediate, and one can achieve (a) by picking a closed point in each of the finitely many schemes $f^{-1}(z)\cap X'_v$, with $z\in Z\cap f(X_v')$. By the surjectivity of f we have

$$\begin{aligned} {Z\subset \bigcup _v f(Z'_v).} \end{aligned}$$

(5.1)

Applying (C) to $X'_v$ and $Z'_v$ we find an at most 1-dimensional closed subscheme $C'_v\subset X'_v$ containing $Z'_v$, such that the pullback of any connected finite étale covering of $X'_v$ to $C_v'$ stays connected. Then $C=\bigcup _v f(C'_v)$ is closed, at most 1-dimensional, and contains Z, by (5.1). It remains to show that for each connected finite étale covering $U\rightarrow X$ the pullback $U\times _X C$ remains connected (then C is connected as well).

To this end fix such a finite étale covering $u:U\rightarrow X$, with U non-empty and connected. Denote by $\Gamma _U$ the graph defined by ${\text {pr}}_1\times {\text {pr}}_2:\pi _0(U'')\rightarrow \pi _0(U')\times \pi _0(U')$, where $U'=U\times _X X'$ and $U''=U\times _{X}X''=U'\times _U U'$. We obtain a surjection of graphs $u:\Gamma _U\rightarrow \Gamma$ and for each edge $\epsilon \in \Gamma _U$ we obtain a finite and étale morphism between connected schemes $U''_{\epsilon }\rightarrow X''_{u(\epsilon )}$ which therefore is surjective. Thus for each edge $e\in \Gamma$ and edge $\epsilon \in \Gamma _U$ mapping to e we can choose a closed point $x_{U,\epsilon }\in U_{\epsilon }''$ with $u(x_{U,\epsilon })=x_e$.

For a vertex $w\in \Gamma _U$ mapping to $v\in \Gamma$ denote by $I_{w}$ the image of $U'_{w}\times _{X'_v} C'_{v}$ under the map

$$\begin{aligned} U'\times _{X'} C'_{v}=U\times _X C'_{v}\longrightarrow U\times _X C_{v},\quad \text {where } C_v=f(C'_v). \end{aligned}$$

The map is induced by the base change with the composition $C'_v\hookrightarrow X'_v\hookrightarrow X'\xrightarrow {f} X$, and therefore is closed and surjective. Hence

$$\begin{aligned} {U\times _X C_{v}=\bigcup _{w\in u^{-1}(v)} I_{w} \quad \text {and}\quad U\times _X C=\bigcup _{w \text { edge in } \Gamma _U} I_w,} \end{aligned}$$

(5.2)

where each $I_{w}$ is closed. By our choice of $C'_v$ the pullback of the connected étale covering $U'_w\rightarrow X'_v$ over $C'_v$ remains connected. Thus $I_w$ is the image of a connected scheme and is hence connected. Let $w_1$ and $w_2$ be the initial and the terminal vertices of an edge $\epsilon \in \Gamma _U$, then $x_{U,\epsilon }\in U''_\epsilon$ maps via the ith projection to points ${\text {pr}}_i(x_{U,\epsilon })$ in $U'_{w_i}\times _{X'_{u(w_i)}} C'_{u(w_i)}$, $i=1,2$, and these points map to same point in U. Thus the intersection $I_{w_1}\cap I_{w_2}$ is non-empty, if $w_1$ and $w_2$ are linked by an edge in $\Gamma _U$. Since the graph $\Gamma _U$ is connected so is the scheme $U\times _X C$. This completes the proof. $\square$

Proposition 5.5

Let X be a connected scheme that is proper over a field k. Then X has property (C).

Proof

We proceed by induction on $n=\dim X$. There is nothing to prove for $n\leqslant 1$. Assume $n\geqslant 2$ and that (C) holds for all connected schemes that are proper over k and have dimension $\leqslant n-1$. Using Lemma 5.3 we can make the following reductions:

• (i) X reduced (using the proper bijection $X_{{\text {red}}}\rightarrow X$);

• (ii) X projective over k (using Chow’s Lemma);

• (iii) X integral (using the proper surjection $\coprod X_i\rightarrow X$, with $X_i$ the irreducible components of X).

According to Kawasaki’s result ([18], Theorem 1.1) there is a proper birational $X'\rightarrow X$ such that the scheme $X'$ is Cohen–Macaulay. Moreover, this Macaulayfication arises as a sequence of blowing-ups. Hence applying Lemma 5.3 one more time, we are reduced to consider the case that X is projective, integral, and Cohen–Macaulay over a field k. Let $Z\subset X$ be a closed subscheme with $\dim (Z)\leqslant 0$. By, e.g., [7], Theorem 5.1, we find an effective ample divisor $D\subset X$ containing Z. By induction the following claim implies that X satisfies (C):

Claim 5.6

Let $X'\rightarrow X$ be an étale covering whose total space is connected. Then the restriction $D'=X'\times _X D$ remains connected.

The argument to prove the claim is similar to [15], Chapter II, Corollary 6.2. Let us recall it for the sake of completeness: Consider the ample invertible sheaf $\mathscr {L}=\mathscr {O}_X(D)$. Since $X'\rightarrow X$ is finite and surjective, the inclusion $D'\subset X'$ remains an effective Cartier divisor, and the corresponding invertible sheaf is the pullback $\mathscr {L}'=\mathscr {L}|X'$, which is still ample. Since X is projective and Cohen–Macaulay, so is $X'$. Let $\omega _{X'}$ be the dualizing sheaf over k. Then $h^1(\mathscr {L}'^{\otimes -t}) = h^{n-1}(\omega _{X'}\otimes \mathscr {L}'^{\otimes t})$ for every integer t. The right hand side vanishes for t sufficiently large, because $\mathscr {L}'$ is ample and $n-1\geqslant 1$. Replacing D by tD, we may assume $H^1(X',\mathscr {L}'^{\otimes -1})=0$. The short exact sequence $0\rightarrow \mathscr {L}^{\otimes -1}\rightarrow \mathscr {O}_{X'}\rightarrow \mathscr {O}_{D'}\rightarrow 0$ thus gives a surjection of rings $H^0(X',\mathscr {O}_{X'})\rightarrow H^0(D',\mathscr {O}_{D'}).$ The term on the left is a finite extension of the ground field k because $X'$ is integral and proper. Hence the above map is bijective, and $D'$ must be connected. $\square$

In view of 2.2 we obtain the following corollary.

Corollary 5.7

Let k be a field and set $K=k^{{\text {sep}}}$. Let X be a connected scheme which is proper over k and let $x_0:{\text {Spec}}(K)\rightarrow X$ be a geometric point. Then there exists a connected, reduced, affine, and 1-dimensional scheme C of finite type over k and a k-morphism $C\rightarrow X$, such that $x_0$ factors via C and the induced map

$$\begin{aligned} \pi _1^\textrm{alg}(C, x_0)\longrightarrow \pi _1^\textrm{alg}(X, x_0) \end{aligned}$$

is surjective.

Proof

By Proposition 5.5 and 2.2 we find a connected closed subscheme $C_1\subset X$ of dimension at most 1, such that $x_0$ factors via $C_1$ and the induced map

$$\begin{aligned} \pi _1^\textrm{alg}(C_1, x_0)\longrightarrow \pi _1^\textrm{alg}(X, x_0) \end{aligned}$$

is surjective. Since passing to the reduced subscheme does not change the fundamental group, we may assume $C_1$ reduced. If $\dim (C_1)=0$, then $C_1={\text {Spec}}(L)$ with L a subfield of K. In this case we can take $C:=\mathbb {A}^1_L$ with map $\mathbb {A}^1_L\rightarrow {\text {Spec}}(L)=C_1\rightarrow X$ and factorization of $x_0$ given by the composition of ${\text {Spec}}(K)\rightarrow {\text {Spec}}(L)$ with the inclusion of the zero section ${\text {Spec}}(L)\hookrightarrow \mathbb {A}^1_L$.

Assume $\dim (C_1)=1$. Note that $C_1$ is quasi-projective, hence we find an affine open $U\subset C_1$ which is connected and contains the singular locus of $C_1$ and the image point of $x_0$. In particular we remove from $C_1$ only finitely many regular closed points, whose local rings are therefore discrete valuation rings. Hence it follows from [22, Tag 0BSC] that $\pi _1^\textrm{alg}(U,x_0)\rightarrow \pi _1^\textrm{alg}(C_1,x_0)$ is surjective and we can take $C=U$. $\square$

7 The non-proper case

In view of Remark 5.2, we consider the following weaker variant of condition (C) in this section.

6.1. Let X be a non-empty connected noetherian scheme. We consider the following property:

• (C*) For each closed subscheme $Z\subset X$ with $\dim (Z)\leqslant 0$ and each finite étale covering $U\rightarrow X$ with connected total space there exists a connected closed subscheme $C\subset X$ with $0\leqslant \dim (C)\leqslant 1$ and $Z\subset C$, such that $C_U=C\times _X U$ remains connected.

Lemma 6.2

Let X be a non-empty connected noetherian scheme. Let $f:X'\rightarrow X$ be a universally closed and surjective morphism from a noetherian scheme $X'$, with connected components $X'_v$. If property (C*) holds for all $X'_v$, then it also holds for X.

We remark, that besides proper maps, all integral morphisms are universally closed, see [11, Proposition (6.1.10)]. Thus the above lemma applies to the normalization map $X'\rightarrow X$ of an integral scheme, and also to the projection from the base-change $X'=X\otimes _kk'$ with respect to any algebraic ground field extension, provided that $X'$ stays noetherian.

Since in Lemma 6.2 the map $f:X'\rightarrow X$ is not assumed to be of finite type, the product $X'\times _X X'$ may not be noetherian and might have infinitely many connected components. Thus the graph constructed in 5.4 might have an infinite set of edges. To deal with this we record the following lemma.

Lemma 6.3

Let $\Gamma$ be a connected oriented graph with a finite set of vertices. Then there exists a finite oriented subgraph $\Gamma '\subset \Gamma$, which has the same set of vertices as $\Gamma$ and which is connected.

Proof

Choose for each pair of vertices $v,w\in \Gamma$ a path $p_{v,w}$ connecting them. Take $\Gamma '$ to be the graph whose set of vertices is equal to the set of vertices of $\Gamma$ and whose edges are given by all the finitely many edges appearing in the paths $p_{v,w}$, for all pairs of vertices (v, w). $\square$

Proof Lemma 6.2

Since f is closed and surjective the image f(Z) of a closed subscheme $Z\subset X'$ is closed and satisfies $\dim f(Z)\leqslant \dim (Z)$, see [12, Proposition (5.4.1)]. In particular, closed points are mapped to closed points.

We assume $\dim X\geqslant 2$. Let $Z\subset X$ be a closed subset with $\dim (Z)\leqslant 0$. Set $X''=X'\times _X X'$. Let $u:U\rightarrow X$ be a finite étale covering with connected total space. We want to find a closed connected at most one dimensional subscheme C which contains Z and such that the pullback of U over C stays connected. To this end we may assume that $u: U\rightarrow X$ is a finite étale Galois covering with Galois group G. We denote by $U'=U\times _X X'$ and $U''= U\times _X X''$ the base changes and by $\Gamma _X$ and $\Gamma _U$ the oriented graphs defined as in 5.4 by ${{\text {pr}}_1}\times {\text {pr}}_2:\pi _0(X'')\rightarrow \pi _0(X')\times \pi _0(X')$ and ${{\text {pr}}_1}\times {\text {pr}}_2:\pi _0(U'')\rightarrow \pi _0(U')\times \pi _0(U')$, respectively. These graphs are connected, since X and U are, and have finite sets of vertices. The map u induces a surjective map of graphs $\Gamma _U\rightarrow \Gamma _X$ again denoted by u. For any vertex $w\in \Gamma _U$ with $v=u(w)\in \Gamma _X$, the morphism u induces a finite étale morphism $U'_w\rightarrow X'_v$.

Let $\Gamma '_U\subset \Gamma _U$ be a finite connected subgraph with the same vertices as $\Gamma _U$, see Lemma 6.3. Choose closed points $x_{U,e}\in U''_e$ for any edge $e\in \Gamma '_U$. Set

$$\begin{aligned} x_{e,i}:= u({\text {pr}}_i(x_{U,e}))\in X'_{v_i},\quad \text {where } v_i=u({\text {pr}}_i(e))\in \Gamma , \, i=1,2. \end{aligned}$$

Since $\Gamma '_U$ is finite we find as in the proof of Lemma 5.3 for each vertex $v\in \Gamma _X$ a 0-dimensional closed subset $Z'_v\subset X'_v$ such that

• (a) $Z\cap f(X_v')\subset f(Z'_v)$ and

• (b) $x_{e,i}\in Z'_v$, for all edges $e\in \Gamma '_U$ with $v=u({\text {pr}}_i(e))$ for $i=1$ or 2.

By the surjectivity of f we have

$$\begin{aligned} {Z\subset \bigcup _v f(Z'_v).} \end{aligned}$$

(6.1)

Fix a vertex $v\in \Gamma$ and choose $w_0\in \Gamma '_U$ mapping to v. Applying (C*) to the finite étale covering $U'_{w_0}\rightarrow X'_v$ we find an at most 1-dimensional connected closed subscheme $C'_{v}\subset X'_v$ containing $Z'_v$, such that the restriction $U'_{w_0}\times _{X'_v} C'_{v}$ remains connected. The base change $U\times _X C'_v\rightarrow C'_v$ is a Galois covering with Galois group G. In particular G acts transitively on the connected components of $U\times _X C'_v$ and we obtain isomorphisms

$$\begin{aligned} {U'_{w}\times _{X'_v} C'_v\cong U'_{w_0}\times _{X'_v} C'_v,\quad \text {for all }w\in \Gamma _U \text { mapping to } v,} \end{aligned}$$

(6.2)

in particular all these schemes are connected. Set

$$\begin{aligned} C_{v}:=f(C'_{v})\quad \text {and}\quad C:=\bigcup _{v} C_{v}. \end{aligned}$$

It follows that $C\subset X$ is closed, non-empty and at most 1-dimensional. By (6.1) and $Z'_v\subset C'_{v}$ we have $Z\subset C$. Moreover, using the choice of $C'_v$ together with (6.2) we can argue in the same way as in the last paragraph of the proof of Lemma 5.3 with $\Gamma _U$ there replaced by $\Gamma '_U$ here to deduce that $U\times _X C$ is connected. $\square$

Lemma 6.4

Let k be an algebraically closed field and X an integral quasi-projective k-scheme of $\dim X\geqslant 2$. Let $Z\subset X$ be a finite (possibly empty) set of closed points and $X'\rightarrow X$ a finite and surjective morphism with $X'$ irreducible. Then there is an integral closed subscheme $H\subset X$ of codimension 1 containing Z, such that $X'\times _X H$ is irreducible.

Proof

This follows from a classical Bertini theorem, where we use a trick of Mumford to ensure that the hyperplane contains Z, see the proof of the Lemma on p. 56 in [17]: Let $f: Y\rightarrow X$ be the blowing up with center Z. Then $\dim f^{-1}(z)\geqslant 1$, for all $z\in Z$. We fix an embedding $Y\hookrightarrow \mathbb {P}^n_k$. Denote by $Y'\subset Y\times _X X'$ an irreducible component which maps birationally onto $X'$. By [16, I, Corollary 6.11, 3)] applied to the quasi-finite morphism $Y'\rightarrow Y\rightarrow \mathbb {P}^n_k$ we find a hyperplane $H_1\subset \mathbb {P}^n_k$ which is not contained in the exceptional locus of f and such that its pullback to $Y'$ is irreducible. (Here we use k algebraically closed, since in loc. cit. $Y'$ is required to be geometrically irreducible and k to be infinite.) Since $f^{-1}(z)$ is closed in $\mathbb {P}^n_k$ and $\dim f^{-1}(z)+\dim H_1\geqslant n$ we find $f^{-1}(z)\cap H_1\ne \emptyset$, for all $z\in Z$. Set $H:=f(H_1\cap Y)_{{\text {red}}}$. Then $Z\subset H$ and the pullback of H to $X'$ is birationally dominated by $H_1\times _{\mathbb {P}^n} Y'$ and hence is irreducible. $\square$

Proposition 6.5

Let X be a connected scheme which is separated and of finite type over a field k. Then X has property (C*).

Proof

We proceed by induction on $d=\dim X$. There is nothing to prove for $d\leqslant 1$. Assume $d\geqslant 2$ and that (C*) holds for all connected schemes that are separated and of finite type over k and have dimension $\leqslant d-1$. Since X is connected all its irreducible components have dimension $\geqslant 1$. Using Lemma 6.2 we can therefore make the following reductions:

• (i) k algebraically closed (since for ${\bar{k}}$ the algebraic closure of k the morphism $X\otimes _k{\bar{k}}\rightarrow X$ is integral, whence universally closed);

• (ii) X reduced (by considering the proper morphism $X_{{\text {red}}}\rightarrow X$);

• (iii) X quasi-projective over k (Chow’s Lemma);

• (iv) X integral (by considering the proper and surjective morphism $\sqcup X_i\rightarrow X$, with $X_i$ the irreducible components of X);

• (v) X normal (by considering the normalization ${\widetilde{X}}\rightarrow X$).

Assume k and X satisfy the conditions above. Let $U\rightarrow X$ be a finite étale map with U connected. By the normality of X the scheme U is normal as well. Thus U is irreducible. Hence the existence of a curve for $U\rightarrow X$ as in (C*) follows directly from Lemma 6.4 and the induction hypothesis. $\square$

Corollary 6.6

Let k be a field and set $K=k^{{\text {sep}}}$. Let X be a connected scheme which is separated and of finite type over k and let $x_0:{\text {Spec}}(K)\rightarrow X$ be a geometric point. Let $X'\rightarrow X$ be a finite étale Galois covering. Then there exists a connected, reduced, affine, and 1-dimensional scheme C of finite type over k and a k-morphism $C\rightarrow X$, such that $x_0$ factors via C and the composite map

$$\begin{aligned} {\pi _1^\textrm{alg}(C, x_0)\longrightarrow \pi _1^\textrm{alg}(X, x_0)\longrightarrow {\text {Aut}}(X'/X)^\textrm{op},} \end{aligned}$$

(6.3)

is surjective. Here the second map is the natural surjection from [9, Exp. V, 4, h)].

Proof

In general the composition (6.3) is surjective if the pullback of $X'$ over C stays connected. Hence the statement follows from Proposition 6.5 the same way Corollary 5.7 follows from Proposition 5.5. $\square$

8 Proof of the main theorem

We prove Theorem 4.4. We start by proving the second statement, i.e., for a proper and connected scheme X over a field k with geometric point $x_0:{\text {Spec}}(k)\rightarrow X$ we want to show that the natural injective group homomorphism $\pi _0\Omega ^\textrm{alg}(X,x_0)\rightarrow \pi _1^\textrm{alg}(X, x_0)$ is surjective as well.

Set $K:=k^{{\text {sep}}}$. By Corollary 5.7 we find a connected, affine, reduced and 1-dimensional k-scheme of finite type C with a morphism $C\rightarrow X$ such that $x_0$ factors via C and the natural $\pi _1^\textrm{alg}(C, x_0)\twoheadrightarrow \pi _1^\textrm{alg}(X,x_0)$ is surjective. We obtain a commutative diagram

(7.1)

in which the vertical arrow on the right is surjective. It therefore remains to show that the top horizontal arrow is surjective. To this end we choose some pro-object $(C_i)_{i\in I}$ in $({{\,\textrm{FinEt}\,}}/C)$ representing $\Phi _{x_0}$, see [9], Exposé V, 4. We write $\textrm{Pro}({{\,\textrm{FinEt}\,}}/C)$ for pro-objects in $({{\,\textrm{FinEt}\,}}/C)$. Since I is a filtered set and the transition maps $C_i\rightarrow C_j$ are finite étale, and therefore affine, we may form the projective limit ${\widetilde{C}}=\varprojlim _{i\in I} C_i$ in the category of C-schemes, see [13], Proposition (8.2.3). For a C-scheme T we obtain a functorial isomorphism

$$\begin{aligned} {\text {Hom}}_C(T, {\widetilde{C}})=\varprojlim _i {\text {Hom}}_C(T, C_i). \end{aligned}$$

In particular an element $a_0\in \varprojlim \Phi _{x_0}(C_i)$ is a morphism $a_0: {\text {Spec}}(K)\rightarrow {\widetilde{C}}$ over $x_0$. Furthermore, by [9], Exposé V, 4, h), we have

$$\begin{aligned} {\text {Aut}}_{\textrm{Pro}({{\,\textrm{FinEt}\,}}/C)}((C_i)_{i})&={\text {Hom}}_{\textrm{Pro}({{\,\textrm{FinEt}\,}}/C)}((C_i)_i, (C_j)_j) \\&= \varprojlim _j {\text {Hom}}_{\textrm{Pro}({{\,\textrm{FinEt}\,}}/C)}((C_i)_i, C_j) \\&=\varprojlim _j {\text {Hom}}_C(C_j, C_j) = \varprojlim _j {\text {Hom}}_C({\widetilde{C}}, C_j) \\&= {\text {Hom}}_C({\widetilde{C}}, {\widetilde{C}}) = {\text {Aut}}_C({\widetilde{C}}). \end{aligned}$$

By loc. cit., the choice of $a_0: {\text {Spec}}(K)\rightarrow {\widetilde{C}}$ over $x_0$ yields a functorial isomorphism

$$\begin{aligned} {\text {Hom}}_C({\widetilde{C}}, U){\mathop {\longrightarrow }\limits ^{\simeq }} \Phi _{x_0}(U), \quad f\longmapsto f\circ a_0,\qquad U\in ({{\,\textrm{FinEt}\,}}/C). \end{aligned}$$

We obtain an isomorphism

$$\begin{aligned} \theta :{\text {Aut}}_C({\widetilde{C}})^\textrm{op}{\mathop {\longrightarrow }\limits ^{\simeq }} {\text {Aut}}(\Phi _{x_0})=\pi _1^\textrm{alg}(C, x_0), \end{aligned}$$

which sends a C-automorphism $\sigma : {\widetilde{C}}\rightarrow {\widetilde{C}}$ to the automorphism $\theta (\sigma )$ of $\Phi _{x_0}$, which on $U\in ({{\,\textrm{FinEt}\,}}/C)$ is given by

$$\begin{aligned} \Phi _{x_0}(U)\ni f\circ a_0\longmapsto f\circ (\sigma \circ a_0)\in \Phi _{x_0}(U), \quad f\in {\text {Hom}}_C({\widetilde{C}}, U). \end{aligned}$$

We claim that for any $\sigma$ the automorphism $\theta (\sigma )$ is in the image of $\pi _0\Omega ^\textrm{alg}(C, x_0)^{\textrm{op}}$. Indeed, by construction ${\widetilde{C}}$ is affine, reduced, connected and 1-dimensional and is simply connected We have the K-rational point $a_0: {\text {Spec}}(K)\rightarrow {\widetilde{C}}$ over $x_0: {\text {Spec}}(K)\rightarrow C$. We note that $a_0$ is a closed immersion. Indeed, for any finite separable field extension L/k a connected component $C_0$ of $C\otimes _k L$ is a finite étale covering of C; hence we have a map ${\widetilde{C}}\rightarrow C_0$. It follows that the algebraic closure of k in $H^0({\widetilde{C}}, \mathscr {O}_{{\widetilde{C}}})$ is equal to $K=k^{{{\text {sep}}}}$, which implies that $a_0$ is a closed immersion. Thus $({\widetilde{C}},a_0, \sigma \circ a_0)$ is an interval scheme in the sense of Definition 3.1 and the map ${\widetilde{C}}\rightarrow C$ induces an algebraic loop $w: ({\widetilde{C}}, a_0, \sigma \circ a_0)\rightarrow (C, x_0, x_0)$. The map (4.3) sends the loop w to the monodromy $\mu _w$ which by construction is equal to $\theta (\sigma )$. This completes the proof of the second part of the theorem.

It remains to show that assuming X is connected and only separated and of finite type over k, then the image of $\pi _0\Omega ^\textrm{alg}(X,x_0)^{\textrm{op}}\rightarrow \pi _1^\textrm{alg}(X,x_0)$ is dense. It suffices to show that for any finite étale Galois covering $X'\rightarrow X$ the composition

$$\begin{aligned} \pi _0\Omega ^\textrm{alg}(X,x_0)^{\textrm{op}}\longrightarrow \pi _1^\textrm{alg}(X,x_0)\longrightarrow {\text {Aut}}(X'/X)^\textrm{op} \end{aligned}$$

is surjective. By Corollary 6.6 we find a curve C as above such that the composition

$$\begin{aligned} \pi _1^\textrm{alg}(C,x_0)\longrightarrow \pi _1^\textrm{alg}(X,x_0)\longrightarrow {\text {Aut}}(X'/X)^\textrm{op} \end{aligned}$$

is surjective. Thus the statement follows from the surjectivity of top horizontal map in (7.1) proved above. $\square$