# Algebraic spaces

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

## Introduction

A quotient can have good local coordinates even when no affine neighbourhood exists at one of its points. Algebraic spaces let us retain those coordinates. The coordinates are allowed to be étale rather than Zariski local, and the object they describe is a sheaf of sets. Automorphisms of an object are not retained in this construction; retaining them will lead to stacks.

We will construct quotients before studying their geometry. The decisive question is whether the relation between two coordinate systems is itself represented by a scheme. After answering it, we will compare finite symmetry, an infinite translation action, and an identification that changes at one point.

We assume the functor of points, fibre products of schemes, the fppf topology, and descent of morphisms. The scheme descent theorem used below is stated precisely in Section 2. We impose no quasi-separatedness condition in the definition of an algebraic space. All groups used in this lesson are abstract groups, acting through their constant group schemes; a profinite group is treated here as an abstract group unless expressly stated otherwise.

Basic references are [Stacks] and [Behrend]. The term algebraic space used here concerns étale presentations. Serre's earlier use of the same words, discussed in [EGA IV, §10.10], concerns a different construction with closed points of Jacobson schemes.

## 1. Local coordinates without an underlying scheme

Fix a scheme \(S\). Work on the big fppf site of \(S\)-schemes, within a fixed universe. A scheme \(V\) is identified with the sheaf

\[
h_V(T)=\operatorname{Hom}_S(T,V).
\]

A transformation \(a:F\to G\) of sheaves is **representable by schemes** if \(T\times_G F\) is represented by a scheme for every scheme \(T\) and every map \(T\to G\). This condition lets us ask geometric questions about \(a\): it is étale, an open immersion, or surjective if every such base change has that property. Here surjectivity is surjectivity of the resulting maps of schemes, not surjectivity of \(F(T)\to G(T)\) for every \(T\).

**Lemma 1.1.** Base change and composition preserve representability by schemes. If a property of scheme morphisms is stable under base change and local on the target for the fppf topology, then its definition for representable transformations can be checked after an fppf cover of any test scheme. If the property is also stable under composition, compositions of representable transformations with that property have the property.

**Proof.** For base change, the test fibre product is still a test fibre product of the original transformation. For \(F\to G\to H\) and \(T\to H\), first form the scheme \(V=T\times_H G\). Then

\[
T\times_H F=V\times_G F
\]

is a scheme. The map to \(T\) is the composition of the two tested morphisms. For the local assertion, pull \(T\times_G F\to T\) back to an fppf covering \(\{T_i\to T\}\) and use the stated locality of the property. These observations prove all the assertions. \(\square\)

The diagonal packages all tests involving a pair of objects. A sheaf \(X\) has representable diagonal if, for every scheme \(T\) and \(a,b\in X(T)\), the equality functor

\[
E_{a,b}(T')=\{f:T'\to T:a|_{T'}=b|_{T'}\}
\]

is represented by a scheme over \(T\). This is exactly the base change of \(X\to X\times_S X\) by \((a,b):T\to X\times_S X\).

**Definition 1.2.** An **algebraic space over \(S\)** is an fppf sheaf \(X\) whose diagonal is representable by schemes and which admits a representable, surjective étale morphism \(q:U\to X\) from a scheme. Such a morphism is an **étale presentation**.

If the diagonal is representable, every map from a scheme \(V\) to \(X\) is representable: \(T\times_X V\) is the base change of the diagonal by \(T\times_S V\to X\times_S X\). Thus the word representable in the last condition is redundant once the diagonal condition has been imposed.

Every scheme is an algebraic space. Its functor of points is an fppf sheaf; its diagonal is a scheme morphism; and its identity is an étale presentation.

For a presentation, the scheme

\[
R=U\times_X U
\]

records equal coordinates. Write \(s,t:R\to U\) for the projections. Both are étale. The map \(j=(s,t):R\to U\times_S U\) is a monomorphism, and for every test scheme its image is an equivalence relation. The relation includes an identity arrow, reversal of arrows, and composition. Because \(j\) is a monomorphism, these arrows, when they exist, are unique.

Conversely, an **étale equivalence relation** consists of schemes \(U,R\) and a monomorphism \(j=(s,t):R\to U\times_S U\) which gives an equivalence relation on every test scheme, with \(s,t\) étale. Its quotient \(U/R\) will always mean the fppf sheafification of the presheaf \(T\mapsto U(T)/R(T)\).

Sheafification matters. A point of \((U/R)(T)\) need not have one coordinate on \(T\). It has coordinates on an fppf cover of \(T\), related on overlaps.

## 2. Constructing the quotient

We use three results about schemes. First, schemes are fppf sheaves, and an fppf-invariant open subscheme descends uniquely to an open subscheme. Second, being étale and being surjective are fppf local on the target. Third, the following effectiveness theorem is available from scheme descent:

**Scheme descent theorem.** Let \(\{T_i\to T\}\) be an fppf covering of a scheme. A descent datum \((V_i\to T_i,\phi_{ij})\) is effective if every \(V_i\to T_i\) is separated and locally quasi-finite. There is then a scheme \(V\to T\) inducing that datum; compatible maps also descend. The theorem does not require quasi-compactness of the \(V_i\).

The precise reference is [Stacks, Tag 02W8]. Its scope is larger than affine descent and smaller than arbitrary descent of schemes. In particular, we will not use a claim that every scheme with fppf descent data descends.

The elementary part of quotient sheafification is useful throughout the proof.

**Lemma 2.1.** Put \(F=U/R\). The map \(U\to F\) is an epimorphism of fppf sheaves, and

\[
U\times_F U\simeq R.
\]

**Proof.** Locally, a section of a sheafification comes from the presheaf being sheafified, and a presheaf quotient section has a representative in \(U\). This proves the first assertion. Two maps \(a,b:T\to U\) have equal images in \(F(T)\) precisely when they are related fppf locally. On a cover \(T_i\to T\), choose the corresponding maps \(r_i:T_i\to R\). Their endpoints agree on overlaps. The monomorphism \(j\) makes the \(r_i\) agree there, and the sheaf property of \(R\) glues them to a unique map \(T\to R\). Thus being related locally is equivalent to being related globally. This identifies the displayed fibre product as a sheaf, hence proves the assertion. \(\square\)

**Theorem 2.2 (étale quotient theorem).** For every étale equivalence relation \(R\rightrightarrows U\) over \(S\), the fppf quotient \(X=U/R\) is an algebraic space. The quotient map \(U\to X\) is representable, surjective and étale, and its relation is the original \(R\).

**Proof.** We first construct quotient neighbourhoods from affine pieces of \(U\), and then show that those neighbourhoods glue.

Suppose first that \(U\) is affine. The morphism \(U\times_S U\to U\times_{\operatorname{Spec}\mathbf Z}U\) is a monomorphism. Since the latter target is affine, \(U\times_S U\) is separated as an absolute scheme. A monomorphism is separated, so \(R\) is separated too. In particular \(s\) and \(t\) are separated. They are locally quasi-finite because they are étale.

Let \(a:T\to X\) be a map from a scheme. Lemma 2.1 gives an fppf cover \(T_i\to T\) and lifts \(a_i:T_i\to U\). On that cover, the sheaf \(V=T\times_X U\) is represented by

\[
V_i=T_i\times_{a_i,U,s}R.
\]

Each \(V_i\to T_i\) is separated and étale. The equality functor defining \(V\) supplies canonical identifications on overlaps, satisfying the cocycle condition. The scheme descent theorem produces a scheme representing \(V\). One can see the representation directly: maps into the descended scheme are compatible maps into the \(V_i\), and these are exactly sections of the sheaf \(V\). Since each \(V_i\to T_i\) is étale and surjective, the descended morphism has these properties. Surjectivity here follows because \(s\) has the identity section. We have proved representability and the presentation condition for \(U\to X\).

We must still prove representability of the diagonal. The morphism \(j:R\to U\times_S U\) is separated and locally quasi-finite. For local finite type, use that its composition with a projection is étale: a morphism whose composition is locally of finite type is locally of finite type. A monomorphism has at most one geometric point in each geometric fibre, so this local finite type morphism is locally quasi-finite.

Given \((a,b):T\to X\times_S X\), take an fppf cover on which both sections have lifts to \(U\). On this cover their equality scheme is the base change of \(j\). It is therefore separated and locally quasi-finite. Apply the same scheme descent theorem to these equality schemes. The descended scheme represents \(E_{a,b}\), by the same sheaf argument. This proves the diagonal condition when \(U\) is affine.

We now return to arbitrary \(U\). For any open subscheme \(A\subset U\), let \(R_A=R\times_{U\times_S U}(A\times_S A)\), and let \(X_A=A/R_A\). Its map to \(X\) is a monomorphism: if two coordinates in \(A\) are related in \(U\), their unique relation arrow lies in \(R_A\).

The map \(X_A\to X\) is represented by open immersions. To prove this without assuming \(X\) algebraic, form the saturation

\[
W_A=t(s^{-1}(A))\subset U.
\]

It is open because étale morphisms are open. Its points are precisely the points related, after a residue-field extension if necessary, to points of \(A\). The equivalence relation identities give \(s^{-1}(W_A)=t^{-1}(W_A)\) as open subschemes.

For \(T\to X\), choose lifts \(a_i:T_i\to U\) on an fppf cover. The opens \(a_i^{-1}(W_A)\) agree on overlaps: the lifts there are endpoints of an arrow in \(R\). They descend to an open \(T_A\subset T\). A section on a test scheme over \(T\) belongs to \(X_A\) exactly when it factors through this open. Indeed, membership in \(W_A\) can be lifted to \(s^{-1}(A)\) fppf locally using the surjective étale map \(s^{-1}(A)\to W_A\); its other endpoint is a coordinate in \(A\). Conversely, a coordinate in \(A\) makes any related coordinate lie in \(W_A\). Thus \(T_A=T\times_X X_A\).

Take an affine open cover \(\{A_\lambda\}\) of \(U\). The quotient opens \(X_{A_\lambda}\) cover \(X\) and, by the affine case, are algebraic spaces. Here they cover even in the Zariski sense appropriate to representable opens: after pulling back to any \(T\to X\), the opens \(T_{A_\lambda}\) cover \(T\).

For completeness, the diagonal of \(X\) is now represented as follows. For a pair of sections on \(T\), cover \(T\) by opens on which the first lies in some \(X_{A_\lambda}\). Its equality with the second is possible only on the open where the second also lies in that same \(X_{A_\lambda}\). There the equality functor is represented by the diagonal of this algebraic space. These schemes glue over their open intersections, since they represent the same equality functor. Thus \(\Delta_X\) is representable.

Finally the maps \(A_\lambda\to X_{A_\lambda}\to X\) are representable étale maps and together are surjective. Their disjoint union is a scheme presentation. Hence \(X\) is algebraic. The original map \(U\to X\) is representable by the diagonal observation in Section 1. On an fppf cover of any test scheme admitting coordinates in \(U\), its base change is a base change of \(s:R\to U\). It is therefore étale and surjective. Lemma 2.1 identifies its relation with \(R\). \(\square\)

*Reference:* [Stacks, Tag 02WW]. The affine neighbourhoods in the proof are used to make scheme descent effective. They do not assert that \(X_{A_\lambda}\) is an affine scheme.

**Corollary 2.3.** An algebraic space is the quotient of any of its étale presentations by the corresponding equivalence relation.

**Proof.** An étale surjection has local sections for the étale topology. Thus \(U\to X\) is an epimorphism of fppf sheaves. Its kernel relation is \(U\times_X U\). Lemma 2.1, or directly the gluing of local coordinates, identifies its quotient with \(X\). \(\square\)

## 3. Products and gluing

**Proposition 3.1.** If \(X\to Z\leftarrow Y\) are morphisms of algebraic spaces over \(S\), then \(X\times_Z Y\) is an algebraic space and has the expected fibre product universal property.

**Proof.** The fibre product \(H\) in sheaves exists. Choose presentations \(U\to X\), \(V\to Y\). Since \(Z\) has representable diagonal, \(W=U\times_Z V\) is a scheme. The map \(W\to H\) is the composition of base changes of the two presentations, so is representable, étale and surjective.

For two sections of \(H\) over a scheme \(T\), their equality is equivalent to the equality of their \(X\)-coordinates and of their \(Y\)-coordinates. The two equality schemes over \(T\) exist by the diagonal conditions of \(X\) and \(Y\). Their scheme fibre product over \(T\) represents the equality functor for \(H\). Thus \(H\) has representable diagonal. Its universal property is its universal property as a sheaf, since the category of algebraic spaces is a full subcategory of sheaves. \(\square\)

**Proposition 3.2.** Algebraic spaces can be glued along open subspaces with specified identifications satisfying the cocycle condition. A disjoint union indexed by a set is an algebraic space, subject to the fixed universe convention.

**Proof.** Construct the glued sheaf by identifying the specified open subfunctors. For any test scheme, membership in an open piece is an open condition and the pieces cover. As in the last part of Theorem 2.2, equality schemes can be built on these opens and glued, establishing the diagonal condition. Choose a scheme presentation of each piece. Their disjoint union is a scheme, and its map to the glued sheaf is representable étale and surjective, because each piece is a representable open. The disjoint union is the case of empty overlaps. \(\square\)

This argument is also why ordinary Zariski gluing remains available after passing from schemes to spaces.

## 4. Symmetry and separation

Suppose an abstract group \(G\) acts on an \(S\)-scheme \(U\). Use the action relation

\[
R=\coprod_{g\in G}U,
\qquad (g,u)\longmapsto (u,g(u)).
\]

This is an equivalence relation precisely when the action has no stabilizers on geometric points. One useful form of that condition is

\[
g(u)=u\text{ and }g|_{\kappa(u)}=\operatorname{id}
\quad\Longrightarrow\quad g=1.
\tag{4.1}
\]

It is possible for a group element to fix an ordinary point while moving its residue field. Condition (4.1) permits this. We call an action satisfying it **free**.

**Proposition 4.1.** Under condition (4.1), the action relation is an étale equivalence relation. Its quotient \(U/G\) is an algebraic space, and \(U\to U/G\) is a torsor for the constant group in the fppf topology. If \(G\) is finite, the quotient map is finite étale and surjective.

**Proof.** A map from a test scheme into \(R\) consists of a locally constant choice of \(g\) and a map to \(U\). Suppose two such maps have the same pair of endpoints. Locally on the test scheme their group labels are constant. At any point \(z\) of a nonempty such piece, equality of endpoints shows that \(g^{-1}h\) fixes the image \(u\) and acts trivially after the injective field map \(\kappa(u)\to\kappa(z)\). It acts trivially on \(\kappa(u)\), and (4.1) gives \(g=h\). The maps to \(U\) then agree too. This proves the monomorphism condition, including disconnected test schemes.

Identity, inversion and multiplication of labels provide reflexivity, symmetry and transitivity. Each projection is, on a component, either the identity of \(U\) or an automorphism, and is therefore étale. Theorem 2.2 applies. Lemma 2.1 gives the torsor identity

\[
G\times U\simeq U\times_{U/G}U.
\]

The quotient map has local sections, and this identity makes the action simply transitive on those local trivializations. If \(G\) is finite, a base change of the quotient map by itself is the finite disjoint union of identity maps on \(U\). It is finite étale. Finiteness and étaleness descend fppf locally on the target; the quotient map is therefore finite étale and surjective. \(\square\)

For an algebraic space \(X\) over \(S\), we use the following separation conditions:

| Condition | Condition on \(\Delta_{X/S}\) |
|---|---|
| Quasi-separated | Quasi-compact |
| Locally separated | An immersion |
| Separated | A closed immersion |

The conditions can be tested on the relation of a presentation. Indeed, \(j:R\to U\times_S U\) is the pullback of \(\Delta_{X/S}\) along the representable étale surjection \(U\times_S U\to X\times_S X\). After any test scheme mapping into the latter sheaf, this gives an étale cover by schemes. Quasi-compactness, immersion and closed immersion descend under these covers. Thus the test does not require that \(U\) itself be separated.

**Proposition 4.2.** For a free action of a finite group, \(U/G\) is quasi-separated over \(S\) if \(U\) is quasi-separated over \(S\), and separated over \(S\) if \(U\) is separated over \(S\).

**Proof.** Each graph of \(g:U\to U\) is obtained from \(\Delta_{U/S}\) by an automorphism of \(U\times_S U\). In the quasi-separated case all these graphs are quasi-compact morphisms, and their finite disjoint union is quasi-compact. This proves the first assertion by the relation test.

In the separated case the graphs are closed immersions. Distinct graphs have disjoint underlying images: a point in their intersection would give, over a common residue-field extension, a geometric point fixed by \(g^{-1}h\), contradicting (4.1). A finite family of disjoint closed subschemes has disjoint-union map a closed immersion. To check this on an affine open of the target, the corresponding ideals are pairwise comaximal, and the Chinese remainder theorem gives the required quotient algebra. Apply the relation test again. \(\square\)

*References:* [Stacks, Tags 02Z2 and 02Z4].

**Proposition 4.3.** A free action of a finite group on an affine scheme \(U=\operatorname{Spec}A\) has affine quotient \(\operatorname{Spec}(A^G)\). The algebra \(A\) is finite locally free over \(A^G\), and the quotient map is finite étale. More generally, if every orbit in a scheme \(U\) is contained in an affine open, then \(U/G\) is a scheme.

**Proof.** Set \(B=A^G\). The action graphs in the absolute affine product \(U\times U\) are pairwise disjoint closed subschemes by freeness. The Chinese remainder theorem therefore makes the map

\[
A\otimes_{\mathbf Z}A\longrightarrow\prod_{g\in G}A,
\qquad a\otimes b\longmapsto (a\,g(b))_g
\]

surjective. Choose a finite tensor sum mapping to the vector which is one in the identity component and zero elsewhere. Write its entries as \(a_i,b_i\in A\); thus

\[
\sum_i a_i g(b_i)=\begin{cases}1&g=1,\\0&g\ne1.\end{cases}
\tag{4.2}
\]

The sum \(\operatorname{tr}(c)=\sum_g g(c)\) belongs to \(B\). Equation (4.2) gives, for every \(c\in A\),

\[
\sum_i a_i\operatorname{tr}(b_i c)=c.
\tag{4.3}
\]

This is a finite dual basis: \(A\) is a direct summand of a finite free \(B\)-module, hence finite projective. Every \(c\in A\) satisfies the monic polynomial \(\prod_g(T-g(c))\) with coefficients in \(B\). The integral inclusion \(B\subset A\) is surjective on spectra by lying over. Consequently this finite projective algebra is faithfully flat.

The induced map

\[
\theta:A\otimes_B A\longrightarrow\prod_g A
\]

is still surjective. It is injective as well. If \(z=\sum_j c_j\otimes d_j\) has zero image, then \(\sum_j c_jg(d_j)=0\) for every \(g\). Expand each \(d_j\) with (4.3). The coefficient of \(1\otimes a_i\) in this expansion is

\[
\sum_j c_j\operatorname{tr}(b_i d_j)
=\sum_g g(b_i)\sum_j c_jg(d_j)=0.
\]

Thus \(z=0\). The faithfully flat map \(\operatorname{Spec}A\to\operatorname{Spec}B\) has the action relation as its fibre relation. Fppf descent of sections of a sheaf therefore identifies its quotient with \(\operatorname{Spec}B\). After base change by \(\operatorname{Spec}A\) it is a disjoint union of identity maps, so it is finite étale. This also identifies its rank as \(|G|\).

For the general assertion, first produce invariant affine neighbourhoods. Let an orbit \(E\) lie in an affine open \(A_0\subset U\). The intersection \(V=\bigcap_g g(A_0)\) is a \(G\)-invariant open containing \(E\), and is an open of \(A_0\). There is a function \(f\) on \(A_0\) such that \(E\subset D(f)\subset V\). Indeed, the ideal defining the closed complement of \(V\) is not contained in any of the finitely many primes in \(E\); finite prime avoidance gives such an \(f\). On \(V\), all translates \(g(f)\) are regular. Their product \(N=\prod_g g(f)\) is invariant and nonzero at every point of \(E\).

The invariant open \(D_V(N)\) is affine: it lies in \(D_{A_0}(f)\), and is the principal open of that affine scheme cut out by the restriction of \(N\). This construction does not assume that intersections of affine opens in \(U\) are affine.

These invariant affine opens cover \(U\). Their quotients are affine by the first part. Each quotient maps to \(U/G\) as a representable open immersion: its pullback to \(U\) is precisely the invariant open, or one can apply the saturation argument in Theorem 2.2. Thus the algebraic space \(U/G\) has a cover by open affine schemes. Gluing these schemes over their open overlaps represents it by a scheme. \(\square\)

Infinite groups behave differently even when every nonidentity element is a very simple automorphism.

## 5. Three quotients that distinguish the definitions

### 5.1. Identify opposite coordinates except at zero

Let \(k\) be a field with \(2\ne0\), and put \(U=\mathbf A^1_k\). Use the relation

\[
R=\mathbf A^1_k\amalg\mathbf G_{m,k},
\quad j|_{\mathbf A^1}(x)=(x,x),
\quad j|_{\mathbf G_m}(x)=(x,-x).
\]

Both projections are étale. The two components have disjoint images: their graphs meet only at zero, which has been removed from the second component. The equivalence relation identities follow from changing sign twice. Thus Theorem 2.2 produces \(X=U/R\).

The map \(j\) is quasi-compact. In fact both components are quasi-compact schemes, and every open subset of either is quasi-compact, because they are Noetherian. The inverse image of an affine open of \(\mathbf A^2_k\) is consequently quasi-compact. Thus \(X\) is quasi-separated.

It is not locally separated. Let \(D\) be the diagonal and \(L\) the line \(y=-x\) in \(\mathbf A^2_k\). The image of \(j\) is \(D\cup(L\setminus\{0\})\), whose underlying set is the closed set \(D\cup L\). Suppose \(j\) were an immersion. Its underlying space would then have the subspace topology of this union. But the diagonal component is open and closed in \(R\). Its image \(D\) is not open in \(D\cup L\): any open neighbourhood of the origin on the line \(L\) contains its generic point, which is in \(L\setminus\{0\}\) and outside \(D\). This contradicts the subspace topology. Hence \(j\) is not an immersion.

Every scheme over \(k\) is locally separated: on an affine neighbourhood of a point its diagonal is closed, and these neighbourhoods give the immersion property for the full diagonal. Therefore \(X\) is not a scheme. The obstruction is visible in the relation, without a search for a hypothetical coordinate ring.

Away from the image of zero, the quotient is \(\mathbf G_m\). The map \(x\mapsto x^2\) is finite étale on \(\mathbf G_m\), and its fibre relation is exactly the identity and sign-change graphs. It represents the quotient there. Every neighbourhood of the image of zero retains the failure of immersion just proved, so none is a scheme.

*Reference:* [Stacks, Tag 02Z1].

### 5.2. An algebraic closure with a discrete Galois action

Let \(U=\operatorname{Spec}\overline{\mathbf Q}\) and let \(G=\operatorname{Gal}(\overline{\mathbf Q}/\mathbf Q)\) act as an abstract group. Its only ordinary point is fixed by every element, but a nonidentity element acts nontrivially on its residue field. Condition (4.1) holds, so \(X=U/G\) is an algebraic space over \(\mathbf Q\).

It is not \(\operatorname{Spec}\mathbf Q\). If it were, its presentation would make \(\operatorname{Spec}\overline{\mathbf Q}\to\operatorname{Spec}\mathbf Q\) étale. A field étale over \(\mathbf Q\) is a finite separable extension; the algebraic closure is infinite. This contradiction already distinguishes the two sheaves.

In fact \(X\) is not a scheme. Here is a useful argument that avoids a classification of finite subgroups of the Galois group. Suppose it were a scheme, and choose an affine open \(V\subset X\) containing the image \(x\) of the unique point of \(U\). The preimage of \(V\) is all of \(U\), so \(U\to V\) is étale. It is locally of finite presentation. Since its source and target are affine and its source has one point, it is also quasi-compact; its separatedness follows from affineness. It is therefore of finite presentation. Its fibre over \(x\) shows that \(\overline{\mathbf Q}\) is finite separable over \(\kappa(x)\).

The map \(U\to X\) is invariant under every element of \(G\), so the embedded field \(\kappa(x)\subset\overline{\mathbf Q}\) lies in the fixed field \(\mathbf Q\). As the map is over \(\mathbf Q\), this embedded field is exactly \(\mathbf Q\). That contradicts the finite-extension conclusion.

The distinction comes from the group used to form the relation. In this example it is the constant group indexed by the underlying set of \(G\), not a group object carrying the profinite topology.

*Reference:* [Stacks, Tag 02Z6].

### 5.3. Translation by integers

Let \(k\) have characteristic zero, and let \(\mathbf Z\) act on \(\mathbf A^1_k\) by \(x\mapsto x+n\). A nonzero translation has no fixed geometric point. At the generic ordinary point it acts nontrivially on \(k(x)\), so condition (4.1) still holds. Put

\[
X=\mathbf A^1_k/\mathbf Z.
\]

The relation is an infinite disjoint union of affine lines. Its entire inverse image over the affine scheme \(\mathbf A^2_k\) is not quasi-compact: the infinitely many nonempty open-and-closed components have no finite subcover. Thus the relation map is not quasi-compact, and \(X\) is not quasi-separated.

A further failure concerns residue fields. The generic point gives \(\gamma:\operatorname{Spec}k(x)\to X\). Unlike a point of a scheme, it cannot factor through a monomorphism from the spectrum of a field. We prove this next, so that the obstruction is explicit.

**Lemma 5.1.** Let a constant abstract group \(G\) act freely on \(U\), and let \(\operatorname{Spec}L\to U/G\) be any field-valued point. For each connected component \(\operatorname{Spec}L'\) of its pullback to \(U\), the extension \(L'/L\) is finite Galois and its Galois group is isomorphic to the subgroup of \(G\) preserving that component.

**Proof.** The pullback \(P\to\operatorname{Spec}L\) is a nonempty étale scheme. It is a disjoint union of spectra of finite separable extensions of \(L\), and \(P\) is a \(G\)-torsor by Proposition 4.1. Let \(H\) preserve the chosen component \(P_0=\operatorname{Spec}L'\). Restricting the torsor identity gives

\[
H\times P_0\simeq P_0\times_L P_0.
\]

The right side is finite over \(P_0\), of degree \([L':L]\), so \(H\) is finite with that cardinality. The isomorphism identifies its elements with all \(L\)-automorphisms of \(L'\). A finite separable extension with as many automorphisms as its degree is Galois. \(\square\)

Suppose \(\gamma\) factored as

\[
\operatorname{Spec}k(x)\longrightarrow\operatorname{Spec}L
\xrightarrow{m}X
\]

with \(m\) a monomorphism. Pull \(m\) back to \(\mathbf A^1_k\), obtaining a monomorphism \(P\to\mathbf A^1_k\) with a lift of the generic point. A component \(\operatorname{Spec}L'\subset P\) therefore receives \(\operatorname{Spec}k(x)\) and maps to the generic point of the affine line. The composite

\[
k(x)\longrightarrow L'\longrightarrow k(x)
\]

is the identity. Since a field map is injective, these maps force \(L'=k(x)\): the second map embeds \(L'\) into \(k(x)\) and its image already contains all of \(k(x)\). Lemma 5.1 says that \(L'/L\) has Galois group a finite subgroup of \(\mathbf Z\), hence is trivial. Thus \(L=L'=k(x)\).

However, restricting the relation to the generic point gives

\[
\operatorname{Spec}k(x)\times_X\operatorname{Spec}k(x)
\simeq\coprod_{n\in\mathbf Z}\operatorname{Spec}k(x).
\]

There is one component for each translation automorphism of the rational function field. The diagonal lands in the component indexed by zero and is not an isomorphism. Therefore \(\gamma\) is not a monomorphism, contradicting the factorization. This explains why the points of general algebraic spaces must be defined by equivalence classes of field-valued maps rather than by monomorphisms from spectra of fields.

*References:* [Stacks, Tags 02Z5 and 02Z7].

## 6. Exercises and solutions

The following exercises revisit the constructions through different tests: equality of coordinates, gluing, and the effect of changing the group. The final exercise develops the affine-neighbourhood question for finite quotients.

**Exercise 6.1 (first steps).** For a scheme \(V\), identify the equality scheme for two maps \(T\to V\). Then construct a presentation of \(\coprod_i X_i\) from presentations \(U_i\to X_i\).

**Solution.** The equality scheme is \(T\times_{V\times_S V}V\), using the diagonal of \(V\). Thus schemes satisfy the diagonal condition, and the identity gives their presentations. For the disjoint union, use \(\coprod_i U_i\). The equality functor between two sections splits over the open-and-closed parts of \(T\) on which both choose the same index. On each such part it is the equality scheme for \(X_i\); on parts with different indices it is empty. These schemes glue, and the disjoint-union presentation is representable étale and surjective.

**Exercise 6.2 (separation).** In the sign-identification example, replace the removed origin in the second graph by an arbitrary finite set of closed points invariant under sign. Require that the set contain zero. Describe the quotient off that set, and show that it is not locally separated at the image of zero when the second graph is nonempty.

**Solution.** On the invariant complement \(A\subset\mathbf A^1\), sign is a free action of the two-element group. The square map identifies \(A/\{\pm1\}\) with the open of \(\mathbf A^1\) obtained by removing the squares of the excluded points, and this open is contained in \(\mathbf G_m\). The restricted relation is its fibre relation. For the full quotient, the second graph remains dense in \(L:y=-x\), while its origin is absent. The image of the diagonal component is not open near the origin in the union of the two lines. The topology argument of Section 5.1 again proves that the relation is not an immersion. In the original example, the same argument also proves quasi-separatedness because both components are Noetherian and quasi-compact.

**Exercise 6.3 (finite torsors).** Let a finite group act freely on \(U\). Find the degree of \(U\to U/G\) and explain why a \((U/G)(T)\)-point need not lift to \(U(T)\).

**Solution.** Proposition 4.1 proves that the map is finite étale; its pullback by itself is \(|G|\) copies of \(U\). Therefore it is finite locally free of constant rank \(|G|\). A point of the quotient over \(T\) pulls back to a \(G\)-torsor on \(T\). It has a lift precisely when that torsor has a section. Such a section trivializes the torsor; a nontrivial torsor has local sections but no global section. For example, a nontrivial quadratic finite separable extension \(L/k\) defines the free Galois action on \(\operatorname{Spec}L\), whose quotient is \(\operatorname{Spec}k\). The latter's identity point does not lift to \(\operatorname{Spec}L(k)\).

**Exercise 6.4 (infinite symmetry).** Replace the integer translation action by translation by the additive group of a nonzero finitely generated subgroup \(\Lambda\subset k\), where \(k\) has characteristic zero. Show that the quotient is not quasi-separated and that its generic point has no factorization through a field-spectrum monomorphism.

**Solution.** The group \(\Lambda\) is infinite and torsion-free. Its action is free on geometric points, and its action on \(k(x)\) is faithful. The relation is the disjoint union of \(\mathbf A^1\) indexed by \(\Lambda\). Its inverse image over \(\mathbf A^2\) is not quasi-compact, proving the first assertion. For the second, the proof after Lemma 5.1 applies unchanged: a hypothetical factorization forces \(L'=k(x)\), and then the finite stabilizer subgroup is trivial. The restricted relation on \(\operatorname{Spec}k(x)\) still has one component for each \(\lambda\in\Lambda\), contradicting the monomorphism. Taking \(\Lambda=\mathbf Z\cdot1\) gives the original integer example.

**Exercise 6.5 (affine neighbourhoods).** Suppose a finite group acts freely on a scheme \(U\) quasi-projective over an affine scheme. Show that every orbit lies in an invariant affine open and that \(U/G\) is a scheme. Why would simply taking the intersection of all translates of an affine open require an extra argument?

**Solution.** A quasi-projective scheme over an affine base has an ample invertible sheaf, and every finite subset lies in an affine open. The precise scheme result is [Stacks, Tag 01ZY]. One way to obtain the affine neighbourhood is to embed \(U\) as a locally closed subscheme of a projective space over the base. Choose homogeneous functions whose standard opens cover the finite set and avoid the boundary of \(U\) in its closure. Homogeneous prime avoidance supplies one homogeneous function in their generated ideal avoiding every prime in the finite set. Its standard open avoids the boundary; its intersection with \(U\) is closed in an affine standard open and is therefore affine.

The homogeneous prime avoidance used here is [Stacks, Tag 00JS]. Apply the affine-neighbourhood result to the orbit, then apply the norm construction in Proposition 4.3 to obtain an invariant affine open. Its quotient is the spectrum of the invariant ring, and these affine quotients give an open scheme cover of \(U/G\). The intersection of the translates is invariant and open, but affineness of an open subscheme does not follow just from its being contained in an affine scheme. In the present quasi-projective case separatedness also gives affineness of finite intersections of affine opens. The norm construction proves the needed refinement even without that shortcut.

## What this lesson does not prove

The scheme effectiveness theorem for separated, locally quasi-finite descent data [Stacks, Tag 02W8] is used as a prerequisite. The scheme descent references for opens, étaleness, surjectivity, finiteness, quasi-compactness, immersion and closed immersion are respectively [Stacks, Tags 03N0, 02VN, 02KV, 02LA, 02KQ, 02YM and 02L6]. For immersions, the locality asserted here is for fppf coverings. We also use the ample finite-set lemma [Stacks, Tag 01ZY] and homogeneous prime avoidance [Stacks, Tag 00JS]. These are statements about schemes, not additional quotient assertions.

## References

- **[Stacks]** The Stacks project, *Algebraic Spaces* and *More on Morphisms*, especially Tags 02WW, 02W8, 02Z1, 02Z2, 02Z4, 02Z5, 02Z6 and 02Z7. The corresponding text is also available in [AI Integrated Stacks Project, Algebraic Spaces](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/spaces.html), an edition with AI-proposed corrections and AI-written additions which has not been reviewed by the Stacks project's maintainers. The [official Stacks project](https://stacks.math.columbia.edu/) retains its own publication and tags.
- **[Behrend]** K. Behrend, *Introduction to algebraic stacks*, lecture notes, 2012. [Author's lecture notes](https://personal.math.ubc.ca/~behrend/math615A/stacksintro.pdf).
- **[EGA IV]** A. Grothendieck and J. Dieudonné, *Éléments de géométrie algébrique IV*, third part, Publications Mathématiques de l'IHÉS 28 (1966), §10.10, for the earlier terminology concerning Serre's algebraic spaces. [Original text](https://www.numdam.org/item/PMIHES_1966__28__5_0/).