Groupoid Schemes

Introduction

This chapter is devoted to generalities concerning groupoid schemes. See for example the beautiful paper [K-M] by Keel and Mori.

Notation

Let \(S\) be a scheme. If \(U\), \(T\) are schemes over \(S\) we denote \(U(T)\) for the set of \(T\)-valued points of \(U\) over \(S\). In a formula: \(U(T) = \Mor_S(T, U)\). We try to reserve the letter \(T\) to denote a “test scheme” over \(S\), as in the discussion that follows. Suppose we are given schemes \(X\), \(Y\) over \(S\) and a morphism of schemes \(f : X \to Y\) over \(S\). For any scheme \(T\) over \(S\) we get an induced map of sets \[f : X(T) \longrightarrow Y(T)\] which as indicated we denote by \(f\) also. In fact this construction is functorial in the scheme \(T/S\). Yoneda’s Lemma, see Categories, Lemma 001P, says that \(f\) determines and is determined by this transformation of functors \(f : h_X \to h_Y\). More generally, we use the same notation for maps between fibre products. For example, if \(X\), \(Y\), \(Z\) are schemes over \(S\), and if \(m : X \times_S Y \to Z \times_S Z\) is a morphism of schemes over \(S\), then we think of \(m\) as corresponding to a collection of maps between \(T\)-valued points \[X(T) \times Y(T) \longrightarrow Z(T) \times Z(T).\] And so on and so forth.

We continue our convention to label projection maps starting with index \(0\), so we have \(\text{pr}_0 : X \times_S Y \to X\) and \(\text{pr}_1 : X \times_S Y \to Y\).

Equivalence relations

Recall that a relation \(R\) on a set \(A\) is just a subset \(R \subset A \times A\). We usually write \(a R b\) to indicate \((a, b) \in R\). We say the relation is transitive if \(a R b, b R c \Rightarrow a R c\). We say the relation is reflexive if \(a R a\) for all \(a \in A\). We say the relation is symmetric if \(a R b \Rightarrow b R a\). A relation is called an equivalence relation if it is transitive, reflexive and symmetric.

In the setting of schemes we are going to relax the notion of a relation a little bit and just require \(R \to A \times A\) to be a map. Here is the definition.

Definition

Let \(S\) be a scheme. Let \(U\) be a scheme over \(S\).

1. A pre-relation on \(U\) over \(S\) is any morphism of schemes \(j : R \to U \times_S U\). In this case we set \(t = \text{pr}_0 \circ j\) and \(s = \text{pr}_1 \circ j\), so that \(j = (t, s)\).

2. A relation on \(U\) over \(S\) is a monomorphism of schemes \(j : R \to U \times_S U\).

3. A pre-equivalence relation is a pre-relation \(j : R \to U \times_S U\) such that the image of \(j : R(T) \to U(T) \times U(T)\) is an equivalence relation for all \(T/S\).

4. We say a morphism \(R \to U \times_S U\) of schemes is an equivalence relation on \(U\) over \(S\) if and only if for every scheme \(T\) over \(S\) the \(T\)-valued points of \(R\) define an equivalence relation on the set of \(T\)-valued points of \(U\).

In other words, an equivalence relation is a pre-equivalence relation such that \(j\) is a relation.

Lemma

Let \(S\) be a scheme. Let \(U\) be a scheme over \(S\). Let \(j : R \to U \times_S U\) be a pre-relation. Let \(g : U' \to U\) be a morphism of schemes. Finally, set \[R' = (U' \times_S U')\times_{U \times_S U} R \xrightarrow{j'} U' \times_S U'\] Then \(j'\) is a pre-relation on \(U'\) over \(S\). If \(j\) is a relation, then \(j'\) is a relation. If \(j\) is a pre-equivalence relation, then \(j'\) is a pre-equivalence relation. If \(j\) is an equivalence relation, then \(j'\) is an equivalence relation.

Proof

Omitted.

Definition

Let \(S\) be a scheme. Let \(U\) be a scheme over \(S\). Let \(j : R \to U \times_S U\) be a pre-relation. Let \(g : U' \to U\) be a morphism of schemes. The pre-relation \(j' : R' \to U' \times_S U'\) is called the restriction, or pullback of the pre-relation \(j\) to \(U'\). In this situation we sometimes write \(R' = R|_{U'}\).

Lemma

Let \(j : R \to U \times_S U\) be a pre-relation. Consider the relation on points of the scheme \(U\) defined by the rule \[x \sim y \Leftrightarrow \exists\ r \in R : t(r) = x, s(r) = y.\] If \(j\) is a pre-equivalence relation then this is an equivalence relation.

Proof

Suppose that \(x \sim y\) and \(y \sim z\). Pick \(r \in R\) with \(t(r) = x\), \(s(r) = y\) and pick \(r' \in R\) with \(t(r') = y\), \(s(r') = z\). Pick a field \(K\) fitting into the following commutative diagram \[\xymatrix{ \kappa(r) \ar[r] & K \\ \kappa(y) \ar[u] \ar[r] & \kappa(r') \ar[u] }\] Denote by \(x_K, y_K, z_K : \Spec(K) \to U\) the morphisms \[\begin{matrix} \Spec(K) \to \Spec(\kappa(r)) \to \Spec(\kappa(x)) \to U \\ \Spec(K) \to \Spec(\kappa(r)) \to \Spec(\kappa(y)) \to U \\ \Spec(K) \to \Spec(\kappa(r')) \to \Spec(\kappa(z)) \to U \end{matrix}\] By construction \((x_K, y_K) \in j(R(K))\) and \((y_K, z_K) \in j(R(K))\). Since \(j\) is a pre-equivalence relation we see that also \((x_K, z_K) \in j(R(K))\). This clearly implies that \(x \sim z\).

The proof that \(\sim\) is reflexive and symmetric is omitted.

Lemma

Let \(j : R \to U \times_S U\) be a pre-relation. Assume

1. \(s, t\) are unramified,

2. for any algebraically closed field \(k\) over \(S\) the map \(R(k) \to U(k) \times U(k)\) is an equivalence relation,

3. there are morphisms \(e : U \to R\), \(i : R \to R\), \(c : R \times_{s, U, t} R \to R\) such that \[\xymatrix{ U \ar[r]_e \ar[d]_\Delta & R \ar[d]_j & R \ar[d]^j \ar[r]_i & R \ar[d]^j & R \times_{s, U, t} R \ar[d]^{j \times j} \ar[r]_c & R \ar[d]^j \\ U \times_S U \ar[r] & U \times_S U & U \times_S U \ar[r]^{flip} & U \times_S U & U \times_S U \times_S U \ar[r]^{\text{pr}_{02}} & U \times_S U }\] are commutative.

Then \(j\) is an equivalence relation.

Proof

By condition (1) and Morphisms, Lemma 02GG we see that \(j\) is unramified. Then \(\Delta_j : R \to R \times_{U \times_S U} R\) is an open immersion by Morphisms, Lemma 02GE. However, then condition (2) says \(\Delta_j\) is bijective on \(k\)-valued points, hence \(\Delta_j\) is an isomorphism, hence \(j\) is a monomorphism. Then it easily follows from the commutative diagrams that \(R(T) \subset U(T) \times U(T)\) is an equivalence relation for all schemes \(T\) over \(S\).

Group schemes

Let us recall that a group is a pair \((G, m)\) where \(G\) is a set, and \(m : G \times G \to G\) is a map of sets with the following properties:

1. (associativity) \(m(g, m(g', g'')) = m(m(g, g'), g'')\) for all \(g, g', g'' \in G\),

2. (identity) there exists a unique element \(e \in G\) (called the identity, unit, or \(1\) of \(G\)) such that \(m(g, e) = m(e, g) = g\) for all \(g \in G\), and

3. (inverse) for all \(g \in G\) there exists an \(i(g) \in G\) such that \(m(g, i(g)) = m(i(g), g) = e\), where \(e\) is the identity.

Thus we obtain a map \(e : \{*\} \to G\) and a map \(i : G \to G\) so that the quadruple \((G, m, e, i)\) satisfies the axioms listed above.

A homomorphism of groups \(\psi : (G, m) \to (G', m')\) is a map of sets \(\psi : G \to G'\) such that \(m'(\psi(g), \psi(g')) = \psi(m(g, g'))\). This automatically insures that \(\psi(e) = e'\) and \(i'(\psi(g)) = \psi(i(g))\). (Obvious notation.) We will use this below.

Definition

Let \(S\) be a scheme.

1. A group scheme over \(S\) is a pair \((G, m)\), where \(G\) is a scheme over \(S\) and \(m : G \times_S G \to G\) is a morphism of schemes over \(S\) with the following property: For every scheme \(T\) over \(S\) the pair \((G(T), m)\) is a group.

2. A morphism \(\psi : (G, m) \to (G', m')\) of group schemes over \(S\) is a morphism \(\psi : G \to G'\) of schemes over \(S\) such that for every \(T/S\) the induced map \(\psi : G(T) \to G'(T)\) is a homomorphism of groups.

Let \((G, m)\) be a group scheme over the scheme \(S\). By the discussion above (and the discussion in Section 022N) we obtain morphisms of schemes over \(S\): (identity) \(e : S \to G\) and (inverse) \(i : G \to G\) such that for every \(T\) the quadruple \((G(T), m, e, i)\) satisfies the axioms of a group listed above.

Let \((G, m)\), \((G', m')\) be group schemes over \(S\). Let \(f : G \to G'\) be a morphism of schemes over \(S\). It follows from the definition that \(f\) is a morphism of group schemes over \(S\) if and only if the following diagram is commutative: \[\xymatrix{ G \times_S G \ar[r]_-{f \times f} \ar[d]_m & G' \times_S G' \ar[d]^{m'} \\ G \ar[r]^f & G' }\]

Lemma

Let \((G, m)\) be a group scheme over \(S\). Let \(S' \to S\) be a morphism of schemes. The pullback \((G_{S'}, m_{S'})\) is a group scheme over \(S'\).

Proof

Omitted.

Definition

Let \(S\) be a scheme. Let \((G, m)\) be a group scheme over \(S\).

1. A closed subgroup scheme of \(G\) is a closed subscheme \(H \subset G\) such that \(m|_{H \times_S H}\) factors through \(H\) and induces a group scheme structure on \(H\) over \(S\).

2. An open subgroup scheme of \(G\) is an open subscheme \(G' \subset G\) such that \(m|_{G' \times_S G'}\) factors through \(G'\) and induces a group scheme structure on \(G'\) over \(S\).

Alternatively, we could say that \(H\) is a closed subgroup scheme of \(G\) if it is a group scheme over \(S\) endowed with a morphism of group schemes \(i : H \to G\) over \(S\) which identifies \(H\) with a closed subscheme of \(G\).

Lemma

Let \(S\) be a scheme. Let \((G, m, e, i)\) be a group scheme over \(S\).

1. A closed subscheme \(H \subset G\) is a closed subgroup scheme if and only if \(e : S \to G\), \(m|_{H \times_S H} : H \times_S H \to G\), and \(i|_H : H \to G\) factor through \(H\).

2. An open subscheme \(H \subset G\) is an open subgroup scheme if and only if \(e : S \to G\), \(m|_{H \times_S H} : H \times_S H \to G\), and \(i|_H : H \to G\) factor through \(H\).

Proof

Looking at \(T\)-valued points this translates into the well known conditions characterizing subsets of groups as subgroups.

Definition

Let \(S\) be a scheme. Let \((G, m)\) be a group scheme over \(S\).

1. We say \(G\) is a smooth group scheme if the structure morphism \(G \to S\) is smooth.

2. We say \(G\) is a flat group scheme if the structure morphism \(G \to S\) is flat.

3. We say \(G\) is a separated group scheme if the structure morphism \(G \to S\) is separated.

Examples of group schemes

Example

Consider the functor which associates to any scheme \(T\) the group \(\Gamma(T, \mathcal{O}_T^*)\) of units in the global sections of the structure sheaf. This is representable by the scheme \[\mathbf{G}_m = \Spec(\mathbf{Z}[x, x^{-1}])\] The morphism giving the group structure is the morphism \[\begin{eqnarray*} \mathbf{G}_m \times \mathbf{G}_m & \to & \mathbf{G}_m \\ \Spec(\mathbf{Z}[x, x^{-1}] \otimes_{\mathbf{Z}} \mathbf{Z}[x, x^{-1}]) & \to & \Spec(\mathbf{Z}[x, x^{-1}]) \\ \mathbf{Z}[x, x^{-1}] \otimes_{\mathbf{Z}} \mathbf{Z}[x, x^{-1}] & \leftarrow & \mathbf{Z}[x, x^{-1}] \\ x \otimes x & \leftarrow & x \end{eqnarray*}\] Hence we see that \(\mathbf{G}_m\) is a group scheme over \(\mathbf{Z}\). For any scheme \(S\) the base change \(\mathbf{G}_{m, S}\) is a group scheme over \(S\) whose functor of points is \[T/S \longmapsto \mathbf{G}_{m, S}(T) = \mathbf{G}_m(T) = \Gamma(T, \mathcal{O}_T^*)\] as before.

Example

Let \(n \in \mathbf{N}\). Consider the functor which associates to any scheme \(T\) the subgroup of \(\Gamma(T, \mathcal{O}_T^*)\) consisting of \(n\)th roots of unity. This is representable by the scheme \[\mu_n = \Spec(\mathbf{Z}[x]/(x^n - 1)).\] The morphism giving the group structure is the morphism \[\begin{eqnarray*} \mu_n \times \mu_n & \to & \mu_n \\ \Spec( \mathbf{Z}[x]/(x^n - 1) \otimes_{\mathbf{Z}} \mathbf{Z}[x]/(x^n - 1)) & \to & \Spec(\mathbf{Z}[x]/(x^n - 1)) \\ \mathbf{Z}[x]/(x^n - 1) \otimes_{\mathbf{Z}} \mathbf{Z}[x]/(x^n - 1) & \leftarrow & \mathbf{Z}[x]/(x^n - 1) \\ x \otimes x & \leftarrow & x \end{eqnarray*}\] Hence we see that \(\mu_n\) is a group scheme over \(\mathbf{Z}\). For any scheme \(S\) the base change \(\mu_{n, S}\) is a group scheme over \(S\) whose functor of points is \[T/S \longmapsto \mu_{n, S}(T) = \mu_n(T) = \{f \in \Gamma(T, \mathcal{O}_T^*) \mid f^n = 1\}\] as before.

Example

Consider the functor which associates to any scheme \(T\) the group \(\Gamma(T, \mathcal{O}_T)\) of global sections of the structure sheaf. This is representable by the scheme \[\mathbf{G}_a = \Spec(\mathbf{Z}[x])\] The morphism giving the group structure is the morphism \[\begin{eqnarray*} \mathbf{G}_a \times \mathbf{G}_a & \to & \mathbf{G}_a \\ \Spec(\mathbf{Z}[x] \otimes_{\mathbf{Z}} \mathbf{Z}[x]) & \to & \Spec(\mathbf{Z}[x]) \\ \mathbf{Z}[x] \otimes_{\mathbf{Z}} \mathbf{Z}[x] & \leftarrow & \mathbf{Z}[x] \\ x \otimes 1 + 1 \otimes x & \leftarrow & x \end{eqnarray*}\] Hence we see that \(\mathbf{G}_a\) is a group scheme over \(\mathbf{Z}\). For any scheme \(S\) the base change \(\mathbf{G}_{a, S}\) is a group scheme over \(S\) whose functor of points is \[T/S \longmapsto \mathbf{G}_{a, S}(T) = \mathbf{G}_a(T) = \Gamma(T, \mathcal{O}_T)\] as before.

Example

Let \(n \geq 1\). Consider the functor which associates to any scheme \(T\) the group \[\text{GL}_n(\Gamma(T, \mathcal{O}_T))\] of invertible \(n \times n\) matrices over the global sections of the structure sheaf. This is representable by the scheme \[\text{GL}_n = \Spec(\mathbf{Z}[\{x_{ij}\}_{1 \leq i, j \leq n}][1/d])\] where \(d = \det((x_{ij}))\) with \((x_{ij})\) the \(n \times n\) matrix with entry \(x_{ij}\) in the \((i, j)\)-spot. The morphism giving the group structure is the morphism \[\begin{eqnarray*} \text{GL}_n \times \text{GL}_n & \to & \text{GL}_n \\ \Spec(\mathbf{Z}[x_{ij}, 1/d] \otimes_{\mathbf{Z}} \mathbf{Z}[x_{ij}, 1/d]) & \to & \Spec(\mathbf{Z}[x_{ij}, 1/d]) \\ \mathbf{Z}[x_{ij}, 1/d] \otimes_{\mathbf{Z}} \mathbf{Z}[x_{ij}, 1/d] & \leftarrow & \mathbf{Z}[x_{ij}, 1/d] \\ \sum_k x_{ik} \otimes x_{kj} & \leftarrow & x_{ij} \end{eqnarray*}\] Hence we see that \(\text{GL}_n\) is a group scheme over \(\mathbf{Z}\). For any scheme \(S\) the base change \(\text{GL}_{n, S}\) is a group scheme over \(S\) whose functor of points is \[T/S \longmapsto \text{GL}_{n, S}(T) = \text{GL}_n(T) =\text{GL}_n(\Gamma(T, \mathcal{O}_T))\] as before.

Example

The determinant defines a morphism of group schemes \[\det : \text{GL}_n \longrightarrow \mathbf{G}_m\] over \(\mathbf{Z}\). By base change it gives a morphism of group schemes \(\text{GL}_{n, S} \to \mathbf{G}_{m, S}\) over any base scheme \(S\).

Example

Let \(G\) be an abstract group. Consider the functor which associates to any scheme \(T\) the group of locally constant maps \(T \to G\) (where \(T\) has the Zariski topology and \(G\) the discrete topology). This is representable by the scheme \[G_{\Spec(\mathbf{Z})} = \coprod\nolimits_{g \in G} \Spec(\mathbf{Z}).\] The morphism giving the group structure is the morphism \[G_{\Spec(\mathbf{Z})} \times_{\Spec(\mathbf{Z})} G_{\Spec(\mathbf{Z})} \longrightarrow G_{\Spec(\mathbf{Z})}\] which maps the component corresponding to the pair \((g, g')\) to the component corresponding to \(gg'\). For any scheme \(S\) the base change \(G_S\) is a group scheme over \(S\) whose functor of points is \[T/S \longmapsto G_S(T) = \{f : T \to G \text{ locally constant}\}\] as before.

Properties of group schemes

In this section we collect some simple properties of group schemes which hold over any base.

Lemma

Let \(S\) be a scheme. Let \(G\) be a group scheme over \(S\). Then \(G \to S\) is separated (resp. quasi-separated) if and only if the identity morphism \(e : S \to G\) is a closed immersion (resp. quasi-compact).

Proof

We recall that by Schemes, Lemma 01KT we have that \(e\) is an immersion which is a closed immersion (resp. quasi-compact) if \(G \to S\) is separated (resp. quasi-separated). For the converse, consider the diagram \[\xymatrix{ G \ar[r]_-{\Delta_{G/S}} \ar[d] & G \times_S G \ar[d]^{(g, g') \mapsto m(i(g), g')} \\ S \ar[r]^e & G }\] It is an exercise in the functorial point of view in algebraic geometry to show that this diagram is cartesian. In other words, we see that \(\Delta_{G/S}\) is a base change of \(e\). Hence if \(e\) is a closed immersion (resp. quasi-compact) so is \(\Delta_{G/S}\), see Schemes, Lemma 01JY (resp. Schemes, Lemma 01K5).

Lemma

Let \(S\) be a scheme. Let \(G\) be a group scheme over \(S\). Let \(T\) be a scheme over \(S\) and let \(\psi : T \to G\) be a morphism over \(S\). If \(T\) is flat over \(S\), then the morphism \[T \times_S G \longrightarrow G, \quad (t, g) \longmapsto m(\psi(t), g)\] is flat. In particular, if \(G\) is flat over \(S\), then \(m : G \times_S G \to G\) is flat.

Proof

Consider the diagram \[\xymatrix{ T \times_S G \ar[rrr]_{(t, g) \mapsto (t, m(\psi(t), g))} & & & T \times_S G \ar[r]_{\text{pr}} \ar[d] & G \ar[d] \\ & & & T \ar[r] & S }\] The left top horizontal arrow is an isomorphism and the square is cartesian. Hence the lemma follows from Morphisms, Lemma 01U9.

Lemma

Let \((G, m, e, i)\) be a group scheme over the scheme \(S\). Denote \(f : G \to S\) the structure morphism. Then there exist canonical isomorphisms \[\Omega_{G/S} \cong f^*\mathcal{C}_{S/G} \cong f^*e^*\Omega_{G/S}\] where \(\mathcal{C}_{S/G}\) denotes the conormal sheaf of the immersion \(e\). In particular, if \(S\) is the spectrum of a field, then \(\Omega_{G/S}\) is a free \(\mathcal{O}_G\)-module.

Proof

By Morphisms, Lemma 01V0 we have \[\Omega_{G \times_S G/G} = \text{pr}_0^*\Omega_{G/S}\] where on the left hand side we view \(G \times_S G\) as a scheme over \(G\) using \(\text{pr}_1\). Let \(\tau : G \times_S G \to G \times_S G\) be the “shearing map” given by \((g, h) \mapsto (m(g, h), h)\) on points. This map is an automorphism of \(G \times_S G\) viewed as a scheme over \(G\) via the projection \(\text{pr}_1\). Combining these two remarks we obtain an isomorphism \[\tau^*\text{pr}_0^*\Omega_{G/S} \to \text{pr}_0^*\Omega_{G/S}\] Since \(\text{pr}_0 \circ \tau = m\) this can be rewritten as an isomorphism \[m^*\Omega_{G/S} \to \text{pr}_0^*\Omega_{G/S}\] Pulling back this isomorphism by \((e \circ f, \text{id}_G) : G \to G \times_S G\) and using that \(m \circ (e \circ f, \text{id}_G) = \text{id}_G\) and \(\text{pr}_0 \circ (e \circ f, \text{id}_G) = e \circ f\) we obtain an isomorphism \[\Omega_{G/S} \to f^*e^*\Omega_{G/S}\] as desired. By Morphisms, Lemma 0474 we have \(\mathcal{C}_{S/G} \cong e^*\Omega_{G/S}\). If \(S\) is the spectrum of a field, then any \(\mathcal{O}_S\)-module on \(S\) is free and the final statement follows.

Lemma

Let \(S\) be a scheme. Let \(G\) be a group scheme over \(S\). Let \(s \in S\). Then the composition \[T_{G/S, e(s)} \oplus T_{G/S, e(s)} = T_{G \times_S G/S, (e(s), e(s))} \rightarrow T_{G/S, e(s)}\] is addition of tangent vectors. Here the \(=\) comes from Varieties, Lemma 0BEB and the right arrow is induced from \(m : G \times_S G \to G\) via Varieties, Lemma 0B2F.

Proof

We will use Varieties, Equation (0BEA) and work with tangent vectors in fibres. An element \(\theta\) in the first factor \(T_{G_s/s, e(s)}\) is the image of \(\theta\) via the map \(T_{G_s/s, e(s)} \to T_{G_s \times G_s/s, (e(s), e(s))}\) coming from \((1, e) : G_s \to G_s \times G_s\). Since \(m \circ (1, e) = 1\) we see that \(\theta\) maps to \(\theta\) by functoriality. Since the map is linear we see that \((\theta_1, \theta_2)\) maps to \(\theta_1 + \theta_2\).

Lemma

Let \(k\) be a field and let \(G\) be a connected commutative group scheme locally of finite type over \(k\). If \(n \geq 1\) is invertible in \(k\), then \[[n] : G \longrightarrow G\] is surjective.

Proof

We may extend the ground field to an algebraic closure. The reduction \(G_{red}\) is a subgroup scheme by Lemma 047R; it is connected and has the same underlying topological space as \(G\). It is smooth by Lemma 047P in positive characteristic and by Lemma 047N in characteristic zero.

The differential of \([n]\) at the identity of \(G_{red}\) is multiplication by \(n\), by repeated application of Lemma 0BF5. It is therefore an isomorphism. Translation shows that \([n]\) is ètale everywhere on \(G_{red}\). Its image is consequently an open subgroup of \(G_{red}\), and it is closed by Lemma 047S. Connectedness gives \([n](G_{red})=G_{red}\). This proves surjectivity on underlying topological spaces for \(G\), which is the assertion.

Lemma

Let \(S\) be a scheme and let \(f : G \to S\) be a group scheme locally of finite presentation. Let \(\mathcal{P}\) be either “smooth” or “flat”. Denote by \(W \subset G\) the locus where \(f\) has property \(\mathcal{P}\), and set \[U = e^{-1}(W) \subset S.\] Then \(U\) is open in \(S\), the open subscheme \(W_U = W \times_S U\) is an open subgroup scheme of \(G_U\), and every section of \(G_U \to U\) factors through \(W_U\).

Proof

The smooth locus is open by the definition of a smooth morphism. The flat locus is open by More on Morphisms, Theorem 0399, applied to \(\mathcal{O}_G\). Thus \(U\) is open. Formation of \(W\) commutes with flat base change. In the smooth case this follows from Morphisms, Lemma 02V4, and in the flat case from More on Morphisms, Lemma 047C.

We may replace \(S\) by \(U\) and assume that \(e(S) \subset W\). In either case \(W \to S\) is flat. Consider the base change \(G_W = W \times_S G\). The property \(\mathcal{P}\) locus in \(G_W\) is \(W \times_S W\). The translation by the universal point \[G_W \longrightarrow G_W, \qquad (w, g) \longmapsto (w, m(w, g))\] is an isomorphism over \(W\), and hence it preserves this locus. It follows that multiplication on \(G\) maps \(W \times_S W\) into \(W\). The inverse morphism of \(G\) preserves \(W\) as well, since it is an isomorphism over \(S\). As \(e(S) \subset W\), this proves that \(W\) is an open subgroup scheme of \(G\).

Let \(\sigma : S \to G\) be a section. Translation by \(\sigma\) is an isomorphism of \(G\) over \(S\) and therefore preserves \(W\). Since it maps \(e(S)\) onto \(\sigma(S)\), the section \(\sigma\) factors through \(W\).

Lemma

In the situation of Lemma groupoids-lemma-smooth-flat-locus-group-scheme, assume that \(f\) has property \(\mathcal{P}\) at every point of \(e(S)\). Then \(f\) has property \(\mathcal{P}\) at every point in the image of any section of \(f\) and at every point of \[G^0 = \bigcup_{s \in S} (G_s)^0,\] where \((G_s)^0\) is the connected component of the identity in \(G_s\).

Proof

Lemma groupoids-lemma-smooth-flat-locus-group-scheme shows that the \(\mathcal{P}\) locus \(W\) is an open subgroup scheme of \(G\) containing every section. For \(s \in S\), the subgroup \(W_s \subset G_s\) is both open and closed. Indeed, if \(x \in G_s \setminus W_s\) and \(K = \kappa(x)\), then translation by the corresponding \(K\)-point of \(G_K\) maps the open \(W_K\) to an open whose image in \(G_s\) is an open neighbourhood of \(x\) disjoint from \(W_s\). Here the projection \(G_K \to G_s\) is open by Morphisms, Lemma 0383. Thus the complement of \(W_s\) is open. Since \(W_s\) contains the identity, it contains the connected component \((G_s)^0\).

Properties of group schemes over a field

In this section we collect some properties of group schemes over a field. In the case of group schemes which are (locally) algebraic over a field we can say a lot more, see Section 0BF6.

Lemma

If \((G, m)\) is a group scheme over a field \(k\), then the multiplication map \(m : G \times_k G \to G\) is open.

Proof

The multiplication map is isomorphic to the projection map \(\text{pr}_0 : G \times_k G \to G\) because the diagram \[\xymatrix{ G \times_k G \ar[d]^m \ar[rrr]_{(g, g') \mapsto (m(g, g'), g')} & & & G \times_k G \ar[d]^{(g, g') \mapsto g} \\ G \ar[rrr]^{\text{id}} & & & G }\] is commutative with isomorphisms as horizontal arrows. The projection is open by Morphisms, Lemma 0383.

Lemma

Let \((G, m)\) be a group scheme over a field \(k\). Let \(U \subset G\) open and \(T \to G\) a morphism of schemes. Then the image of the composition \(T \times_k U \to G \times_k G \to G\) is open.

Proof

For any field extension \(K/k\) the morphism \(G_K \to G\) is open (Morphisms, Lemma 0383). Every point \(\xi\) of \(T \times_k U\) is the image of a morphism \((t, u) : \Spec(K) \to T \times_k U\) for some \(K\). Then the image of \(T_K \times_K U_K = (T \times_k U)_K \to G_K\) contains the translate \(t \cdot U_K\) which is open. Combining these facts we see that the image of \(T \times_k U \to G\) contains an open neighbourhood of the image of \(\xi\). Since \(\xi\) was arbitrary we win.

Lemma

Let \(G\) be a group scheme over a field. Then \(G\) is a separated scheme.

Proof

Say \(S = \Spec(k)\) with \(k\) a field, and let \(G\) be a group scheme over \(S\). By Lemma 047G we have to show that \(e : S \to G\) is a closed immersion. By Morphisms, Lemma 01TE the image of \(e : S \to G\) is a closed point of \(G\). It is clear that \(\mathcal{O}_G \to e_*\mathcal{O}_S\) is surjective, since \(e_*\mathcal{O}_S\) is a skyscraper sheaf supported at the neutral element of \(G\) with value \(k\). We conclude that \(e\) is a closed immersion by Schemes, Lemma 01LD.

Lemma

Let \(G\) be a group scheme over a field \(k\). Then

1. every local ring \(\mathcal{O}_{G, g}\) of \(G\) has a unique minimal prime ideal,

2. there is exactly one irreducible component \(Z\) of \(G\) passing through \(e\), and

3. \(Z\) is geometrically irreducible over \(k\).

Proof

For any point \(g \in G\) there exists a field extension \(K/k\) and a \(K\)-valued point \(g' \in G(K)\) mapping to \(g\). If we think of \(g'\) as a \(K\)-rational point of the group scheme \(G_K\), then we see that \(\mathcal{O}_{G, g} \to \mathcal{O}_{G_K, g'}\) is a faithfully flat local ring map (as \(G_K \to G\) is flat, and a local flat ring map is faithfully flat, see Algebra, Lemma 00HR). The result for \(\mathcal{O}_{G_K, g'}\) implies the result for \(\mathcal{O}_{G, g}\), see Algebra, Lemma 00FK. Hence in order to prove (1) it suffices to prove it for \(k\)-rational points \(g\) of \(G\). In this case translation by \(g\) defines an automorphism \(G \to G\) which maps \(e\) to \(g\). Hence \(\mathcal{O}_{G, g} \cong \mathcal{O}_{G, e}\). In this way we see that (2) implies (1), since irreducible components passing through \(e\) correspond one to one with minimal prime ideals of \(\mathcal{O}_{G, e}\).

In order to prove (2) and (3) it suffices to prove (2) when \(k\) is algebraically closed. In this case, let \(Z_1\), \(Z_2\) be two irreducible components of \(G\) passing through \(e\). Since \(k\) is algebraically closed the closed subscheme \(Z_1 \times_k Z_2 \subset G \times_k G\) is irreducible too, see Varieties, Lemma 038F. Hence \(m(Z_1 \times_k Z_2)\) is contained in an irreducible component of \(G\). On the other hand it contains \(Z_1\) and \(Z_2\) since \(m|_{e \times G} = \text{id}_G\) and \(m|_{G \times e} = \text{id}_G\). We conclude \(Z_1 = Z_2\) as desired.

Remark

Warning: The result of Lemma 047M does not mean that every irreducible component of \(G/k\) is geometrically irreducible. For example the group scheme \(\mu_{3, \mathbf{Q}} = \Spec(\mathbf{Q}[x]/(x^3 - 1))\) over \(\mathbf{Q}\) has two irreducible components corresponding to the factorization \(x^3 - 1 = (x - 1)(x^2 + x + 1)\). The first factor corresponds to the irreducible component passing through the identity, and the second irreducible component is not geometrically irreducible over \(\Spec(\mathbf{Q})\).

Lemma

Let \(G\) be a group scheme over a perfect field \(k\). Then the reduction \(G_{red}\) of \(G\) is a closed subgroup scheme of \(G\).

Proof

Omitted. Hint: Use that \(G_{red} \times_k G_{red}\) is reduced by Varieties, Lemmas 020I and 035Z.

Lemma

Let \(k\) be a field. Let \(\psi : G' \to G\) be a morphism of group schemes over \(k\). If \(\psi(G')\) is open in \(G\), then \(\psi(G')\) is closed in \(G\).

Proof

Let \(U = \psi(G') \subset G\). Let \(Z = G \setminus \psi(G') = G \setminus U\) with the reduced induced closed subscheme structure. By Lemma 0B7N the image of \[Z \times_k G' \longrightarrow Z \times_k U \longrightarrow G\] is open (the first arrow is surjective). On the other hand, since \(\psi\) is a homomorphism of group schemes, the image of \(Z \times_k G' \to G\) is contained in \(Z\) (because translation by \(\psi(g')\) preserves \(U\) for all points \(g'\) of \(G'\); small detail omitted). Hence \(Z \subset G\) is an open subset (although not necessarily an open subscheme). Thus \(U = \psi(G')\) is closed.

Lemma

Let \(i : G' \to G\) be an immersion of group schemes over a field \(k\). Then \(i\) is a closed immersion, i.e., \(i(G')\) is a closed subgroup scheme of \(G\).

Proof

To show that \(i\) is a closed immersion it suffices to show that \(i(G')\) is a closed subset of \(G\). Let \(k \subset k'\) be a perfect extension of \(k\). If \(i(G'_{k'}) \subset G_{k'}\) is closed, then \(i(G') \subset G\) is closed by Morphisms, Lemma 02JY (as \(G_{k'} \to G\) is flat, quasi-compact and surjective). Hence we may and do assume \(k\) is perfect. We will use without further mention that products of reduced schemes over \(k\) are reduced. We may replace \(G'\) and \(G\) by their reductions, see Lemma 047R. Let \(\overline{G'} \subset G\) be the closure of \(i(G')\) viewed as a reduced closed subscheme. By Varieties, Lemma 047B we conclude that \(\overline{G'} \times_k \overline{G'}\) is the closure of the image of \(G' \times_k G' \to G \times_k G\). Hence \[m\Big(\overline{G'} \times_k \overline{G'}\Big) \subset \overline{G'}\] as \(m\) is continuous. It follows that \(\overline{G'} \subset G\) is a (reduced) closed subgroup scheme. By Lemma 047S we see that \(i(G') \subset \overline{G'}\) is also closed which implies that \(i(G') = \overline{G'}\) as desired.

Lemma

Let \(G\) be a group scheme over a field \(k\). If \(G\) is irreducible, then \(G\) is quasi-compact.

Proof

Suppose that \(K/k\) is a field extension. If \(G_K\) is quasi-compact, then \(G\) is too as \(G_K \to G\) is surjective. By Lemma 047M we see that \(G_K\) is irreducible. Hence it suffices to prove the lemma after replacing \(k\) by some extension. Choose \(K\) to be an algebraically closed field extension of very large cardinality. Then by Varieties, Lemma 0479, we see that \(G_K\) is a Jacobson scheme all of whose closed points have residue field equal to \(K\). In other words we may assume \(G\) is a Jacobson scheme all of whose closed points have residue field \(k\).

Let \(U \subset G\) be a nonempty affine open. Let \(g \in G(k)\). Then \(gU \cap U \not = \emptyset\). Hence we see that \(g\) is in the image of the morphism \[U \times_{\Spec(k)} U \longrightarrow G, \quad (u_1, u_2) \longmapsto u_1u_2^{-1}\] Since the image of this morphism is open (Lemma 047K) we see that the image is all of \(G\) (because \(G\) is Jacobson and closed points are \(k\)-rational). Since \(U\) is affine, so is \(U \times_{\Spec(k)} U\). Hence \(G\) is the image of a quasi-compact scheme, hence quasi-compact.

Lemma

Let \(G\) be a group scheme over a field \(k\). If \(G\) is connected, then \(G\) is irreducible.

Proof

By Varieties, Lemma 04KV we see that \(G\) is geometrically connected. If we show that \(G_K\) is irreducible for some field extension \(K/k\), then the lemma follows. Hence we may apply Varieties, Lemma 0479 to reduce to the case where \(k\) is algebraically closed, \(G\) is a Jacobson scheme, and all the closed points are \(k\)-rational.

Let \(Z \subset G\) be the unique irreducible component of \(G\) passing through the neutral element, see Lemma 047M. Endowing \(Z\) with the reduced induced closed subscheme structure, we see that \(Z \times_k Z\) is reduced and irreducible (Varieties, Lemmas 035Z and 038F). We conclude that \(m|_{Z \times_k Z} : Z \times_k Z \to G\) factors through \(Z\). Hence \(Z\) becomes a closed subgroup scheme of \(G\).

To get a contradiction, assume there exists another irreducible component \(Z' \subset G\). Then \(Z \cap Z' = \emptyset\) by Lemma 047M. By Lemma 0B7P we see that \(Z\) is quasi-compact. Thus we may choose a quasi-compact open \(U \subset G\) with \(Z \subset U\) and \(U \cap Z' = \emptyset\). The image \(W\) of \(Z \times_k U \to G\) is open in \(G\) by Lemma 0B7N. On the other hand, \(W\) is quasi-compact as the image of a quasi-compact space. We claim that \(W\) is closed.

Proof of the claim. Since \(W\) is quasi-compact and \(G\) is separated (Lemma 047L), the inclusion \(W \to G\) is quasi-compact by Schemes, Lemma 03GI. We conclude that points in the closure of \(W\) are specializations of points of \(W\) (Morphisms, Lemma 02JQ). Thus we have to show that any irreducible component \(Z'' \subset G\) of \(G\) which meets \(W\) is contained in \(W\). As \(G\) is Jacobson and closed points are rational, \(Z'' \cap W\) has a rational point \(g \in Z''(k) \cap W(k)\) and hence \(Z'' = Zg\). But \(W = m(Z \times_k W)\) by construction, so \(Z'' \cap W \not = \emptyset\) implies \(Z'' \subset W\).

By the claim \(W \subset G\) is an open and closed subset of \(G\). Now \(W \cap Z' = \emptyset\) since otherwise by the argument given in the preceding paragraph we would get \(Z' = Zg\) for some \(g \in W(k)\). Then as \(Z\) is a subgroup we could even pick \(g \in U(k)\) which would contradict \(Z' \cap U = \emptyset\). Hence \(W \subset G\) is a proper open and closed subset which contradicts the assumption that \(G\) is connected.

Proposition

Let \(G\) be a group scheme over a field \(k\). There exists a canonical closed subgroup scheme \(G^0 \subset G\) with the following properties

1. \(G^0 \to G\) is a flat closed immersion,

2. \(G^0 \subset G\) is the connected component of the identity,

3. \(G^0\) is geometrically irreducible, and

4. \(G^0\) is quasi-compact.

Proof

Let \(G^0\) be the connected component of the identity with its canonical scheme structure (Morphisms, Definition 04PX). To show that \(G^0\) is a closed subgroup scheme we will use the criterion of Lemma 0G8L. The morphism \(e : \Spec(k) \to G\) factors through \(G^0\) as we chose \(G^0\) to be the connected component of \(G\) containing \(e\). Since \(i : G \to G\) is an automorphism fixing \(e\), we see that \(i\) sends \(G^0\) into itself. By Varieties, Lemma 056R the scheme \(G^0\) is geometrically connected over \(k\). Thus \(G^0 \times_k G^0\) is connected (Varieties, Lemma 0385). Thus \(m(G^0 \times_k G^0) \subset G^0\) set theoretically. Thus \(m|_{G^0 \times_k G^0} : G^0 \times_k G^0 \to G\) factors through \(G^0\) by Morphisms, Lemma 04PW. Hence \(G^0\) is a closed subgroup scheme of \(G\). By Lemma 0B7Q we see that \(G^0\) is irreducible. By Lemma 047M we see that \(G^0\) is geometrically irreducible. By Lemma 0B7P we see that \(G^0\) is quasi-compact.

Lemma

Let \(k\) be a field and let \(G\) be a commutative group scheme of finite type over \(k\). Let \(\overline{k}\) be an algebraic closure of \(k\). Assume that \(G^0_{\overline{k}}\) contains no closed subgroup scheme isomorphic to \(\mathbf{G}_{a, \overline{k}}\). Then \((G^0)_{red}\) is a smooth connected closed subgroup scheme of \(G\). Moreover, \[((G^0)_{red})_{\overline{k}}=(G^0_{\overline{k}})_{red}\] as closed subschemes of \(G_{\overline{k}}\).

Proof

Set \(\overline{G}=G_{\overline{k}}\) and \(H=(\overline{G}^0)_{red}\). Since \(\overline{k}\) is perfect, \(H\) is a closed subgroup scheme of \(\overline{G}\) by Lemma 047R, and it is smooth by Lemma 047P in positive characteristic and by Lemma 047N in characteristic zero.

The standard structure theorem for connected commutative algebraic groups says that the hypothesis on additive subgroups makes \(H\) an extension of an abelian variety by a torus. Consequently multiplication by every positive integer on \(H\) is surjective. This is immediate for a torus, follows from Lemma 0BFG for an abelian variety, and passes to an extension in the fppf topology.

By the standard quotient theorem for algebraic groups over a field, the fppf quotient \(Q=\overline{G}/H\) is represented by a group scheme. It is finite: its underlying space is the finite set of connected components of \(\overline{G}\) and its identity component is the finite infinitesimal quotient of \(\overline{G}^0\) by its reduction. Choose \(m>0\) annihilating \(Q\). Such an \(m\) exists because the finite ètale quotient of \(Q\) has finite exponent and, in positive characteristic, the connected part of \(Q\) is killed by a power of the characteristic. It follows that \[[m] : \overline{G}\longrightarrow\overline{G}\] factors through \(H\). Its restriction to \(H\) is surjective, so the scheme theoretic image of \([m]\) is exactly \(H\); here we also use that \(H\) is reduced and Morphisms, Lemma 056B.

Let \(Z\subset G\) be the scheme theoretic image of \([m]:G\to G\). The morphism \([m]\) is quasi-compact. Hence formation of its scheme theoretic image commutes with the flat base change \(\Spec(\overline{k})\to\Spec(k)\) by Morphisms, Lemma 081I. Thus \(Z_{\overline{k}}=H\). In particular \(Z\) is smooth and geometrically connected over \(k\). The identity, inverse, and multiplication maps make \(Z\) a closed subgroup scheme because this can be checked after the faithfully flat extension \(\overline{k}/k\). Finally \(Z\) has the identity component of \(G\) as its underlying space and is reduced. Therefore \(Z=(G^0)_{red}\), and the displayed base change identity follows.

Lemma

Let \(k\) be a field. Let \(T = \Spec(A)\) where \(A\) is a directed colimit of algebras which are finite products of copies of \(k\). For any scheme \(X\) over \(k\) we have \(|T \times_k X| = |T| \times |X|\) as topological spaces.

Proof

By taking an affine open covering we reduce to the case of an affine \(X\). Say \(X = \Spec(B)\). Write \(A = \colim A_i\) with \(A_i = \prod_{t \in T_i} k\) and \(T_i\) finite. Then \(T_i = |\Spec(A_i)|\) with the discrete topology and the transition morphisms \(A_i \to A_{i'}\) are given by set maps \(T_{i'} \to T_i\). Thus \(|T| = \lim T_i\) as a topological space, see Limits, Lemma 01YY. Similarly we have \[\begin{align*} |T \times_k X| & = |\Spec(A \otimes_k B)| \\ & = |\Spec(\colim A_i \otimes_k B)| \\ & = \lim |\Spec(A_i \otimes_k B)| \\ & = \lim |\Spec(\prod\nolimits_{t \in T_i} B)| \\ & = \lim T_i \times |X| \\ & = (\lim T_i) \times |X| \\ & = |T| \times |X| \end{align*}\] by the lemma above and the fact that limits commute with limits.

The following lemma says that in fact we can put a “algebraic profinite family of points” in an affine open. We urge the reader to read Lemma 0B7S first.

Lemma

Let \(k\) be an algebraically closed field. Let \(G\) be a group scheme over \(k\). Assume that \(G\) is Jacobson and that all closed points are \(k\)-rational. Let \(T = \Spec(A)\) where \(A\) is a directed colimit of algebras which are finite products of copies of \(k\). For any morphism \(f : T \to G\) there exists an affine open \(U \subset G\) containing \(f(T)\).

Proof

Let \(G^0 \subset G\) be the closed subgroup scheme found in Proposition 0B7R. The first two paragraphs serve to reduce to the case \(G = G^0\).

Observe that \(T\) is a directed inverse limit of finite topological spaces (Limits, Lemma 01YY), hence profinite as a topological space (Topology, Definition 08ZX). Let \(W \subset G\) be a quasi-compact open containing the image of \(T \to G\). After replacing \(W\) by the image of \(G^0 \times W \to G \times G \to G\) we may assume that \(W\) is invariant under the action of left translation by \(G^0\), see Lemma 0B7N. Consider the composition \[\psi = \pi \circ f : T \xrightarrow{f} W \xrightarrow{\pi} \pi_0(W)\] The space \(\pi_0(W)\) is profinite (Topology, Lemma 0906 and Properties, Lemma 094L). Let \(F_\xi \subset T\) be the fibre of \(T \to \pi_0(W)\) over \(\xi \in \pi_0(W)\). Suppose that for every \(\xi \in \pi_0(W)\) we have found affine open \(U_\xi \subset W\) such that \(f(F_\xi) \subset U_\xi\). As \(\psi\) is closed by Topology, Lemma 08YD, \(\psi(T \setminus f^{-1}(U_\xi))\) is a closed subset of \(\pi_0(W)\) that does not contain \(\xi\). Hence its complement \(V_\xi\) is an open neighborhood of \(\xi\) and \(\psi^{-1}(V_\xi) \subseteq f^{-1}(U_\xi)\). The open covering of \(\pi_0(W)\) by the \(V_\xi\) may be refined by a finite covering \(\pi_0(W) = V_1 \amalg \ldots \amalg V_n\) of pairwise disjoint open and closed subsets (Topology, Lemma 08ZZ). For every \(i = 1, \ldots, n\) pick \(\xi_i \in \pi_0(W)\) such that \(V_i \subset V_{\xi_i}\). As \(V_i\) is closed and \(U_{\xi_i}\) affine, \(U_{\xi_i} \cap \pi^{-1}(V_i)\) is affine as well. Hence replacing \(U_{\xi_i}\) by \(U_{\xi_i} \cap \pi^{-1}(V_i)\) we have \(\psi^{-1}(V_i) = f^{-1}(U_{\xi_i})\). As the \(V_i\)’s cover \(\pi_0(W)\), the \(f^{-1}(U_{\xi_i})\)’s cover \(T\) and so \(f(T) \subseteq U = \bigcup_i U_{\xi_i}\). Then \(U\) is the disjoint union of finitely many affine opens and hence affine as desired.

Let \(Z\) be a connected component of \(G\) which meets \(f(T)\). Then \(Z\) has a \(k\)-rational point \(z\) (because all residue fields of the scheme \(T\) are isomorphic to \(k\)). Hence \(Z = G^0 z\). By our choice of \(W\), we see that \(Z \subset W\). The argument in the preceding paragraph reduces us to the problem of finding an affine open neighbourhood of \(f(T) \cap Z\) in \(W\). After translation by a rational point we may assume that \(Z = G^0\) (details omitted). Observe that the scheme theoretic inverse image \(T' = f^{-1}(G^0) \subset T\) is a closed subscheme, which has the same type. After replacing \(T\) by \(T'\) we may assume that \(f(T) \subset G^0\). Choose an affine open neighbourhood \(U \subset W\) of \(e \in G\), so that in particular \(U \cap G^0\) is nonempty. We will show there exists a \(g \in G^0(k)\) such that \(f(T) \subset g^{-1}U\). This will finish the proof as \(g^{-1}U \subset W\) by the left \(G^0\)-invariance of \(W\).

The arguments in the preceding two paragraphs allow us to pass to \(G^0\) and reduce the problem to the following: Assume \(G\) is irreducible and \(U \subset G\) an affine open neighbourhood of \(e\). Show that \(f(T) \subset g^{-1}U\) for some \(g \in G(k)\). Consider the morphism \[U \times_k T \longrightarrow G \times_k T,\quad (u, t) \longrightarrow (uf(t)^{-1}, t)\] which is an open immersion (because the extension of this morphism to \(G \times_k T \to G \times_k T\) is an isomorphism). By our assumption on \(T\) we see that we have \(|U \times_k T| = |U| \times |T|\) and similarly for \(G \times_k T\), see Lemma 0B7T. Hence the image of the displayed open immersion is a finite union of boxes \(\bigcup_{i = 1, \ldots, n} U_i \times V_i\) with \(V_i \subset T\) and \(U_i \subset G\) quasi-compact open. This means that the possible opens \(Uf(t)^{-1}\), \(t \in T\) are finite in number, say \(Uf(t_1)^{-1}, \ldots, Uf(t_r)^{-1}\). Since \(G\) is irreducible the intersection \[Uf(t_1)^{-1} \cap \ldots \cap Uf(t_r)^{-1}\] is nonempty and since \(G\) is Jacobson with closed points \(k\)-rational, we can choose a \(k\)-valued point \(g \in G(k)\) of this intersection. Then we see that \(g \in Uf(t)^{-1}\) for all \(t \in T\) which means that \(f(t) \in g^{-1}U\) as desired.

Remark

If \(G\) is a group scheme over a field, is there always a quasi-compact open and closed subgroup scheme? By Proposition 0B7R this question is only interesting if \(G\) has infinitely many connected components (geometrically).

Lemma

Let \(G\) be a group scheme over a field. There exists an open and closed subscheme \(G' \subset G\) which is a countable union of affines.

Proof

Let \(e \in U(k)\) be a quasi-compact open neighbourhood of the identity element. By replacing \(U\) by \(U \cap i(U)\) we may assume that \(U\) is invariant under the inverse map. As \(G\) is separated this is still a quasi-compact set. Set \[G' = \bigcup\nolimits_{n \geq 1} m_n(U \times_k \ldots \times_k U)\] where \(m_n : G \times_k \ldots \times_k G \to G\) is the \(n\)-slot multiplication map \((g_1, \ldots, g_n) \mapsto m(m(\ldots (m(g_1, g_2), g_3), \ldots ), g_n)\). Each of these maps are open (see Lemma 047K) hence \(G'\) is an open subgroup scheme. By Lemma 047S it is also a closed subgroup scheme.

Properties of algebraic group schemes

Recall that a scheme over a field \(k\) is (locally) algebraic if it is (locally) of finite type over \(\Spec(k)\), see Varieties, Definition 06LG. This is the sense of algebraic we are using in the title of this section.

Lemma

Let \(k\) be a field. Let \(G\) be a locally algebraic group scheme over \(k\). Then \(G\) is equidimensional and \(\dim(G) = \dim_g(G)\) for all \(g \in G\). For any closed point \(g \in G\) we have \(\dim(G) = \dim(\mathcal{O}_{G, g})\).

Proof

Let us first prove that \(\dim_g(G) = \dim_{g'}(G)\) for any pair of points \(g, g' \in G\). By Morphisms, Lemma 02FY we may extend the ground field at will. Hence we may assume that both \(g\) and \(g'\) are defined over \(k\). Hence there exists an automorphism of \(G\) mapping \(g\) to \(g'\), whence the equality. By Morphisms, Lemma 02FX we have \(\dim_g(G) = \dim(\mathcal{O}_{G, g}) + \text{trdeg}_k(\kappa(g))\). On the other hand, the dimension of \(G\) (or any open subset of \(G\)) is the supremum of the dimensions of the local rings of \(G\), see Properties, Lemma 02IZ. Clearly this is maximal for closed points \(g\) in which case \(\text{trdeg}_k(\kappa(g)) = 0\) (by the Hilbert Nullstellensatz, see Morphisms, Section 01T9). Hence the lemma follows.

The following result is sometimes referred to as Cartier’s theorem.

Lemma

Let \(k\) be a field of characteristic \(0\). Let \(G\) be a locally algebraic group scheme over \(k\). Then the structure morphism \(G \to \Spec(k)\) is smooth, i.e., \(G\) is a smooth group scheme.

Proof

By Lemma 047I the module of differentials of \(G\) over \(k\) is free. Hence smoothness follows from Varieties, Lemma 04QN.

Remark

Any group scheme over a field of characteristic \(0\) is reduced, see [Perrin-thesis, I, Theorem 1.1 and I, Corollary 3.9, and II, Theorem 2.4] and also [Perrin, Proposition 4.2.8]. This was a question raised in [Oort, page 80]. We have seen in Lemma 047N that this holds when the group scheme is locally of finite type.

Lemma

Let \(k\) be a perfect field of characteristic \(p > 0\) (see Lemma 047N for the characteristic zero case). Let \(G\) be a locally algebraic group scheme over \(k\). If \(G\) is reduced then the structure morphism \(G \to \Spec(k)\) is smooth, i.e., \(G\) is a smooth group scheme.

Proof

By Lemma 047I the sheaf \(\Omega_{G/k}\) is free. Hence the lemma follows from Varieties, Lemma 04QP.

Remark

Let \(k\) be a field of characteristic \(p > 0\). Let \(\alpha \in k\) be an element which is not a \(p\)th power. The closed subgroup scheme \[G = V(x^p + \alpha y^p) \subset \mathbf{G}_{a, k}^2\] is reduced and irreducible but not smooth (not even normal).

The following lemma is a special case of Lemma 0B7U with a somewhat easier proof.

Lemma

Let \(k\) be an algebraically closed field. Let \(G\) be a locally algebraic group scheme over \(k\). Let \(g_1, \ldots, g_n \in G(k)\) be \(k\)-rational points. Then there exists an affine open \(U \subset G\) containing \(g_1, \ldots, g_n\).

Proof

We first argue by induction on \(n\) that we may assume all \(g_i\) are on the same connected component of \(G\). Namely, if not, then we can find a decomposition \(G = W_1 \amalg W_2\) with \(W_i\) open in \(G\) and (after possibly renumbering) \(g_1, \ldots, g_r \in W_1\) and \(g_{r + 1}, \ldots, g_n \in W_2\) for some \(0 < r < n\). By induction we can find affine opens \(U_1\) and \(U_2\) of \(G\) with \(g_1, \ldots, g_r \in U_1\) and \(g_{r + 1}, \ldots, g_n \in U_2\). Then \[g_1, \ldots, g_n \in (U_1 \cap W_1) \cup (U_2 \cap W_2)\] is a solution to the problem. Thus we may assume \(g_1, \ldots, g_n\) are all on the same connected component of \(G\). Translating by \(g_1^{-1}\) we may assume \(g_1, \ldots, g_n \in G^0\) where \(G^0 \subset G\) is as in Proposition 0B7R. Choose an affine open neighbourhood \(U\) of \(e\), in particular \(U \cap G^0\) is nonempty. Since \(G^0\) is irreducible we see that \[G^0 \cap (Ug_1^{-1} \cap \ldots \cap Ug_n^{-1})\] is nonempty. Since \(G \to \Spec(k)\) is locally of finite type, also \(G^0 \to \Spec(k)\) is locally of finite type, hence any nonempty open has a \(k\)-rational point. Thus we can pick \(g \in G^0(k)\) with \(g \in Ug_i^{-1}\) for all \(i\). Then \(g_i \in g^{-1}U\) for all \(i\) and \(g^{-1}U\) is the affine open we were looking for.

Lemma

Let \(k\) be a field. Let \(G\) be an algebraic group scheme over \(k\). Then \(G\) is quasi-projective over \(k\).

Proof

By Varieties, Lemma 0BDC we may assume that \(k\) is algebraically closed. Let \(G^0 \subset G\) be the connected component of \(G\) as in Proposition 0B7R. Then every other connected component of \(G\) has a \(k\)-rational point and hence is isomorphic to \(G^0\) as a scheme. Since \(G\) is quasi-compact and Noetherian, there are finitely many of these connected components. Thus we reduce to the case discussed in the next paragraph.

Let \(G\) be a connected algebraic group scheme over an algebraically closed field \(k\). If the characteristic of \(k\) is zero, then \(G\) is smooth over \(k\) by Lemma 047N. If the characteristic of \(k\) is \(p > 0\), then we let \(H = G_{red}\) be the reduction of \(G\). By Divisors, Proposition 0BD4 it suffices to show that \(H\) has an ample invertible sheaf. (For an algebraic scheme over \(k\) having an ample invertible sheaf is equivalent to being quasi-projective over \(k\), see for example the very general More on Morphisms, Lemma 0B42.) By Lemma 047R we see that \(H\) is a group scheme over \(k\). By Lemma 047P we see that \(H\) is smooth over \(k\). This reduces us to the situation discussed in the next paragraph.

Let \(G\) be a quasi-compact irreducible smooth group scheme over an algebraically closed field \(k\). Observe that the local rings of \(G\) are regular and hence UFDs (Varieties, Lemma 056S and More on Algebra, Lemma 0AG0). The complement of a nonempty affine open of \(G\) is the support of an effective Cartier divisor \(D\). This follows from Divisors, Lemma 0BCW. (Observe that \(G\) is separated by Lemma 047L.) We conclude there exists an effective Cartier divisor \(D \subset G\) such that \(G \setminus D\) is affine. We will use below that for any \(n \geq 1\) and \(g_1, \ldots, g_n \in G(k)\) the complement \(G \setminus \bigcup D g_i\) is affine. Namely, it is the intersection of the affine opens \(G \setminus Dg_i \cong G \setminus D\) in the separated scheme \(G\).

We may choose the top row of the diagram \[\xymatrix{ G & U \ar[l]_j \ar[r]^\pi & \mathbf{A}^d_k \\ & W \ar[r]^{\pi'} \ar[u] & V \ar[u] }\] such that \(U \not = \emptyset\), \(j : U \to G\) is an open immersion, and \(\pi\) is étale, see Morphisms, Lemma 054L. There is a nonempty affine open \(V \subset \mathbf{A}^d_k\) such that with \(W = \pi^{-1}(V)\) the morphism \(\pi' = \pi|_W : W \to V\) is finite étale. In particular \(\pi'\) is finite locally free, say of degree \(n\). Consider the effective Cartier divisor \[\mathcal{D} = \{(g, w) \mid m(g, j(w)) \in D\} \subset G \times W\] (This is the restriction to \(G \times W\) of the pullback of \(D \subset G\) under the flat morphism \(m : G \times G \to G\).) Consider the closed subset¹ \(T = (1 \times \pi')(\mathcal{D}) \subset G \times V\). Since \(\pi'\) is finite locally free, every irreducible component of \(T\) has codimension \(1\) in \(G \times V\). Since \(G \times V\) is smooth over \(k\) we conclude these components are effective Cartier divisors (Divisors, Lemma 0AGA and lemmas cited above) and hence \(T\) is the support of an effective Cartier divisor \(E\) in \(G \times V\). If \(v \in V(k)\), then \((\pi')^{-1}(v) = \{w_1, \ldots, w_n\} \subset W(k)\) and we see that \[E_v = \bigcup\nolimits_{i = 1, \ldots, n} D j(w_i)^{-1}\] in \(G\) set theoretically. In particular we see that \(G \setminus E_v\) is affine open (see above). Moreover, if \(g \in G(k)\), then there exists a \(v \in V\) such that \(g \not \in E_v\). Namely, the set \(W'\) of \(w \in W\) such that \(g \not \in Dj(w)^{-1}\) is nonempty open and it suffices to pick \(v\) such that the fibre of \(W' \to V\) over \(v\) has \(n\) elements.

Consider the invertible sheaf \(\mathcal{M} = \mathcal{O}_{G \times V}(E)\) on \(G \times V\). By Varieties, Lemma 0BEH the isomorphism class \(\mathcal{L}\) of the restriction \(\mathcal{M}_v = \mathcal{O}_G(E_v)\) is independent of \(v \in V(k)\). On the other hand, for every \(g \in G(k)\) we can find a \(v\) such that \(g \not \in E_v\) and such that \(G \setminus E_v\) is affine. Thus the canonical section (Divisors, Definition 01WX) of \(\mathcal{O}_G(E_v)\) corresponds to a section \(s_v\) of \(\mathcal{L}\) which does not vanish at \(g\) and such that \(G_{s_v}\) is affine. This means that \(\mathcal{L}\) is ample by definition (Properties, Definition 01PS).

Lemma

Let \(k\) be a field. Let \(G\) be a locally algebraic group scheme over \(k\). Then the center of \(G\) is a closed subgroup scheme of \(G\).

Proof

Let \(\text{Aut}(G)\) denote the contravariant functor on the category of schemes over \(k\) which associates to \(S/k\) the set of automorphisms of the base change \(G_S\) as a group scheme over \(S\). There is a natural transformation \[G \longrightarrow \text{Aut}(G),\quad g \longmapsto \text{inn}_g\] sending an \(S\)-valued point \(g\) of \(G\) to the inner automorphism of \(G\) determined by \(g\). The center \(C\) of \(G\) is by definition the kernel of this transformation, i.e., the functor which to \(S\) associates those \(g \in G(S)\) whose associated inner automorphism is trivial. The statement of the lemma is that this functor is representable by a closed subgroup scheme of \(G\).

Choose an integer \(n \geq 1\). Let \(G_n \subset G\) be the \(n\)th infinitesimal neighbourhood of the identity element \(e\) of \(G\). For every scheme \(S/k\) the base change \(G_{n, S}\) is the \(n\)th infinitesimal neighbourhood of \(e_S : S \to G_S\). Thus we see that there is a natural transformation \(\text{Aut}(G) \to \text{Aut}(G_n)\) where the right hand side is the functor of automorphisms of \(G_n\) as a scheme (\(G_n\) isn’t in general a group scheme). Observe that \(G_n\) is the spectrum of an artinian local ring \(A_n\) with residue field \(k\) which has finite dimension as a \(k\)-vector space (Varieties, Lemma 06LH). Since every automorphism of \(G_n\) induces in particular an invertible linear map \(A_n \to A_n\), we obtain transformations of functors \[G \to \text{Aut}(G) \to \text{Aut}(G_n) \to \text{GL}(A_n)\] The final group valued functor is representable, see Example 022W, and the last arrow is visibly injective. Thus for every \(n\) we obtain a closed subgroup scheme \[H_n = \Ker(G \to \text{Aut}(G_n)) = \Ker(G \to \text{GL}(A_n)).\] As a first approximation we set \(H = \bigcap_{n \geq 1} H_n\) (scheme theoretic intersection). This is a closed subgroup scheme which contains the center \(C\).

Let \(h\) be an \(S\)-valued point of \(H\) with \(S\) locally Noetherian. Then the automorphism \(\text{inn}_h\) induces the identity on all the closed subschemes \(G_{n, S}\). Consider the equalizer \(K \subset G_S\) of \(\text{inn}_h\) and \(\text{id}_{G_S}\). This is a closed subgroup scheme of \(G_S\) over \(S\) containing the closed subschemes \(G_{n, S}\) for \(n \geq 1\). This implies that \(K\) contains an open neighbourhood of \(e(S) \subset G_S\), see Algebra, Remark 00IR. Let \(G^0 \subset G\) be as in Proposition 0B7R. Since \(G^0\) is geometrically irreducible, we conclude that \(K\) contains \(G^0_S\) (for any nonempty open \(U \subset G^0_{k'}\) and any field extension \(k'/k\) we have \(U \cdot U^{-1} = G^0_{k'}\), see proof of Lemma 0B7P). Applying this with \(S = H\) we find that \(G^0\) and \(H\) are subgroup schemes of \(G\) whose points commute: for any scheme \(S\) and any \(S\)-valued points \(g \in G^0(S)\), \(h \in H(S)\) we have \(gh = hg\) in \(G(S)\).

Assume that \(k\) is algebraically closed. Then we can pick a \(k\)-valued point \(g_i\) in each irreducible component \(G_i\) of \(G\). Observe that in this case the connected components of \(G\) are the irreducible components of \(G\) and are the translates of \(G^0\) by our \(g_i\). We claim that \[C = H \cap \bigcap\nolimits_i \operatorname{Eq}(\text{inn}_{g_i}, \text{id}_G) \quad (\text{scheme theoretic intersection})\] Namely, \(C\) is contained in the right hand side. On the other hand, every \(S\)-valued point \(h\) of the right hand side commutes with \(G^0\) and with \(g_i\) hence with everything in \(G = \bigcup G^0g_i\).

The case of a general base field \(k\) follows from the result for the algebraic closure \(\overline{k}\) by descent. Namely, let \(A \subset G_{\overline{k}}\) be the closed subgroup scheme representing the center of \(G_{\overline{k}}\). Then we have \[A \times_{\Spec(k)} \Spec(\overline{k}) = \Spec(\overline{k}) \times_{\Spec(k)} A\] as closed subschemes of \(G_{\overline{k} \otimes_k \overline{k}}\) by the functorial nature of the center. Hence we see that \(A\) descends to a closed subgroup scheme \(Z \subset G\) by Descent, Lemma 03I0 (and Descent, Lemma 02L6). Then \(Z\) represents \(C\) (small argument omitted) and the proof is complete.

Abelian varieties

An excellent reference for this material is Mumford’s book on abelian varieties, see [AVar]. We encourage the reader to look there. There are many equivalent definitions; here is one.

Definition

Let \(k\) be a field. An abelian variety is a group scheme over \(k\) which is also a proper, geometrically integral variety over \(k\)².

We prove a few lemmas about this notion and then we collect all the results together in Proposition 03RP.

Lemma

Let \(k\) be a field. Let \(A\) be an abelian variety over \(k\). Then \(A\) is projective.

Proof

This follows from Lemma 0BF7 and More on Morphisms, Lemma 0B45.

Lemma

Let \(k\) be a field. Let \(A\) be an abelian variety over \(k\). For any field extension \(K/k\) the base change \(A_K\) is an abelian variety over \(K\).

Proof

Omitted. Note that this is why we insisted on \(A\) being geometrically integral; without that condition this lemma (and many others below) would be wrong.

Lemma

Let \(k\) be a field. Let \(A\) be an abelian variety over \(k\). Then \(A\) is smooth over \(k\).

Proof

If \(k\) is perfect then this follows from Lemma 047N (characteristic zero) and Lemma 047P (positive characteristic). We can reduce the general case to this case by descent for smoothness (Descent, Lemma 02VL) and going to the perfect closure using Lemma 0BFB.

Lemma

An abelian variety is an abelian group scheme, i.e., the group law is commutative.

Proof

Let \(k\) be a field. Let \(A\) be an abelian variety over \(k\). By Lemma 0BFB we may replace \(k\) by its algebraic closure. Consider the morphism \[h : A \times_k A \longrightarrow A \times_k A,\quad (x, y) \longmapsto (x, xyx^{-1}y^{-1})\] This is a morphism over \(A\) via the first projection on either side. Let \(e \in A(k)\) be the unit. Then we see that \(h|_{e \times A}\) is constant with value \((e, e)\). By More on Morphisms, Lemma 0AH8 there exists an open neighbourhood \(U \subset A\) of \(e\) such that \(h|_{U \times A}\) factors through some \(Z \subset U \times A\) finite over \(U\). This means that for \(x \in U(k)\) the morphism \(A \to A\), \(y \mapsto xyx^{-1}y^{-1}\) takes finitely many values. Of course this means it is constant with value \(e\). Thus \((x, y) \mapsto xyx^{-1}y^{-1}\) is constant with value \(e\) on \(U \times A\) which implies that the group law on \(A\) is abelian.

Lemma

Let \(k\) be a field. Let \(A\) be an abelian variety over \(k\). Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_A\)-module. Then there is an isomorphism \[m_{1, 2, 3}^*\mathcal{L} \otimes m_1^*\mathcal{L} \otimes m_2^*\mathcal{L} \otimes m_3^*\mathcal{L} \cong m_{1, 2}^*\mathcal{L} \otimes m_{1, 3}^*\mathcal{L} \otimes m_{2, 3}^*\mathcal{L}\] of invertible modules on \(A \times_k A \times_k A\) where \(m_{i_1, \ldots, i_t} : A \times_k A \times_k A \to A\) is the morphism \((x_1, x_2, x_3) \mapsto \sum x_{i_j}\).

Proof

Apply the theorem of the cube (More on Morphisms, Theorem 0BF4) to the difference \[\mathcal{M} = m_{1, 2, 3}^*\mathcal{L} \otimes m_1^*\mathcal{L} \otimes m_2^*\mathcal{L} \otimes m_3^*\mathcal{L} \otimes m_{1, 2}^*\mathcal{L}^{\otimes -1} \otimes m_{1, 3}^*\mathcal{L}^{\otimes -1} \otimes m_{2, 3}^*\mathcal{L}^{\otimes -1}\] This works because the restriction of \(\mathcal{M}\) to \(A \times A \times e = A \times A\) is equal to \[n_{1, 2}^*\mathcal{L} \otimes n_1^*\mathcal{L} \otimes n_2^*\mathcal{L} \otimes n_{1, 2}^*\mathcal{L}^{\otimes -1} \otimes n_1^*\mathcal{L}^{\otimes -1} \otimes n_2^*\mathcal{L}^{\otimes -1} \cong \mathcal{O}_{A \times_k A}\] where \(n_{i_1, \ldots, i_t} : A \times_k A \to A\) is the morphism \((x_1, x_2) \mapsto \sum x_{i_j}\). Similarly for \(A \times e \times A\) and \(e \times A \times A\).

Lemma

Let \(k\) be a field. Let \(A\) be an abelian variety over \(k\). Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_A\)-module. Then \[[n]^*\mathcal{L} \cong \mathcal{L}^{\otimes n(n + 1)/2} \otimes ([-1]^*\mathcal{L})^{\otimes n(n - 1)/2}\] where \([n] : A \to A\) sends \(x\) to \(x + x + \ldots + x\) with \(n\) summands and where \([-1] : A \to A\) is the inverse of \(A\).

Proof

Consider the morphism \(A \to A \times_k A \times_k A\), \(x \mapsto (x, x, -x)\) where \(-x = [-1](x)\). Pulling back the relation of Lemma 0BFE we obtain \[\mathcal{L} \otimes \mathcal{L} \otimes \mathcal{L} \otimes [-1]^*\mathcal{L} \cong [2]^*\mathcal{L}\] which proves the result for \(n = 2\). By induction assume the result holds for \(1, 2, \ldots, n\). Then consider the morphism \(A \to A \times_k A \times_k A\), \(x \mapsto (x, x, [n - 1]x)\). Pulling back the relation of Lemma 0BFE we obtain \[[n + 1]^*\mathcal{L} \otimes \mathcal{L} \otimes \mathcal{L} \otimes [n - 1]^*\mathcal{L} \cong [2]^*\mathcal{L} \otimes [n]^*\mathcal{L} \otimes [n]^*\mathcal{L}\] and the result follows by elementary arithmetic.

Lemma

Let \(k\) be a field. Let \(A\) be an abelian variety over \(k\). Let \(d \in \mathbf{Z}\) be nonzero, and let \([d] : A \to A\) be multiplication by \(d\). Then \([d]\) is finite locally free of degree \(d^{2\dim(A)}\).

Proof

By Lemma 0BFA (and More on Morphisms, Lemma 0B45) we see that \(A\) has an ample invertible module \(\mathcal{L}\). Since \([-1] : A \to A\) is an automorphism, we see that \([-1]^*\mathcal{L}\) is an ample invertible \(\mathcal{O}_A\)-module as well. Thus \(\mathcal{N} = \mathcal{L} \otimes [-1]^*\mathcal{L}\) is ample, see Properties, Lemma 0890. Since \(\mathcal{N} \cong [-1]^*\mathcal{N}\) we see that \([d]^*\mathcal{N} \cong \mathcal{N}^{\otimes d^2}\) by Lemma 0BFF.

To get a contradiction, let \(C\) be a proper curve contained in a fibre of \([d]\). Then \(\mathcal{N}^{\otimes d^2}|_C \cong \mathcal{O}_C\) is an ample invertible \(\mathcal{O}_C\)-module of degree \(0\) which contradicts Varieties, Lemma 0B5X for example. (You can also use Varieties, Lemma 0BEV.) Thus every fibre of \([d]\) has dimension \(0\) and hence \([d]\) is finite for example by Cohomology of Schemes, Lemma 02OG. Moreover, since \(A\) is smooth over \(k\) by Lemma 0BFC we see that \([d] : A \to A\) is flat by Algebra, Lemma 00R4 (we also use that schemes smooth over fields are regular and that regular rings are Cohen-Macaulay, see Varieties, Lemma 056S and Algebra, Lemma 00NQ). Thus \([d]\) is finite flat hence finite locally free by Morphisms, Lemma 02KB.

Finally, we come to the formula for the degree. By Varieties, Lemma 0BEX we see that \[\deg_{\mathcal{N}^{\otimes d^2}}(A) = \deg([d]) \deg_\mathcal{N}(A)\] Since the degree of \(A\) with respect to \(\mathcal{N}^{\otimes d^2}\), respectively \(\mathcal{N}\) is the coefficient of \(n^{\dim(A)}\) in the polynomial \[n \longmapsto \chi(A, \mathcal{N}^{\otimes nd^2}),\quad \text{respectively}\quad n \longmapsto \chi(A, \mathcal{N}^{\otimes n})\] we see that \(\deg([d]) = d^{2 \dim(A)}\).

Lemma

Let \(k\) be a field. Let \(A\) be a nonzero abelian variety over \(k\). Then \([d] : A \to A\) is étale if and only if \(d\) is invertible in \(k\).

Proof

If \(d = 0\), then the fibre of \([d]\) over the unit is the positive-dimensional variety \(A\), so \([d]\) is not étale. Thus we may assume \(d \not= 0\). Observe that \([d](x + y) = [d](x) + [d](y)\). Since translation by a point is an automorphism of \(A\), we see that the set of points where \([d] : A \to A\) is étale is either empty or equal to \(A\) (some details omitted). Thus it suffices to check whether \([d]\) is étale at the unit \(e \in A(k)\). Since we know that \([d]\) is finite locally free (Lemma 0BFG) to see that it is étale at \(e\) is equivalent to proving that \(\text{d}[d] : T_{A/k, e} \to T_{A/k, e}\) is injective. See Varieties, Lemma 0B2G and Morphisms, Lemma 02GV. By Lemma 0BF5 we see that \(\text{d}[d]\) is given by multiplication by \(d\) on \(T_{A/k, e}\).

Lemma

Let \(k\) be a field of characteristic \(p > 0\). Let \(A\) be an abelian variety of dimension \(g\) over \(k\). The fibre of \([p] : A \to A\) over \(0\) has at most \(p^g\) distinct points.

Proof

To prove this, we may and do replace \(k\) by the algebraic closure. By Lemma 0BF5 the derivative of \([p]\) is multiplication by \(p\) as a map \(T_{A/k, e} \to T_{A/k, e}\) and hence is zero (compare with proof of Lemma 0BFH). Since \([p]\) commutes with translation we conclude that the derivative of \([p]\) is everywhere zero, i.e., that the induced map \([p]^*\Omega_{A/k} \to \Omega_{A/k}\) is zero. Looking at generic points, we find that the corresponding map \([p]^* : k(A) \to k(A)\) of function fields induces the zero map on \(\Omega_{k(A)/k}\). Let \(t_1, \ldots, t_g\) be a p-basis of \(k(A)\) over \(k\) (More on Algebra, Definition 07P1 and Lemma 07P2). Then \([p]^*(t_i)\) has a \(p\)th root by Algebra, Lemma 031W. We conclude that \(k(A)[x_1, \ldots, x_g]/(x_1^p - t_1, \ldots, x_g^p - t_g)\) is a subextension of \([p]^* : k(A) \to k(A)\). Thus we can find an affine open \(U \subset A\) such that \(t_i \in \mathcal{O}_A(U)\) and \(x_i \in \mathcal{O}_A([p]^{-1}(U))\). We obtain a factorization \[[p]^{-1}(U) \xrightarrow{\pi_1} \Spec(\mathcal{O}(U)[x_1, \ldots, x_g]/(x_1^p - t_1, \ldots, x_g^p - t_g)) \xrightarrow{\pi_2} U\] of \([p]\) over \(U\). After shrinking \(U\) we may assume that \(\pi_1\) is finite locally free (for example by generic flatness – actually it is already finite locally free in our case). By Lemma 0BFG we see that \([p]\) has degree \(p^{2g}\). Since \(\pi_2\) has degree \(p^g\) we see that \(\pi_1\) has degree \(p^g\) as well. The morphism \(\pi_2\) is a universal homeomorphism hence the fibres are singletons. We conclude that the (set theoretic) fibres of \([p]^{-1}(U) \to U\) are the fibres of \(\pi_1\). Hence they have at most \(p^g\) elements. Since \([p]\) is a homomorphism of group schemes over \(k\), the fibre of \([p] : A(k) \to A(k)\) has the same cardinality for every \(a \in A(k)\) and the proof is complete.

Proposition

Let \(A\) be an abelian variety over a field \(k\). Then

1. \(A\) is projective over \(k\),

2. \(A\) is a commutative group scheme,

3. the morphism \([n] : A \to A\) is surjective for all \(n \geq 1\),

4. if \(k\) is algebraically closed, then \(A(k)\) is a divisible abelian group,

5. \(A[n] = \Ker([n] : A \to A)\) is a finite group scheme of degree \(n^{2\dim A}\) over \(k\),

6. \(A[n]\) is étale over \(k\) if and only if \(n \in k^*\),

7. if \(n \in k^*\) and \(k\) is algebraically closed, then \(A(k)[n] \cong (\mathbf{Z}/n\mathbf{Z})^{\oplus 2\dim(A)}\),

8. if \(k\) is algebraically closed of characteristic \(p > 0\), then there exists an integer \(0 \leq f \leq \dim(A)\) such that \(A(k)[p^m] \cong (\mathbf{Z}/p^m\mathbf{Z})^{\oplus f}\) for all \(m \geq 1\).

Proof

Part (1) follows from Lemma 0BFA. Part (2) follows from Lemma 0BFD. Part (3) follows from Lemma 0BFG. If \(k\) is algebraically closed then surjective morphisms of varieties over \(k\) induce surjective maps on \(k\)-rational points, hence (4) follows from (3). Part (5) follows from Lemma 0BFG and the fact that a base change of a finite locally free morphism of degree \(N\) is a finite locally free morphism of degree \(N\). Part (6) follows from Lemma 0BFH. Namely, if \(n\) is invertible in \(k\), then \([n]\) is étale and hence \(A[n]\) is étale over \(k\). On the other hand, if \(n\) is not invertible in \(k\), then \([n]\) is not étale at \(e\) and it follows that \(A[n]\) is not étale over \(k\) at \(e\) (use Morphisms, Lemmas 02GV and 0475).

Assume \(k\) is algebraically closed. Set \(g = \dim(A)\). Proof of (7). Let \(\ell\) be a prime number which is invertible in \(k\). Then we see that \[A[\ell](k) = A(k)[\ell]\] is a finite abelian group, annihilated by \(\ell\), of order \(\ell^{2g}\). It follows that it is isomorphic to \((\mathbf{Z}/\ell\mathbf{Z})^{2g}\) by the structure theory for finite abelian groups. Next, we consider the short exact sequence \[0 \to A(k)[\ell] \to A(k)[\ell^2] \xrightarrow{\ell} A(k)[\ell] \to 0\] Arguing similarly as above we conclude that \(A(k)[\ell^2] \cong (\mathbf{Z}/\ell^2\mathbf{Z})^{2g}\). By induction on the exponent we find that \(A(k)[\ell^m] \cong (\mathbf{Z}/\ell^m\mathbf{Z})^{2g}\). For composite integers \(n\) prime to the characteristic of \(k\) we take primary parts and we find the correct shape of the \(n\)-torsion in \(A(k)\). The proof of (8) proceeds in exactly the same way, using that Lemma 0C0Y gives \(A(k)[p] \cong (\mathbf{Z}/p\mathbf{Z})^{\oplus f}\) for some \(0 \leq f \leq g\).

Remark

Let \(k\) be a field. There are \(2 \times 4 \times 2 = 16\) equivalent definitions of abelian varieties. Let

• projective, proper,

• geometrically irreducible, irreducible, geometrically connected, connected,

• smooth, geometrically reduced

be three sets of properties, pick one from each of them, and let \(A\) be a group scheme over \(k\) with the chosen properties over \(k\). Then \(A\) is an abelian variety. If we pick the options “proper, geometrically irreducible, geometrically reduced”, then we recover Definition 03RO (use Varieties, Lemma 038K). The weakest possible options would be “proper, connected, and geometrically reduced”, see for example Morphisms, Lemma 01WC and Varieties, Lemma 056T. So say \(A\) is a proper, connected, and geometrically reduced group scheme over \(k\). Then \(A\) is geometrically irreducible by Lemmas 0B7Q and 047M and hence an abelian variety. Finally, if \(A/k\) is an abelian variety, then it is projective and smooth over \(k\) (Proposition 03RP), whence satisfies the strongest possible options "projective, geometrically irreducible, smooth".

Actions of group schemes

Let \((G, m)\) be a group and let \(V\) be a set. Recall that a (left) action of \(G\) on \(V\) is given by a map \(a : G \times V \to V\) such that

1. (associativity) \(a(m(g, g'), v) = a(g, a(g', v))\) for all \(g, g' \in G\) and \(v \in V\), and

2. (identity) \(a(e, v) = v\) for all \(v \in V\).

We also say that \(V\) is a \(G\)-set (this usually means we drop the \(a\) from the notation – which is abuse of notation). A map of \(G\)-sets \(\psi : V \to V'\) is any set map such that \(\psi(a(g, v)) = a(g, \psi(v))\) for all \(v \in V\).

Definition

Let \(S\) be a scheme. Let \((G, m)\) be a group scheme over \(S\).

1. An action of \(G\) on the scheme \(X/S\) is a morphism \(a : G \times_S X \to X\) over \(S\) such that for every \(T/S\) the map \(a : G(T) \times X(T) \to X(T)\) defines the structure of a \(G(T)\)-set on \(X(T)\).

2. Suppose that \(X\), \(Y\) are schemes over \(S\) each endowed with an action of \(G\). An equivariant or more precisely a \(G\)-equivariant morphism \(\psi : X \to Y\) is a morphism of schemes over \(S\) such that for every \(T/S\) the map \(\psi : X(T) \to Y(T)\) is a morphism of \(G(T)\)-sets.

In situation (1) this means that the diagrams [03LD]\[\begin{equation} \vcenter{ \xymatrix{ G \times_S G \times_S X \ar[r]_-{1_G \times a} \ar[d]_{m \times 1_X} & G \times_S X \ar[d]^a \\ G \times_S X \ar[r]^a & X } } \quad\quad \vcenter{ \xymatrix{ G \times_S X \ar[r]_-a & X \\ X\ar[u]^{e \times 1_X} \ar[ru]_{1_X} } } \end{equation}\] are commutative. In situation (2) this just means that the diagram \[\xymatrix{ G \times_S X \ar[r]_-{\text{id} \times \psi} \ar[d]_a & G \times_S Y \ar[d]^a \\ X \ar[r]^\psi & Y }\] commutes.

Definition

Let \(S\), \(G \to S\), and \(X \to S\) as in Definition 022Z. Let \(a : G \times_S X \to X\) be an action of \(G\) on \(X/S\). We say the action is free if for every scheme \(T\) over \(S\) the action \(a : G(T) \times X(T) \to X(T)\) is a free action of the group \(G(T)\) on the set \(X(T)\).

Lemma

Situation as in Definition 07S1, The action \(a\) is free if and only if \[G \times_S X \to X \times_S X, \quad (g, x) \mapsto (a(g, x), x)\] is a monomorphism.

Proof

Immediate from the definitions.

Principal homogeneous spaces

In Cohomology on Sites, Definition 03AH we have defined a torsor for a sheaf of groups on a site. Suppose \(\tau \in \{Zariski, \etale, smooth, syntomic, fppf\}\) is a topology and \((G, m)\) is a group scheme over \(S\). Since \(\tau\) is stronger than the canonical topology (see Descent, Lemma 023Q) we see that \(\underline{G}\) (see Sites, Definition 00WR) is a sheaf of groups on \((\Sch/S)_\tau\). Hence we already know what it means to have a torsor for \(\underline{G}\) on \((\Sch/S)_\tau\). A special situation arises if this sheaf is representable. In the following definitions we define directly what it means for the representing scheme to be a \(G\)-torsor.

Definition

Let \(S\) be a scheme. Let \((G, m)\) be a group scheme over \(S\). Let \(X\) be a scheme over \(S\), and let \(a : G \times_S X \to X\) be an action of \(G\) on \(X\).

1. We say \(X\) is a pseudo \(G\)-torsor or that \(X\) is formally principally homogeneous under \(G\) if the induced morphism of schemes \(G \times_S X \to X \times_S X\), \((g, x) \mapsto (a(g, x), x)\) is an isomorphism of schemes over \(S\).

2. A pseudo \(G\)-torsor \(X\) is called trivial if there exists a \(G\)-equivariant isomorphism \(G \to X\) over \(S\) where \(G\) acts on \(G\) by left multiplication.

It is clear that if \(S' \to S\) is a morphism of schemes then the pullback \(X_{S'}\) of a pseudo \(G\)-torsor over \(S\) is a pseudo \(G_{S'}\)-torsor over \(S'\).

Lemma

In the situation of Definition 0498.

1. The scheme \(X\) is a pseudo \(G\)-torsor if and only if for every scheme \(T\) over \(S\) the set \(X(T)\) is either empty or the action of the group \(G(T)\) on \(X(T)\) is simply transitive.

2. A pseudo \(G\)-torsor \(X\) is trivial if and only if the morphism \(X \to S\) has a section.

Proof

Omitted.

Definition

Let \(S\) be a scheme. Let \((G, m)\) be a group scheme over \(S\). Let \(X\) be a pseudo \(G\)-torsor over \(S\).

1. We say \(X\) is a principal homogeneous space or a \(G\)-torsor if there exists a fpqc covering³ \(\{S_i \to S\}_{i \in I}\) such that each \(X_{S_i} \to S_i\) has a section (i.e., is a trivial pseudo \(G_{S_i}\)-torsor).

2. Let \(\tau \in \{Zariski, \etale, smooth, syntomic, fppf\}\). We say \(X\) is a \(G\)-torsor in the \(\tau\) topology, or a \(\tau\) \(G\)-torsor, or simply a \(\tau\) torsor if there exists a \(\tau\) covering \(\{S_i \to S\}_{i \in I}\) such that each \(X_{S_i} \to S_i\) has a section.

3. If \(X\) is a \(G\)-torsor, then we say that it is quasi-isotrivial if it is a torsor for the étale topology.

4. If \(X\) is a \(G\)-torsor, then we say that it is locally trivial if it is a torsor for the Zariski topology.

We sometimes say “let \(X\) be a \(G\)-torsor over \(S\)” to indicate that \(X\) is a scheme over \(S\) equipped with an action of \(G\) which turns it into a principal homogeneous space over \(S\). Next we show that this agrees with the notation introduced earlier when both apply.

Lemma

Let \(S\) be a scheme. Let \((G, m)\) be a group scheme over \(S\). Let \(X\) be a scheme over \(S\), and let \(a : G \times_S X \to X\) be an action of \(G\) on \(X\). Let \(\tau \in \{Zariski, \etale, smooth, syntomic, fppf\}\). Then \(X\) is a \(G\)-torsor in the \(\tau\)-topology if and only if \(\underline{X}\) is a \(\underline{G}\)-torsor on \((\Sch/S)_\tau\).

Proof

Omitted.

Remark

Let \((G, m)\) be a group scheme over the scheme \(S\). In this situation we have the following natural types of questions:

1. If \(X \to S\) is a pseudo \(G\)-torsor and \(X \to S\) is surjective, then is \(X\) necessarily a \(G\)-torsor?

2. Is every \(\underline{G}\)-torsor on \((\Sch/S)_{fppf}\) representable? In other words, does every \(\underline{G}\)-torsor come from a fppf \(G\)-torsor?

3. Is every \(G\)-torsor an fppf (resp. smooth, resp. étale, resp. Zariski) torsor?

In general the answers to these questions are no. To get a positive answer we need to impose additional conditions on \(G \to S\). For example: If \(S\) is the spectrum of a field, then the answer to (1) is yes because then \(\{X \to S\}\) is a fpqc covering trivializing \(X\). If \(G \to S\) is affine, then the answer to (2) is yes (this follows from Descent, Lemma 0245). If \(G = \text{GL}_{n, S}\) then the answer to (3) is yes and in fact any \(\text{GL}_{n, S}\)-torsor is locally trivial (this follows from Descent, Lemma 05B2).

Lemma

Let \(G \to S\) be a smooth group scheme. Every \(G\)-torsor over \(S\) is a \(G\)-torsor in the ètale topology.

Proof

Let \(X\) be a \(G\)-torsor. There is an fpqc covering \(\{S_i \to S\}\) such that \(X_{S_i} \cong G_{S_i}\). Hence \(X \to S\) is smooth by Descent, Lemma 02VL. It is surjective because this is true after the covering \(\{S_i \to S\}\). For every \(s \in S\), More on Morphisms, Lemma 055U gives an ètale neighbourhood \(S' \to S\) of \(s\) and a section \(S' \to X\). These neighbourhoods form an ètale covering of \(S\).

Lemma

Let \(S\) be a scheme and let \(n \geq 1\). Write \[\text{SL}_{n, S} = \Ker(\det : \text{GL}_{n, S} \to \mathbf{G}_{m, S})\] and let \(\text{Sp}_{2n, S} \subset \text{GL}_{2n, S}\) be the group scheme preserving the standard perfect alternating form. Every torsor under \(\text{SL}_{n, S}\) or \(\text{Sp}_{2n, S}\) is locally trivial. The same is true for torsors under finite products of these groups and general linear groups.

Proof

Let \(P\) be an \(\text{SL}_{n, S}\)-torsor. Choose an fpqc covering trivializing \(P\). The transition matrices belong to \(\text{SL}_n\). By fpqc descent they define a finite locally free \(\mathcal{O}_S\)-module \(\mathcal{E}\) of rank \(n\) together with an isomorphism \[\epsilon : \wedge^n\mathcal{E} \longrightarrow \mathcal{O}_S.\] Here we use Descent, Proposition 023T and Lemma 05B2. The same descent identifies \(P\) with the scheme of frames \((e_1, \ldots, e_n)\) of \(\mathcal{E}\) such that \(\epsilon(e_1 \wedge \ldots \wedge e_n) = 1\). Zariski locally \(\mathcal{E}\) is free. If the value of \(\epsilon\) on a chosen frame is \(u \in \mathcal{O}_S^*\), then replacing \(e_1\) by \(u^{-1}e_1\) gives a section of \(P\).

The same argument for an \(\text{Sp}_{2n, S}\)-torsor gives a finite locally free module \(\mathcal{E}\) of rank \(2n\) with a perfect alternating form \(b\), and identifies the torsor with the scheme of symplectic frames. Such frames exist Zariski locally. Indeed, over a local ring choose a basis vector \(e\). Since \(b\) is perfect, there is an \(f\) such that \(b(e, f)\) is a unit; rescale \(f\) so that \(b(e, f) = 1\). The span of \(e, f\) is a nondegenerate direct summand, and its orthogonal complement is again finite free with a perfect alternating form. Induction produces a symplectic basis. The resulting basis over a local ring spreads out to an open neighbourhood.

Finally, the projections of a trivializing cocycle for a torsor under a finite product give torsors under the factors. Their fibre product has the original cocycle. A common Zariski refinement which trivializes the factor torsors therefore trivializes the original torsor.

Lemma

Let \(H \to G\) be a homomorphism of affine group schemes over \(S\). Assume the fppf quotient \(H \backslash G\) is represented by a scheme \(Q\) and \(G \to Q\) is an \(H\)-torsor. Assume also that every \(G_T\)-torsor is Zariski locally trivial for every scheme \(T\) over \(S\).

Let \(\mathcal{C}\) be either the class of finite faithfully flat morphisms or the class of surjective ètale morphisms. If \(G \to Q\) is, Zariski locally on \(Q\), trivialized by a morphism in \(\mathcal{C}\), then every \(H_T\)-torsor is, Zariski locally on \(T\), trivialized by a morphism in \(\mathcal{C}\).

Proof

Let \(P\) be an \(H_T\)-torsor. Choose an fpqc covering trivializing \(P\) and view its \(H\)-valued transition functions as \(G\)-valued transition functions. Affine fpqc descent, Descent, Lemma 0245, produces a \(G_T\)-torsor \(E\) and an \(H\)-equivariant morphism \(P \to E\). Zariski locally on \(T\) the torsor \(E\) is trivial, so we may assume that \(E = G_T\).

The \(H\)-equivariant morphism \(P \to G_T\) descends to a morphism \(q : T \to Q\). Moreover, the diagram \[\xymatrix{ P \ar[r] \ar[d] & G \ar[d] \\ T \ar[r]^q & Q }\] is cartesian. This can be checked after the fpqc covering \(P \to T\), where it is the standard description of the quotient. Pulling back the assumed local trivializations of \(G \to Q\) along \(q\) proves the result.

Remark

In Lemma groupoids-lemma-subgroup-torsor-local-triviality, one may take \(G = \text{GL}_{n, S}\) by Remark 049C. Inversion identifies \(H \backslash G\) with \(G/H\), exchanging left and right actions. Thus the lemma applies in either quotient convention to a flat closed subgroup \(H \subset \text{GL}_{n, S}\) whenever the quotient and the displayed universal \(H\)-torsor exist. It gives both the finite faithfully flat and the ètale conclusions discussed in the reference.

Equivariant quasi-coherent sheaves

We think of “functions” as dual to “space”. Thus for a morphism of spaces the map on functions goes the other way. Moreover, we think of the sections of a sheaf of modules as “functions”. This leads us naturally to the direction of the arrows chosen in the following definition.

Definition

Let \(S\) be a scheme, let \((G, m)\) be a group scheme over \(S\), and let \(a : G \times_S X \to X\) be an action of the group scheme \(G\) on \(X/S\). A \(G\)-equivariant quasi-coherent \(\mathcal{O}_X\)-module, or simply an equivariant quasi-coherent \(\mathcal{O}_X\)-module, is a pair \((\mathcal{F}, \alpha)\), where \(\mathcal{F}\) is a quasi-coherent \(\mathcal{O}_X\)-module, and \(\alpha\) is a \(\mathcal{O}_{G \times_S X}\)-module map \[\alpha : a^*\mathcal{F} \longrightarrow \text{pr}_1^*\mathcal{F}\] where \(\text{pr}_1 : G \times_S X \to X\) is the projection such that

1. the diagram \[\xymatrix{ (1_G \times a)^*\text{pr}_1^*\mathcal{F} \ar[r]_-{\text{pr}_{12}^*\alpha} & \text{pr}_2^*\mathcal{F} \\ (1_G \times a)^*a^*\mathcal{F} \ar[u]^{(1_G \times a)^*\alpha} \ar@{=}[r] & (m \times 1_X)^*a^*\mathcal{F} \ar[u]_{(m \times 1_X)^*\alpha} }\] is commutative in the category of \(\mathcal{O}_{G \times_S G \times_S X}\)-modules, and

2. the pullback \[(e \times 1_X)^*\alpha : \mathcal{F} \longrightarrow \mathcal{F}\] is the identity map.

For explanation compare with the relevant diagrams of Equation (03LD).

Note that the commutativity of the first diagram guarantees that \((e \times 1_X)^*\alpha\) is an idempotent operator on \(\mathcal{F}\), and hence condition (2) is just the condition that it is an isomorphism.

Lemma

Let \(S\) be a scheme. Let \(G\) be a group scheme over \(S\). Let \(f : Y \to X\) be a \(G\)-equivariant morphism between \(S\)-schemes endowed with \(G\)-actions. The rule \((\mathcal{F}, \alpha) \mapsto (f^*\mathcal{F}, (1_G \times f)^*\alpha)\) defines a functor from the category of \(G\)-equivariant quasi-coherent \(\mathcal{O}_X\)-modules to the category of \(G\)-equivariant quasi-coherent \(\mathcal{O}_Y\)-modules.

Proof

Omitted.

Let us give an example.

Example

Let \(A\) be a \(\mathbf{Z}\)-graded ring, i.e., \(A\) comes with a direct sum decomposition \(A = \bigoplus_{n \in \mathbf{Z}} A_n\) and \(A_n \cdot A_m \subset A_{n + m}\). Set \(X = \Spec(A)\). Then we obtain a \(\mathbf{G}_m\)-action \[a : \mathbf{G}_m \times X \longrightarrow X\] by the ring map \(\mu : A \to A \otimes \mathbf{Z}[x, x^{-1}]\), \(f \mapsto f \otimes x^{\deg(f)}\). Namely, to check this we have to verify commutativity of the diagram \[\xymatrix{ A \ar[r]_\mu \ar[d]_\mu & A \otimes \mathbf{Z}[x, x^{-1}] \ar[d]^{\mu \otimes 1} \\ A \otimes \mathbf{Z}[x, x^{-1}] \ar[r]^-{1 \otimes m} & A \otimes \mathbf{Z}[x, x^{-1}] \otimes \mathbf{Z}[x, x^{-1}] }\] where \(m(x) = x \otimes x\), see Example 022U. This is immediately clear when evaluating on a homogeneous element. Suppose that \(M\) is a graded \(A\)-module. Then we obtain a \(\mathbf{G}_m\)-equivariant quasi-coherent \(\mathcal{O}_X\)-module \(\mathcal{F} = \widetilde{M}\) by using \(\alpha\) as in Definition 03LF corresponding to the \(A \otimes \mathbf{Z}[x, x^{-1}]\)-module map \[M \otimes_{A, \mu} (A \otimes \mathbf{Z}[x, x^{-1}]) \longrightarrow M \otimes_{A, \text{id}_A \otimes 1} (A \otimes \mathbf{Z}[x, x^{-1}])\] sending \(m \otimes 1 \otimes 1\) to \(m \otimes 1 \otimes x^{\deg(m)}\) for \(m \in M\) homogeneous.

Lemma

Let \(a : \mathbf{G}_m \times X \to X\) be an action on an affine scheme. Then \(X\) is the spectrum of a \(\mathbf{Z}\)-graded ring and the action is as in Example 0EKJ.

Proof

Let \(f \in A = \Gamma(X, \mathcal{O}_X)\). Then we can write \[a^\sharp(f) = \sum\nolimits_{n \in \mathbf{Z}} f_n \otimes x^n \quad\text{in}\quad A \otimes \mathbf{Z}[x, x^{-1}] = \Gamma(\mathbf{G}_m \times X, \mathcal{O}_{\mathbf{G}_m \times X})\] as a finite sum with \(f_n\) in \(A\) uniquely determined. Thus we obtain maps \(A \to A\), \(f \mapsto f_n\). Since \(a\) is an action, if we evaluate at \(x = 1\), we see \(f = \sum f_n\). Since \(a\) is an action we find that \[\sum (f_n)_m \otimes x^m \otimes x^n = \sum f_n \otimes x^n \otimes x^n\] (compare with computation in Example 0EKJ). Thus \((f_n)_m = 0\) if \(n \not = m\) and \((f_n)_n = f_n\). Thus if we set \[A_n = \{f \in A \mid f_n = f\}\] then we get \(A = \sum A_n\). On the other hand, the sum has to be direct since \(f = 0\) implies \(f_n = 0\) in the situation above.

Lemma

Let \(A\) be a graded ring. Let \(X = \Spec(A)\) with action \(a : \mathbf{G}_m \times X \to X\) as in Example 0EKJ. Let \(\mathcal{F}\) be a \(\mathbf{G}_m\)-equivariant quasi-coherent \(\mathcal{O}_X\)-module. Then \(M = \Gamma(X, \mathcal{F})\) has a canonical grading such that it is a graded \(A\)-module and such that the isomorphism \(\widetilde{M} \to \mathcal{F}\) (Schemes, Lemma 01IA) is an isomorphism of \(\mathbf{G}_m\)-equivariant modules where the \(\mathbf{G}_m\)-equivariant structure on \(\widetilde{M}\) is the one from Example 0EKJ.

Proof

You can either prove this by repeating the arguments of Lemma 0EKK for the module \(M\). Alternatively, you can consider the scheme \((X', \mathcal{O}_{X'}) = (X, \mathcal{O}_X \oplus \mathcal{F})\) where \(\mathcal{F}\) is viewed as an ideal of square zero. There is a natural action \(a' : \mathbf{G}_m \times X' \to X'\) defined using the action on \(X\) and on \(\mathcal{F}\). Then apply Lemma 0EKK to \(X'\) and conclude. (The nice thing about this argument is that it immediately shows that the grading on \(A\) and \(M\) are compatible, i.e., that \(M\) is a graded \(A\)-module.) Details omitted.

Groupoids

Recall that a groupoid is a category in which every morphism is an isomorphism, see Categories, Definition 0018. Hence a groupoid has a set of objects \(\text{Ob}\), a set of arrows \(\text{Arrows}\), a source and target map \(s, t : \text{Arrows} \to \text{Ob}\), and a composition law \(c : \text{Arrows} \times_{s, \text{Ob}, t} \text{Arrows} \to \text{Arrows}\). These maps satisfy exactly the following axioms

1. (associativity) \(c \circ (1, c) = c \circ (c, 1)\) as maps \(\text{Arrows} \times_{s, \text{Ob}, t} \text{Arrows} \times_{s, \text{Ob}, t} \text{Arrows} \to \text{Arrows}\),

2. (identity) there exists a map \(e : \text{Ob} \to \text{Arrows}\) such that

   1. \(s \circ e = t \circ e = \text{id}\) as maps \(\text{Ob} \to \text{Ob}\),

   2. \(c \circ (1, e \circ s) = c \circ (e \circ t, 1) = 1\) as maps \(\text{Arrows} \to \text{Arrows}\),

3. (inverse) there exists a map \(i : \text{Arrows} \to \text{Arrows}\) such that

   1. \(s \circ i = t\), \(t \circ i = s\) as maps \(\text{Arrows} \to \text{Ob}\), and

   2. \(c \circ (1, i) = e \circ t\) and \(c \circ (i, 1) = e \circ s\) as maps \(\text{Arrows} \to \text{Arrows}\).

If this is the case the maps \(e\) and \(i\) are uniquely determined and \(i\) is a bijection. Note that if \((\text{Ob}', \text{Arrows}', s', t', c')\) is a second groupoid category, then a functor \(f : (\text{Ob}, \text{Arrows}, s, t, c) \to (\text{Ob}', \text{Arrows}', s', t', c')\) is given by a pair of set maps \(f : \text{Ob} \to \text{Ob}'\) and \(f : \text{Arrows} \to \text{Arrows}'\) such that \(s' \circ f = f \circ s\), \(t' \circ f = f \circ t\), and \(c' \circ (f, f) = f \circ c\). The compatibility with identity and inverse is automatic. We will use this below. (Warning: The compatibility with identity has to be imposed in the case of general categories.)

Definition

Let \(S\) be a scheme.

1. A groupoid scheme over \(S\), or simply a groupoid over \(S\) is a quintuple \((U, R, s, t, c)\) where \(U\) and \(R\) are schemes over \(S\), and \(s, t : R \to U\) and \(c : R \times_{s, U, t} R \to R\) are morphisms of schemes over \(S\) with the following property: For any scheme \(T\) over \(S\) the quintuple \[(U(T), R(T), s, t, c)\] is a groupoid category in the sense described above.

2. A morphism \(f : (U, R, s, t, c) \to (U', R', s', t', c')\) of groupoid schemes over \(S\) is given by morphisms of schemes \(f : U \to U'\) and \(f : R \to R'\) with the following property: For any scheme \(T\) over \(S\) the maps \(f\) define a functor from the groupoid category \((U(T), R(T), s, t, c)\) to the groupoid category \((U'(T), R'(T), s', t', c')\).

Let \((U, R, s, t, c)\) be a groupoid over \(S\). Note that, by the remarks preceding the definition and the Yoneda lemma, there are unique morphisms of schemes \(e : U \to R\) and \(i : R \to R\) over \(S\) such that for every scheme \(T\) over \(S\) the induced map \(e : U(T) \to R(T)\) is the identity, and \(i : R(T) \to R(T)\) is the inverse of the groupoid category. The septuple \((U, R, s, t, c, e, i)\) satisfies commutative diagrams corresponding to each of the axioms (1), (2)(a), (2)(b), (3)(a) and (3)(b) above, and conversely given a septuple with this property the quintuple \((U, R, s, t, c)\) is a groupoid scheme. Note that \(i\) is an isomorphism, and \(e\) is a section of both \(s\) and \(t\). Moreover, given a groupoid scheme over \(S\) we denote \[j = (t, s) : R \longrightarrow U \times_S U\] which is compatible with our conventions in Section 022O above. We sometimes say “let \((U, R, s, t, c, e, i)\) be a groupoid over \(S\)” to stress the existence of identity and inverse.

Lemma

Given a groupoid scheme \((U, R, s, t, c)\) over \(S\) the morphism \(j : R \to U \times_S U\) is a pre-equivalence relation.

Proof

Omitted. This is a nice exercise in the definitions.

Lemma

Given an equivalence relation \(j : R \to U \times_S U\) over \(S\) there is a unique way to extend it to a groupoid \((U, R, s, t, c)\) over \(S\).

Proof

Omitted. This is a nice exercise in the definitions.

Lemma

Let \(S\) be a scheme. Let \((U, R, s, t, c)\) be a groupoid over \(S\). In the commutative diagram \[\xymatrix{ & U & \\ R \ar[d]_s \ar[ru]^t & R \times_{s, U, t} R \ar[l]^-{\text{pr}_0} \ar[d]^{\text{pr}_1} \ar[r]_-c & R \ar[d]^s \ar[lu]_t \\ U & R \ar[l]_t \ar[r]^s & U }\] the two lower squares are fibre product squares. Moreover, the triangle on top (which is really a square) is also cartesian.

Proof

Omitted. Exercise in the definitions and the functorial point of view in algebraic geometry.

Lemma

Let \(S\) be a scheme. Let \((U, R, s, t, c, e, i)\) be a groupoid over \(S\). The diagram [03C7]\[\begin{equation} \xymatrix{ R \times_{t, U, t} R \ar@<1ex>[r]^-{\text{pr}_1} \ar@<-1ex>[r]_-{\text{pr}_0} \ar[d]_{(\text{pr}_0, c \circ (i, 1))} & R \ar[r]^t \ar[d]^{\text{id}_R} & U \ar[d]^{\text{id}_U} \\ R \times_{s, U, t} R \ar@<1ex>[r]^-c \ar@<-1ex>[r]_-{\text{pr}_0} \ar[d]_{\text{pr}_1} & R \ar[r]^t \ar[d]^s & U \\ R \ar@<1ex>[r]^s \ar@<-1ex>[r]_t & U } \end{equation}\] is commutative. The two top rows are isomorphic via the vertical maps given. The two lower left squares are cartesian.

Proof

The commutativity of the diagram follows from the axioms of a groupoid. Note that, in terms of groupoids, the top left vertical arrow assigns to a pair of morphisms \((\alpha, \beta)\) with the same target, the pair of morphisms \((\alpha, \alpha^{-1} \circ \beta)\). In any groupoid this defines a bijection between \(\text{Arrows} \times_{t, \text{Ob}, t} \text{Arrows}\) and \(\text{Arrows} \times_{s, \text{Ob}, t} \text{Arrows}\). Hence the second assertion of the lemma. The last assertion follows from Lemma 02YE.

Lemma

Let \((U, R, s, t, c)\) be a groupoid over a scheme \(S\). Let \(S' \to S\) be a morphism. Then the base changes \(U' = S' \times_S U\), \(R' = S' \times_S R\) endowed with the base changes \(s'\), \(t'\), \(c'\) of the morphisms \(s, t, c\) form a groupoid scheme \((U', R', s', t', c')\) over \(S'\) and the projections determine a morphism \((U', R', s', t', c') \to (U, R, s, t, c)\) of groupoid schemes over \(S\).

Proof

Omitted. Hint: \(R' \times_{s', U', t'} R' = S' \times_S (R \times_{s, U, t} R)\).

Quasi-coherent sheaves on groupoids

See the introduction of Section 03LE for our choices in direction of arrows.

Definition

Let \(S\) be a scheme, let \((U, R, s, t, c)\) be a groupoid scheme over \(S\). A quasi-coherent module on \((U, R, s, t, c)\) is a pair \((\mathcal{F}, \alpha)\), where \(\mathcal{F}\) is a quasi-coherent \(\mathcal{O}_U\)-module, and \(\alpha\) is a \(\mathcal{O}_R\)-module map \[\alpha : t^*\mathcal{F} \longrightarrow s^*\mathcal{F}\] such that

1. the diagram \[\xymatrix{ & \text{pr}_1^*t^*\mathcal{F} \ar[r]_-{\text{pr}_1^*\alpha} & \text{pr}_1^*s^*\mathcal{F} \ar@{=}[rd] & \\ \text{pr}_0^*s^*\mathcal{F} \ar@{=}[ru] & & & c^*s^*\mathcal{F} \\ & \text{pr}_0^*t^*\mathcal{F} \ar[lu]^{\text{pr}_0^*\alpha} \ar@{=}[r] & c^*t^*\mathcal{F} \ar[ru]_{c^*\alpha} }\] is a commutative diagram in the category of \(\mathcal{O}_{R \times_{s, U, t} R}\)-modules, and

2. the pullback \[e^*\alpha : \mathcal{F} \longrightarrow \mathcal{F}\] is the identity map.

Compare with the commutative diagrams of Lemma 02YE.

The commutativity of the first diagram forces the operator \(e^*\alpha\) to be idempotent. Hence the second condition can be reformulated as saying that \(e^*\alpha\) is an isomorphism. In fact, the condition implies that \(\alpha\) is an isomorphism.

Lemma

Let \(S\) be a scheme, let \((U, R, s, t, c)\) be a groupoid scheme over \(S\). If \((\mathcal{F}, \alpha)\) is a quasi-coherent module on \((U, R, s, t, c)\) then \(\alpha\) is an isomorphism.

Proof

Pull back the commutative diagram of Definition 03LI by the morphism \((i, 1) : R \to R \times_{s, U, t} R\). Then we see that \(\alpha \circ i^*\alpha = s^*e^*\alpha\). Pulling back by the morphism \((1, i)\) we obtain the relation \(i^*\alpha \circ \alpha = t^*e^*\alpha\). By the second assumption these morphisms are the identity. Hence \(i^*\alpha\) is an inverse of \(\alpha\).

Lemma

Let \(S\) be a scheme. Consider a morphism \(f : (U, R, s, t, c) \to (U', R', s', t', c')\) of groupoid schemes over \(S\). Then pullback \(f^*\) given by \[(\mathcal{F}, \alpha) \mapsto (f^*\mathcal{F}, f^*\alpha)\] defines a functor from the category of quasi-coherent sheaves on \((U', R', s', t', c')\) to the category of quasi-coherent sheaves on \((U, R, s, t, c)\).

Proof

Omitted.

Lemma

Let \(S\) be a scheme. Consider a morphism \(f : (U, R, s, t, c) \to (U', R', s', t', c')\) of groupoid schemes over \(S\). Assume that

1. \(f : U \to U'\) is quasi-compact and quasi-separated,

2. the square \[\xymatrix{ R \ar[d]_t \ar[r]_f & R' \ar[d]^{t'} \\ U \ar[r]^f & U' }\] is cartesian, and

3. \(s'\) and \(t'\) are flat.

Then pushforward \(f_*\) given by \[(\mathcal{F}, \alpha) \mapsto (f_*\mathcal{F}, f_*\alpha)\] defines a functor from the category of quasi-coherent sheaves on \((U, R, s, t, c)\) to the category of quasi-coherent sheaves on \((U', R', s', t', c')\) which is right adjoint to pullback as defined in Lemma 03LJ.

Proof

Since \(U \to U'\) is quasi-compact and quasi-separated we see that \(f_*\) transforms quasi-coherent sheaves into quasi-coherent sheaves (Schemes, Lemma 01LC). Moreover, since the squares \[\vcenter{ \xymatrix{ R \ar[d]_t \ar[r]_f & R' \ar[d]^{t'} \\ U \ar[r]^f & U' } } \quad\text{and}\quad \vcenter{ \xymatrix{ R \ar[d]_s \ar[r]_f & R' \ar[d]^{s'} \\ U \ar[r]^f & U' } }\] are cartesian we find that \((t')^*f_*\mathcal{F} = f_*t^*\mathcal{F}\) and \((s')^*f_*\mathcal{F} = f_*s^*\mathcal{F}\) , see Cohomology of Schemes, Lemma 02KH. Thus it makes sense to think of \(f_*\alpha\) as a map \((t')^*f_*\mathcal{F} \to (s')^*f_*\mathcal{F}\). A similar argument shows that \(f_*\alpha\) satisfies the cocycle condition. The functor is adjoint to the pullback functor since pullback and pushforward on modules on ringed spaces are adjoint. Some details omitted.

Lemma

Let \(S\) be a scheme. Let \((U, R, s, t, c)\) be a groupoid scheme over \(S\). The category of quasi-coherent modules on \((U, R, s, t, c)\) has colimits.

Proof

Let \(i \mapsto (\mathcal{F}_i, \alpha_i)\) be a diagram over the index category \(\mathcal{I}\). We can form the colimit \(\mathcal{F} = \colim \mathcal{F}_i\) which is a quasi-coherent sheaf on \(U\), see Schemes, Section 01LA. Since colimits commute with pullback we see that \(s^*\mathcal{F} = \colim s^*\mathcal{F}_i\) and similarly \(t^*\mathcal{F} = \colim t^*\mathcal{F}_i\). Hence we can set \(\alpha = \colim \alpha_i\). We omit the proof that \((\mathcal{F}, \alpha)\) is the colimit of the diagram in the category of quasi-coherent modules on \((U, R, s, t, c)\).

Lemma

Let \(S\) be a scheme. Let \((U, R, s, t, c)\) be a groupoid scheme over \(S\). If \(s\), \(t\) are flat, then the category of quasi-coherent modules on \((U, R, s, t, c)\) is abelian.

Proof

Let \(\varphi : (\mathcal{F}, \alpha) \to (\mathcal{G}, \beta)\) be a homomorphism of quasi-coherent modules on \((U, R, s, t, c)\). Since \(s\) is flat we see that \[0 \to s^*\Ker(\varphi) \to s^*\mathcal{F} \to s^*\mathcal{G} \to s^*\Coker(\varphi) \to 0\] is exact and similarly for pullback by \(t\). Hence \(\alpha\) and \(\beta\) induce isomorphisms \(\kappa : t^*\Ker(\varphi) \to s^*\Ker(\varphi)\) and \(\lambda : t^*\Coker(\varphi) \to s^*\Coker(\varphi)\) which satisfy the cocycle condition. Then it is straightforward to verify that \((\Ker(\varphi), \kappa)\) and \((\Coker(\varphi), \lambda)\) are a kernel and cokernel in the category of quasi-coherent modules on \((U, R, s, t, c)\). Moreover, the condition \(\Coim(\varphi) = \Im(\varphi)\) follows because it holds over \(U\).

Colimits of quasi-coherent modules

In this section we prove some technical results saying that under suitable assumptions every quasi-coherent module on a groupoid is a filtered colimit of “small” quasi-coherent modules.

Lemma

Let \((U, R, s, t, c)\) be a groupoid scheme over \(S\). Assume \(s, t\) are flat, quasi-compact, and quasi-separated. For any quasi-coherent module \(\mathcal{G}\) on \(U\), there exists a canonical isomorphism \(\alpha : t^*s_*t^*\mathcal{G} \to s^*s_*t^*\mathcal{G}\) which turns \((s_*t^*\mathcal{G}, \alpha)\) into a quasi-coherent module on \((U, R, s, t, c)\). This construction defines a functor \[\QCoh(\mathcal{O}_U) \longrightarrow \QCoh(U, R, s, t, c)\] which is a right adjoint to the forgetful functor \((\mathcal{F}, \beta) \mapsto \mathcal{F}\).

Proof

The pushforward of a quasi-coherent module along a quasi-compact and quasi-separated morphism is quasi-coherent, see Schemes, Lemma 01LC. Hence \(s_*t^*\mathcal{G}\) is quasi-coherent. With notation as in Lemma 02YE we have \[t^*s_*t^*\mathcal{G} = \text{pr}_{1, *}\text{pr}_0^*t^*\mathcal{G} = \text{pr}_{1, *}c^*t^*\mathcal{G} = s^*s_*t^*\mathcal{G}\] The middle equality holds because \(t \circ c = t \circ \text{pr}_0\) as morphisms \(R \times_{s, U, t} R \to U\), and the first and last equalities hold because we know that base change and pushforward commute in these steps by Cohomology of Schemes, Lemma 02KH.

To verify the cocycle condition of Definition 03LI for \(\alpha\) and the adjointness property we describe the construction \(\mathcal{G} \mapsto (s_*t^*\mathcal{G}, \alpha)\) in another way. Consider the groupoid scheme \((R, R \times_{t, U, t} R, \text{pr}_1, \text{pr}_0, \text{pr}_{02})\) associated to the equivalence relation \(R \times_{t, U, t} R\) on \(R\), see Lemma 0233. There is a morphism \[f : (R, R \times_{t, U, t} R, \text{pr}_1, \text{pr}_0, \text{pr}_{02}) \longrightarrow (U, R, s, t, c)\] of groupoid schemes given by \(s : R \to U\) and \(R \times_{t, U, t} R \to R\) given by \((r_0, r_1) \mapsto r_0^{-1} \circ r_1\); we omit the verification of the commutativity of the required diagrams. Since \(t, s : R \to U\) are quasi-compact, quasi-separated, and flat, and since we have a cartesian square \[\xymatrix{ R \times_{t, U, t} R \ar[d]_{\text{pr}_0} \ar[rr]_-{(r_0, r_1) \mapsto r_0^{-1} \circ r_1} & & R \ar[d]^t \\ R \ar[rr]^s & & U }\] by Lemma 03C6 it follows that Lemma 09VH applies to \(f\). Thus pushforward and pullback of quasi-coherent modules along \(f\) are adjoint functors. To finish the proof we will identify these functors with the functors described above. To do this, note that \[t^* : \QCoh(\mathcal{O}_U) \longrightarrow \QCoh(R, R \times_{t, U, t} R, \text{pr}_1, \text{pr}_0, \text{pr}_{02})\] is an equivalence by the theory of descent of quasi-coherent sheaves as \(\{t : R \to U\}\) is an fpqc covering, see Descent, Proposition 023T.

Pushforward along \(f\) precomposed with the equivalence \(t^*\) sends \(\mathcal{G}\) to \((s_*t^*\mathcal{G}, \alpha)\); we omit the verification that the isomorphism \(\alpha\) obtained in this fashion is the same as the one constructed above.

Pullback along \(f\) postcomposed with the inverse of the equivalence \(t^*\) sends \((\mathcal{F}, \beta)\) to the descent relative to \(\{t : R \to U\}\) of the module \(s^*\mathcal{F}\) endowed with the descent datum \(\gamma\) on \(R \times_{t, U, t} R\) which is the pullback of \(\beta\) by \((r_0, r_1) \mapsto r_0^{-1} \circ r_1\). Consider the isomorphism \(\beta : t^*\mathcal{F} \to s^*\mathcal{F}\). The canonical descent datum (Descent, Definition 023D) on \(t^*\mathcal{F}\) relative to \(\{t : R \to U\}\) translates via \(\beta\) into the map \[\text{pr}_0^*s^*\mathcal{F} \xrightarrow{\text{pr}_0^*\beta^{-1}} \text{pr}_0^*t^*\mathcal{F} \xrightarrow{can} \text{pr}_1^*t^*\mathcal{F} \xrightarrow{\text{pr}_1^*\beta} \text{pr}_1^*s^*\mathcal{F}\] Since \(\beta\) satisfies the cocycle condition, this is equal to the pullback of \(\beta\) by \((r_0, r_1) \mapsto r_0^{-1} \circ r_1\). To see this take the actual cocycle relation in Definition 03LI and pull it back by the morphism \((\text{pr}_0, c \circ (i, 1)) : R \times_{t, U, t} R \to R \times_{s, U, t} R\) which also plays a role in the commutative diagram of Lemma 03C6. It follows that \((s^*\mathcal{F}, \gamma)\) is isomorphic to \((t^*\mathcal{F}, can)\). All in all, we conclude that pullback by \(f\) postcomposed with the inverse of the equivalence \(t^*\) is isomorphic to the forgetful functor \((\mathcal{F}, \beta) \mapsto \mathcal{F}\).

Remark

In the situation of Lemma 07TR denote \[F : \QCoh(U, R, s, t, c) \to \QCoh(\mathcal{O}_U),\quad (\mathcal{F}, \beta) \mapsto \mathcal{F}\] the forgetful functor and denote \[G : \QCoh(\mathcal{O}_U) \to \QCoh(U, R, s, t, c),\quad \mathcal{G} \mapsto (s_*t^*\mathcal{G}, \alpha)\] the right adjoint constructed in the lemma. Then the unit \(\eta : \text{id} \to G \circ F\) of the adjunction evaluated on \((\mathcal{F}, \beta)\) is given by the map \[\mathcal{F} \to s_*s^*\mathcal{F} \xrightarrow{\beta^{-1}} s_*t^*\mathcal{F}\] We omit the verification.

Lemma

Let \(f : Y \to X\) be a morphism of schemes. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module, let \(\mathcal{G}\) be a quasi-coherent \(\mathcal{O}_Y\)-module, and let \(\varphi : \mathcal{G} \to f^*\mathcal{F}\) be a module map. Assume

1. \(\varphi\) is injective,

2. \(f\) is quasi-compact, quasi-separated, flat, and surjective,

3. \(X\), \(Y\) are locally Noetherian, and

4. \(\mathcal{G}\) is a coherent \(\mathcal{O}_Y\)-module.

Then \(\mathcal{F} \cap f_*\mathcal{G}\) defined as the pullback \[\xymatrix{ \mathcal{F} \ar[r] & f_*f^*\mathcal{F} \\ \mathcal{F} \cap f_*\mathcal{G} \ar[u] \ar[r] & f_*\mathcal{G} \ar[u] }\] is a coherent \(\mathcal{O}_X\)-module.

Proof

We will freely use the characterization of coherent modules of Cohomology of Schemes, Lemma 01XZ as well as the fact that coherent modules form a Serre subcategory of \(\QCoh(\mathcal{O}_X)\), see Cohomology of Schemes, Lemma 01Y1. If \(f\) has a section \(\sigma\), then we see that \(\mathcal{F} \cap f_*\mathcal{G}\) is contained in the image of \(\sigma^*\mathcal{G} \to \sigma^*f^*\mathcal{F} = \mathcal{F}\), hence coherent. In general, to show that \(\mathcal{F} \cap f_*\mathcal{G}\) is coherent, it suffices to show that \(f^*(\mathcal{F} \cap f_*\mathcal{G})\) is coherent (see Descent, Lemma 05AZ). Since \(f\) is flat this is equal to \(f^*\mathcal{F} \cap f^*f_*\mathcal{G}\). Since \(f\) is flat, quasi-compact, and quasi-separated we see \(f^*f_*\mathcal{G} = p_*q^*\mathcal{G}\) where \(p, q : Y \times_X Y \to Y\) are the projections, see Cohomology of Schemes, Lemma 02KH. Since \(p\) has a section we win.

Let \(S\) be a scheme. Let \((U, R, s, t, c)\) be a groupoid in schemes over \(S\). Assume that \(U\) is locally Noetherian. In the lemma below we say that a quasi-coherent sheaf \((\mathcal{F}, \alpha)\) on \((U, R, s, t, c)\) is coherent if \(\mathcal{F}\) is a coherent \(\mathcal{O}_U\)-module.

Lemma

Let \((U, R, s, t, c)\) be a groupoid scheme over \(S\). Assume that

1. \(U\), \(R\) are Noetherian,

2. \(s, t\) are flat, quasi-compact, and quasi-separated.

Then every quasi-coherent module \((\mathcal{F}, \beta)\) on \((U, R, s, t, c)\) is a filtered colimit of coherent modules.

Proof

We will use the characterization of Cohomology of Schemes, Lemma 01XZ of coherent modules on locally Noetherian scheme without further mention. We can write \(\mathcal{F} = \colim \mathcal{H}_i\) as the filtered colimit of coherent submodules \(\mathcal{H}_i \subset \mathcal{F}\), see Cohomology of Schemes, Lemma 0GN6. Given a quasi-coherent sheaf \(\mathcal{H}\) on \(U\) we denote \((s_*t^*\mathcal{H}, \alpha)\) the quasi-coherent sheaf on \((U, R, s, t, c)\) of Lemma 07TR. Consider the adjunction map \((\mathcal{F}, \beta) \to (s_*t^*\mathcal{F}, \alpha)\) in \(\QCoh(U, R, s, t, c)\), see Remark 0GNF. Set \[(\mathcal{F}_i, \beta_i) = (\mathcal{F}, \beta) \times_{(s_*t^*\mathcal{F}, \alpha)} (s_*t^*\mathcal{H}_i, \alpha)\] in \(\QCoh(U, R, s, t, c)\). Since restriction to \(U\) is an exact functor on \(\QCoh(U, R, s, t, c)\) by the proof of Lemma 077S we obtain a pullback diagram \[\xymatrix{ \mathcal{F} \ar[r] & s_*t^*\mathcal{F} \\ \mathcal{F}_i \ar[r] \ar[u] & s_*t^*\mathcal{H}_i \ar[u] }\] in other words \(\mathcal{F}_i = \mathcal{F} \cap s_*t^*\mathcal{H}_i\). By the description of the adjunction map in Remark 0GNF this diagram is isomorphic to the diagram \[\xymatrix{ \mathcal{F} \ar[r] & s_*s^*\mathcal{F} \\ \mathcal{F}_i \ar[r] \ar[u] & s_*t^*\mathcal{H}_i \ar[u] }\] where the right vertical arrow is the result of applying \(s_*\) to the map \[t^*\mathcal{H}_i \to t^*\mathcal{F} \xrightarrow{\beta} s^*\mathcal{F}\] This arrow is injective as \(t\) is a flat morphism. It follows that \(\mathcal{F}_i\) is coherent by Lemma 07TT. Finally, because \(s\) is quasi-compact and quasi-separated we see that \(s_*\) commutes with colimits (see Cohomology of Schemes, Lemma 07TB). Hence \(s_*t^*\mathcal{F} = \colim s_*t^*\mathcal{H}_i\) and hence \((\mathcal{F}, \beta) = \colim (\mathcal{F}_i, \beta_i)\) as desired.

Here is a curious lemma that is useful when working with groupoids on fields. In fact, this is the standard argument to prove that any representation of an algebraic group is a colimit of finite dimensional representations.

Lemma

Let \((U, R, s, t, c)\) be a groupoid scheme over \(S\). Assume that

1. \(U\), \(R\) are affine,

2. there exist \(e_i \in \mathcal{O}_R(R)\) such that every element \(g \in \mathcal{O}_R(R)\) can be uniquely written as \(\sum s^*(f_i)e_i\) for some \(f_i \in \mathcal{O}_U(U)\).

Then every quasi-coherent module \((\mathcal{F}, \alpha)\) on \((U, R, s, t, c)\) is a filtered colimit of finite type quasi-coherent modules.

Proof

The assumption means that \(\mathcal{O}_R(R)\) is a free \(\mathcal{O}_U(U)\)-module via \(s\) with basis \(e_i\). Hence for any quasi-coherent \(\mathcal{O}_U\)-module \(\mathcal{G}\) we see that \(s^*\mathcal{G}(R) = \bigoplus_i \mathcal{G}(U)e_i\). We will write \(s(-)\) to indicate pullback of sections by \(s\) and similarly for other morphisms. Let \((\mathcal{F}, \alpha)\) be a quasi-coherent module on \((U, R, s, t, c)\). Let \(\sigma \in \mathcal{F}(U)\). By the above we can write \[\alpha(t(\sigma)) = \sum s(\sigma_i) e_i\] for some unique \(\sigma_i \in \mathcal{F}(U)\) (all but finitely many are zero of course). We can also write \[c(e_i) = \sum \text{pr}_1(f_{ij}) \text{pr}_0(e_j)\] as functions on \(R \times_{s, U, t}R\). Then the commutativity of the diagram in Definition 03LI means that \[\sum \text{pr}_1(\alpha(t(\sigma_i))) \text{pr}_0(e_i) = \sum \text{pr}_1(s(\sigma_i)f_{ij}) \text{pr}_0(e_j)\] (calculation omitted). Picking off the coefficients of \(\text{pr}_0(e_l)\) we see that \(\alpha(t(\sigma_l)) = \sum s(\sigma_i)f_{il}\). Hence the submodule \(\mathcal{G} \subset \mathcal{F}\) generated by the elements \(\sigma_i\) defines a finite type quasi-coherent module preserved by \(\alpha\). Hence it is a subobject of \(\mathcal{F}\) in \(\QCoh(U, R, s, t, c)\). This submodule contains \(\sigma\) (as one sees by pulling back the first relation by \(e\)). Hence we win.

We suggest the reader skip the rest of this section. Let \(S\) be a scheme. Let \((U, R, s, t, c)\) be a groupoid in schemes over \(S\). Let \(\kappa\) be a cardinal. In the following we will say that a quasi-coherent sheaf \((\mathcal{F}, \alpha)\) on \((U, R, s, t, c)\) is \(\kappa\)-generated if \(\mathcal{F}\) is a \(\kappa\)-generated \(\mathcal{O}_U\)-module, see Properties, Definition 077L.

Lemma

Let \((U, R, s, t, c)\) be a groupoid scheme over \(S\). Let \(\kappa\) be a cardinal. There exists a set \(T\) and a family \((\mathcal{F}_t, \alpha_t)_{t \in T}\) of \(\kappa\)-generated quasi-coherent modules on \((U, R, s, t, c)\) such that every \(\kappa\)-generated quasi-coherent module on \((U, R, s, t, c)\) is isomorphic to one of the \((\mathcal{F}_t, \alpha_t)\).

Proof

For each quasi-coherent module \(\mathcal{F}\) on \(U\) there is a (possibly empty) set of maps \(\alpha : t^*\mathcal{F} \to s^*\mathcal{F}\) such that \((\mathcal{F}, \alpha)\) is a quasi-coherent module on \((U, R, s, t, c)\). By Properties, Lemma 077M there exists a set of isomorphism classes of \(\kappa\)-generated quasi-coherent \(\mathcal{O}_U\)-modules.

Lemma

Let \((U, R, s, t, c)\) be a groupoid scheme over \(S\). Assume that \(s, t\) are flat. There exists a cardinal \(\kappa\) such that every quasi-coherent module \((\mathcal{F}, \alpha)\) on \((U, R, s, t, c)\) is the directed colimit of its \(\kappa\)-generated quasi-coherent submodules.

Proof

In the statement of the lemma and in this proof a submodule of a quasi-coherent module \((\mathcal{F}, \alpha)\) is a quasi-coherent submodule \(\mathcal{G} \subset \mathcal{F}\) such that \(\alpha(t^*\mathcal{G}) = s^*\mathcal{G}\) as subsheaves of \(s^*\mathcal{F}\). This makes sense because since \(s, t\) are flat the pullbacks \(s^*\) and \(t^*\) are exact, i.e., preserve subsheaves. The proof will be a repeat of the proof of Properties, Lemma 077N. We urge the reader to read that proof first.

Choose an affine open covering \(U = \bigcup_{i \in I} U_i\). For each pair \(i, j\) choose affine open coverings \[U_i \cap U_j = \bigcup\nolimits_{k \in I_{ij}} U_{ijk} \quad\text{and}\quad t^{-1}(U_i) \cap s^{-1}(U_j) = \bigcup\nolimits_{k \in J_{ij}} W_{ijk}.\] Write \(U_i = \Spec(A_i)\), \(U_{ijk} = \Spec(A_{ijk})\), \(W_{ijk} = \Spec(B_{ijk})\). Let \(\kappa\) be any infinite cardinal \(\geq\) the cardinality of any of the sets \(I\), \(I_{ij}\), \(J_{ij}\).

Let \((\mathcal{F}, \alpha)\) be a quasi-coherent module on \((U, R, s, t, c)\). Set \(M_i = \mathcal{F}(U_i)\), \(M_{ijk} = \mathcal{F}(U_{ijk})\). Note that \[M_i \otimes_{A_i} A_{ijk} = M_{ijk} = M_j \otimes_{A_j} A_{ijk}\] and that \(\alpha\) gives isomorphisms \[\alpha|_{W_{ijk}} : M_i \otimes_{A_i, t} B_{ijk} \longrightarrow M_j \otimes_{A_j, s} B_{ijk}\] see Schemes, Lemma 01I9. Using the axiom of choice we choose a map \[(i, j, k, m) \mapsto S(i, j, k, m)\] which associates to every \(i, j \in I\), \(k \in I_{ij}\) or \(k \in J_{ij}\) and \(m \in M_i\) a finite subset \(S(i, j, k, m) \subset M_j\) such that we have \[m \otimes 1 = \sum\nolimits_{m' \in S(i, j, k, m)} m' \otimes a_{m'} \quad\text{or}\quad \alpha(m \otimes 1) = \sum\nolimits_{m' \in S(i, j, k, m)} m' \otimes b_{m'}\] in \(M_{ijk}\) or \(M_j \otimes_{A_j, s} B_{ijk}\), respectively, for some \(a_{m'} \in A_{ijk}\) or \(b_{m'} \in B_{ijk}\). Moreover, let’s agree that \(S(i, i, k, m) = \{m\}\) for all \(i \in I\), \(k \in I_{ii}\), and \(m \in M_i\). Fix such a collection \(S(i, j, k, m)\)

Given a family \(\mathcal{S} = (S_i)_{i \in I}\) of subsets \(S_i \subset M_i\) of cardinality at most \(\kappa\) we set \(\mathcal{S}' = (S'_i)\) where \[S'_j = \bigcup\nolimits_{(i, j, k, m)\text{ such that }m \in S_i} S(i, j, k, m)\] Note that \(S_i \subset S'_i\). Note that \(S'_i\) has cardinality at most \(\kappa\) because it is a union over a set of cardinality at most \(\kappa\) of finite sets. Set \(\mathcal{S}^{(0)} = \mathcal{S}\), \(\mathcal{S}^{(1)} = \mathcal{S}'\) and by induction \(\mathcal{S}^{(n + 1)} = (\mathcal{S}^{(n)})'\). Then set \(\mathcal{S}^{(\infty)} = \bigcup_{n \geq 0} \mathcal{S}^{(n)}\). Writing \(\mathcal{S}^{(\infty)} = (S^{(\infty)}_i)\) we see that for any element \(m \in S^{(\infty)}_i\) the image of \(m\) in \(M_{ijk}\) can be written as a finite sum \(\sum m' \otimes a_{m'}\) with \(m' \in S_j^{(\infty)}\). In this way we see that setting \[N_i = A_i\text{-submodule of }M_i\text{ generated by }S^{(\infty)}_i\] we have \[N_i \otimes_{A_i} A_{ijk} = N_j \otimes_{A_j} A_{ijk} \quad\text{and}\quad \alpha(N_i \otimes_{A_i, t} B_{ijk}) = N_j \otimes_{A_j, s} B_{ijk}\] as submodules of \(M_{ijk}\) or \(M_j \otimes_{A_j, s} B_{ijk}\). Thus there exists a quasi-coherent submodule \(\mathcal{G} \subset \mathcal{F}\) with \(\mathcal{G}(U_i) = N_i\) such that \(\alpha(t^*\mathcal{G}) = s^*\mathcal{G}\) as submodules of \(s^*\mathcal{F}\). In other words, \((\mathcal{G}, \alpha|_{t^*\mathcal{G}})\) is a submodule of \((\mathcal{F}, \alpha)\). Moreover, by construction \(\mathcal{G}\) is \(\kappa\)-generated.

Let \(\{(\mathcal{G}_t, \alpha_t)\}_{t \in T}\) be the set of \(\kappa\)-generated quasi-coherent submodules of \((\mathcal{F}, \alpha)\). If \(t, t' \in T\) then \(\mathcal{G}_t + \mathcal{G}_{t'}\) is also a \(\kappa\)-generated quasi-coherent submodule as it is the image of the map \(\mathcal{G}_t \oplus \mathcal{G}_{t'} \to \mathcal{F}\). Hence the system (ordered by inclusion) is directed. The arguments above show that every section of \(\mathcal{F}\) over \(U_i\) is in one of the \(\mathcal{G}_t\) (because we can start with \(\mathcal{S}\) such that the given section is an element of \(S_i\)). Hence \(\colim_t \mathcal{G}_t \to \mathcal{F}\) is both injective and surjective as desired.

Groupoids and group schemes

There are many ways to construct a groupoid out of an action \(a\) of a group \(G\) on a set \(V\). We choose the one where we think of an element \(g \in G\) as an arrow with source \(v\) and target \(a(g, v)\). This leads to the following construction for group actions of schemes.

Lemma

Let \(S\) be a scheme. Let \(Y\) be a scheme over \(S\). Let \((G, m)\) be a group scheme over \(Y\) with identity \(e_G\) and inverse \(i_G\). Let \(X/Y\) be a scheme over \(Y\) and let \(a : G \times_Y X \to X\) be an action of \(G\) on \(X/Y\). Then we get a groupoid scheme \((U, R, s, t, c, e, i)\) over \(S\) in the following manner:

1. We set \(U = X\), and \(R = G \times_Y X\).

2. We set \(s : R \to U\) equal to \((g, x) \mapsto x\).

3. We set \(t : R \to U\) equal to \((g, x) \mapsto a(g, x)\).

4. We set \(c : R \times_{s, U, t} R \to R\) equal to \(((g, x), (g', x')) \mapsto (m(g, g'), x')\).

5. We set \(e : U \to R\) equal to \(x \mapsto (e_G(x), x)\).

6. We set \(i : R \to R\) equal to \((g, x) \mapsto (i_G(g), a(g, x))\).

Proof

Omitted. Hint: It is enough to show that this works on the set level. For this use the description above the lemma describing \(g\) as an arrow from \(v\) to \(a(g, v)\).

Lemma

Let \(S\) be a scheme. Let \(Y\) be a scheme over \(S\). Let \((G, m)\) be a group scheme over \(Y\). Let \(X\) be a scheme over \(Y\) and let \(a : G \times_Y X \to X\) be an action of \(G\) on \(X\) over \(Y\). Let \((U, R, s, t, c)\) be the groupoid scheme constructed in Lemma 0234. The rule \((\mathcal{F}, \alpha) \mapsto (\mathcal{F}, \alpha)\) defines an equivalence of categories between \(G\)-equivariant \(\mathcal{O}_X\)-modules and the category of quasi-coherent modules on \((U, R, s, t, c)\).

Proof

The assertion makes sense because \(t = a\) and \(s = \text{pr}_1\) as morphisms \(R = G \times_Y X \to X\), see Definitions 03LF and 03LI. Using the translation in Lemma 0234 the commutativity requirements of the two definitions match up exactly.

The stabilizer group scheme

Given a groupoid scheme we get a group scheme as follows.

Lemma

Let \(S\) be a scheme. Let \((U, R, s, t, c)\) be a groupoid over \(S\). The scheme \(G\) defined by the cartesian square \[\xymatrix{ G \ar[r] \ar[d] & R \ar[d]^{j = (t, s)} \\ U \ar[r]^-{\Delta} & U \times_S U }\] is a group scheme over \(U\) with composition law \(m\) induced by the composition law \(c\).

Proof

This is true because in a groupoid category the set of self maps of any object forms a group.

Since \(\Delta\) is an immersion we see that \(G = j^{-1}(\Delta_{U/S})\) is a locally closed subscheme of \(R\). Thinking of it in this way, the structure morphism \(j^{-1}(\Delta_{U/S}) \to U\) is induced by either \(s\) or \(t\) (it is the same), and \(m\) is induced by \(c\).

Definition

Let \(S\) be a scheme. Let \((U, R, s, t, c)\) be a groupoid over \(S\). The group scheme \(j^{-1}(\Delta_{U/S})\to U\) is called the stabilizer of the groupoid scheme \((U, R, s, t, c)\).

In the literature the stabilizer group scheme is often denoted \(S\) (because the word stabilizer starts with an “s” presumably); we cannot do this since we have already used \(S\) for the base scheme.

Lemma

Let \(S\) be a scheme. Let \((U, R, s, t, c)\) be a groupoid over \(S\), and let \(G/U\) be its stabilizer. Denote \(R_t/U\) the scheme \(R\) seen as a scheme over \(U\) via the morphism \(t : R \to U\). There is a canonical left action \[a : G \times_U R_t \longrightarrow R_t\] induced by the composition law \(c\).

Proof

In terms of points over \(T/S\) we define \(a(g, r) = c(g, r)\).

Lemma

Let \(S\) be a scheme. Let \((U, R, s, t, c)\) be a groupoid scheme over \(S\). Let \(G\) be the stabilizer group scheme of \(R\). Let \[G_0 = G \times_{U, \text{pr}_0} (U \times_S U) = G \times_S U\] as a group scheme over \(U \times_S U\). The action of \(G\) on \(R\) of Lemma 0237 induces an action of \(G_0\) on \(R\) over \(U \times_S U\) which turns \(R\) into a pseudo \(G_0\)-torsor over \(U \times_S U\).

Proof

This is true because in a groupoid category \(\mathcal{C}\) the set \(\Mor_\mathcal{C}(x, y)\) is a principal homogeneous set under the group \(\Mor_\mathcal{C}(y, y)\).

Lemma

Let \(S\) be a scheme. Let \((U, R, s, t, c)\) be a groupoid scheme over \(S\). Let \(p \in U \times_S U\) be a point. Denote \(R_p\) the scheme theoretic fibre of \(j = (t, s) : R \to U \times_S U\). If \(R_p \not = \emptyset\), then the action \[G_{0, \kappa(p)} \times_{\kappa(p)} R_p \longrightarrow R_p\] (see Lemma 04Q2) turns \(R_p\) into a \(G_{\kappa(p)}\)-torsor over \(\kappa(p)\).

Proof

The action is a pseudo-torsor by the lemma cited in the statement. And if \(R_p\) is not the empty scheme, then \(\{R_p \to p\}\) is an fpqc covering which trivializes the pseudo-torsor.

Restricting groupoids

Consider a (usual) groupoid \(\mathcal{C} = (\text{Ob}, \text{Arrows}, s, t, c)\). Suppose we have a map of sets \(g : \text{Ob}' \to \text{Ob}\). Then we can construct a groupoid \(\mathcal{C}' = (\text{Ob}', \text{Arrows}', s', t', c')\) by thinking of a morphism between elements \(x', y'\) of \(\text{Ob}'\) as a morphism in \(\mathcal{C}\) between \(g(x'), g(y')\). In other words we set \[\text{Arrows}' = \text{Ob}' \times_{g, \text{Ob}, t} \text{Arrows} \times_{s, \text{Ob}, g} \text{Ob}'.\] with obvious choices for \(s'\), \(t'\), and \(c'\). There is a canonical functor \(\mathcal{C}' \to \mathcal{C}\) which is fully faithful, but not necessarily essentially surjective. This groupoid \(\mathcal{C}'\) endowed with the functor \(\mathcal{C}' \to \mathcal{C}\) is called the restriction of the groupoid \(\mathcal{C}\) to \(\text{Ob}'\).

Lemma

Let \(S\) be a scheme. Let \((U, R, s, t, c)\) be a groupoid scheme over \(S\). Let \(g : U' \to U\) be a morphism of schemes. Consider the following diagram \[\xymatrix{ R' \ar[d] \ar[r] \ar@/_3pc/[dd]_{t'} \ar@/^1pc/[rr]^{s'}& R \times_{s, U} U' \ar[r] \ar[d] & U' \ar[d]^g \\ U' \times_{U, t} R \ar[d] \ar[r] & R \ar[r]^s \ar[d]_t & U \\ U' \ar[r]^g & U }\] where all the squares are fibre product squares. Then there is a canonical composition law \(c' : R' \times_{s', U', t'} R' \to R'\) such that \((U', R', s', t', c')\) is a groupoid scheme over \(S\) and such that \(U' \to U\), \(R' \to R\) defines a morphism \((U', R', s', t', c') \to (U, R, s, t, c)\) of groupoid schemes over \(S\). Moreover, for any scheme \(T\) over \(S\) the functor of groupoids \[(U'(T), R'(T), s', t', c') \to (U(T), R(T), s, t, c)\] is the restriction (see above) of \((U(T), R(T), s, t, c)\) via the map \(U'(T) \to U(T)\).

Proof

Omitted.

Definition

Let \(S\) be a scheme. Let \((U, R, s, t, c)\) be a groupoid scheme over \(S\). Let \(g : U' \to U\) be a morphism of schemes. The morphism of groupoids \((U', R', s', t', c') \to (U, R, s, t, c)\) constructed in Lemma 02VB is called the restriction of \((U, R, s, t, c)\) to \(U'\). We sometimes use the notation \(R' = R|_{U'}\) in this case.

Lemma

The notions of restricting groupoids and (pre-)equivalence relations defined in Definitions 02VC and 02V9 agree via the constructions of Lemmas 0232 and 0233.

Proof

What we are saying here is that \(R'\) of Lemma 02VB is also equal to \[R' = (U' \times_S U')\times_{U \times_S U} R \longrightarrow U' \times_S U'\] In fact this might have been a clearer way to state that lemma.

Lemma

Let \(S\) be a scheme. Let \((U, R, s, t, c)\) be a groupoid scheme over \(S\). Let \(g : U' \to U\) be a morphism of schemes. Let \((U', R', s', t', c')\) be the restriction of \((U, R, s, t, c)\) via \(g\). Let \(G\) be the stabilizer of \((U, R, s, t, c)\) and let \(G'\) be the stabilizer of \((U', R', s', t', c')\). Then \(G'\) is the base change of \(G\) by \(g\), i.e., there is a canonical identification \(G' = U' \times_{g, U} G\).

Proof

Omitted.

Invariant subschemes

In this section we discuss briefly the notion of an invariant subscheme.

Definition

Let \((U, R, s, t, c)\) be a groupoid scheme over the base scheme \(S\).

1. A subset \(W \subset U\) is set-theoretically \(R\)-invariant if \(t(s^{-1}(W)) \subset W\).

2. An open \(W \subset U\) is \(R\)-invariant if \(t(s^{-1}(W)) \subset W\).

3. A closed subscheme \(Z \subset U\) is called \(R\)-invariant if \(t^{-1}(Z) = s^{-1}(Z)\). Here we use the scheme theoretic inverse image, see Schemes, Definition 01JV.

4. A monomorphism of schemes \(T \to U\) is \(R\)-invariant if \(T \times_{U, t} R = R \times_{s, U} T\) as schemes over \(R\).

For subsets and open subschemes \(W \subset U\) the \(R\)-invariance is also equivalent to requiring that \(s^{-1}(W) = t^{-1}(W)\) as subsets of \(R\). If \(W \subset U\) is an \(R\)-equivariant open subscheme then the restriction of \(R\) to \(W\) is just \(R_W = s^{-1}(W) = t^{-1}(W)\). Similarly, if \(Z \subset U\) is an \(R\)-invariant closed subscheme, then the restriction of \(R\) to \(Z\) is just \(R_Z = s^{-1}(Z) = t^{-1}(Z)\).

Lemma

Let \(S\) be a scheme. Let \((U, R, s, t, c)\) be a groupoid scheme over \(S\).

1. For any subset \(W \subset U\) the subset \(t(s^{-1}(W))\) is set-theoretically \(R\)-invariant.

2. If \(s\) and \(t\) are open, then for every open \(W \subset U\) the open \(t(s^{-1}(W))\) is an \(R\)-invariant open subscheme.

3. If \(s\) and \(t\) are open and quasi-compact, then \(U\) has an open covering consisting of \(R\)-invariant quasi-compact open subschemes.

Proof

Part (1) follows from Lemmas 022Q and 0232, namely, \(t(s^{-1}(W))\) is the set of points of \(U\) equivalent to a point of \(W\). Next, assume \(s\) and \(t\) open and \(W \subset U\) open. Since \(t\) is open the set \(W' = t(s^{-1}(W))\) is an open subset of \(U\). Finally, assume that \(s\), \(t\) are both open and quasi-compact. Then, if \(W \subset U\) is a quasi-compact open, then also \(W' = t(s^{-1}(W))\) is a quasi-compact open, and invariant by the discussion above. Letting \(W\) range over all affine opens of \(U\) we see (3).

Lemma

Let \(S\) be a scheme. Let \((U, R, s, t, c)\) be a groupoid scheme over \(S\). Assume \(s\) and \(t\) quasi-compact and flat and \(U\) quasi-separated. Let \(W \subset U\) be quasi-compact open. Then \(t(s^{-1}(W))\) is an intersection of a nonempty family of quasi-compact open subsets of \(U\).

Proof

Note that \(s^{-1}(W)\) is quasi-compact open in \(R\). As a continuous map \(t\) maps the quasi-compact subset \(s^{-1}(W)\) to a quasi-compact subset \(t(s^{-1}(W))\). As \(t\) is flat and \(s^{-1}(W)\) is closed under generalization, so is \(t(s^{-1}(W))\), see (Morphisms, Lemma 03HV and Topology, Lemma 0065). Pick a quasi-compact open \(W' \subset U\) containing \(t(s^{-1}(W))\). By Properties, Lemma 094L we see that \(W'\) is a spectral space (here we use that \(U\) is quasi-separated). Then the lemma follows from Topology, Lemma 0A31 applied to \(t(s^{-1}(W)) \subset W'\).

Lemma

Assumptions and notation as in Lemma 0APA. There exists an \(R\)-invariant open \(V \subset U\) and a quasi-compact open \(W'\) such that \(W \subset V \subset W' \subset U\).

Proof

Set \(E = t(s^{-1}(W))\). Recall that \(E\) is set-theoretically \(R\)-invariant (Lemma 03LO). By Lemma 0APA there exists a quasi-compact open \(W'\) containing \(E\). Let \(Z = U \setminus W'\) and consider \(T = t(s^{-1}(Z))\). Observe that \(Z \subset T\) and that \(E \cap T = \emptyset\) because \(s^{-1}(E) = t^{-1}(E)\) is disjoint from \(s^{-1}(Z)\). Since \(T\) is the image of the closed subset \(s^{-1}(Z) \subset R\) under the quasi-compact morphism \(t : R \to U\) we see that any point \(\xi\) in the closure \(\overline{T}\) is the specialization of a point of \(T\), see Morphisms, Lemma 02JQ (and Morphisms, Lemma 01R8 to see that the scheme theoretic image is the closure of the image). Say \(\xi' \leadsto \xi\) with \(\xi' \in T\). Suppose that \(r \in R\) and \(s(r) = \xi\). Since \(s\) is flat we can find a specialization \(r' \leadsto r\) in \(R\) such that \(s(r') = \xi'\) (Morphisms, Lemma 03HV). Then \(t(r') \leadsto t(r)\). We conclude that \(t(r') \in T\) as \(T\) is set-theoretically invariant by Lemma 03LO. Thus \(\overline{T}\) is a set-theoretically \(R\)-invariant closed subset and \(V = U \setminus \overline{T}\) is the open we are looking for. It is contained in \(W'\) which finishes the proof.

Quotient sheaves

Let \(\tau \in \{Zariski, \etale, fppf, smooth, syntomic\}\). Let \(S\) be a scheme. Let \(j : R \to U \times_S U\) be a pre-relation over \(S\). Say \(U, R, S\) are objects of a \(\tau\)-site \(\Sch_\tau\) (see Topologies, Section 020M). Then we can consider the functors \[h_U, h_R : (\Sch/S)_\tau^{opp} \longrightarrow \textit{Sets}.\] These are sheaves, see Descent, Lemma 023Q. The morphism \(j\) induces a map \(j : h_R \to h_U \times h_U\). For each object \(T \in \Ob((\Sch/S)_\tau)\) we can take the equivalence relation \(\sim_T\) generated by \(j(T) : R(T) \to U(T) \times U(T)\) and consider the quotient. Hence we get a presheaf [02VF]\[\begin{equation} (\Sch/S)_\tau^{opp} \longrightarrow \textit{Sets}, \quad T \longmapsto U(T)/\sim_T \end{equation}\]

Definition

Let \(\tau\), \(S\), and the pre-relation \(j : R \to U \times_S U\) be as above. In this setting the quotient sheaf \(U/R\) associated to \(j\) is the sheafification of the presheaf (02VF) in the \(\tau\)-topology. If \(j : R \to U \times_S U\) comes from the action of a group scheme \(G/S\) on \(U\) as in Lemma 0234 then we sometimes denote the quotient sheaf \(U/G\).

This means exactly that the diagram \[\xymatrix{ h_R \ar@<1ex>[r] \ar@<-1ex>[r] & h_U \ar[r] & U/R }\] is a coequalizer diagram in the category of sheaves of sets on \((\Sch/S)_\tau\). Using the Yoneda embedding we may view \((\Sch/S)_\tau\) as a full subcategory of sheaves on \((\Sch/S)_\tau\) and hence identify schemes with representable functors. Using this abuse of notation we will often depict the diagram above simply \[\xymatrix{ R \ar@<1ex>[r]^s \ar@<-1ex>[r]_t & U \ar[r] & U/R }\] We will mostly work with the fppf topology when considering quotient sheaves of groupoids/equivalence relations.

Definition

In the situation of Definition 02VG. We say that the pre-relation \(j\) has a representable quotient if the sheaf \(U/R\) is representable. We will say a groupoid \((U, R, s, t, c)\) has a representable quotient if the quotient \(U/R\) with \(j = (t, s)\) is representable.

The following lemma characterizes schemes \(M\) representing the quotient. It applies for example if \(\tau = fppf\), \(U \to M\) is flat, of finite presentation and surjective, and \(R \cong U \times_M U\).

Lemma

In the situation of Definition 02VG. Assume there is a scheme \(M\), and a morphism \(U \to M\) such that

1. the morphism \(U \to M\) equalizes \(s, t\),

2. the morphism \(U \to M\) induces a surjection of sheaves \(h_U \to h_M\) in the \(\tau\)-topology, and

3. the induced map \((t, s) : R \to U \times_M U\) induces a surjection of sheaves \(h_R \to h_{U \times_M U}\) in the \(\tau\)-topology.

In this case \(M\) represents the quotient sheaf \(U/R\).

Proof

Condition (1) says that \(h_U \to h_M\) factors through \(U/R\). Condition (2) says that \(U/R \to h_M\) is surjective as a map of sheaves. Condition (3) says that \(U/R \to h_M\) is injective as a map of sheaves. Hence the lemma follows.

The following lemma is wrong if we do not require \(j\) to be a pre-equivalence relation (but just a pre-relation say).

Lemma

Let \(\tau \in \{Zariski, \etale, fppf, smooth, syntomic\}\). Let \(S\) be a scheme. Let \(j : R \to U \times_S U\) be a pre-equivalence relation over \(S\). Assume \(U, R, S\) are objects of a \(\tau\)-site \(\Sch_\tau\). For \(T \in \Ob((\Sch/S)_\tau)\) and \(a, b \in U(T)\) the following are equivalent:

1. \(a\) and \(b\) map to the same element of \((U/R)(T)\), and

2. there exists a \(\tau\)-covering \(\{f_i : T_i \to T\}\) of \(T\) and morphisms \(r_i : T_i \to R\) such that \(a \circ f_i = s \circ r_i\) and \(b \circ f_i = t \circ r_i\).

In other words, in this case the map of \(\tau\)-sheaves \[h_R \longrightarrow h_U \times_{U/R} h_U\] is surjective.

Proof

Omitted. Hint: The reason this works is that the presheaf (02VF) in this case is really given by \(T \mapsto U(T)/j(R(T))\) as \(j(R(T)) \subset U(T) \times U(T)\) is an equivalence relation, see Definition 022P.

Lemma

Let \(\tau \in \{Zariski, \etale, fppf, smooth, syntomic\}\). Let \(S\) be a scheme. Let \(j : R \to U \times_S U\) be a pre-equivalence relation over \(S\) and \(g : U' \to U\) a morphism of schemes over \(S\). Let \(j' : R' \to U' \times_S U'\) be the restriction of \(j\) to \(U'\). Assume \(U, U', R, S\) are objects of a \(\tau\)-site \(\Sch_\tau\). The map of quotient sheaves \[U'/R' \longrightarrow U/R\] is injective. If \(g\) defines a surjection \(h_{U'} \to h_U\) of sheaves in the \(\tau\)-topology (for example if \(\{g : U' \to U\}\) is a \(\tau\)-covering), then \(U'/R' \to U/R\) is an isomorphism.

Proof

Suppose \(\xi, \xi' \in (U'/R')(T)\) are sections which map to the same section of \(U/R\). Then we can find a \(\tau\)-covering \(\mathcal{T} = \{T_i \to T\}\) of \(T\) such that \(\xi|_{T_i}, \xi'|_{T_i}\) are given by \(a_i, a_i' \in U'(T_i)\). By Lemma 045Y and the axioms of a site we may after refining \(\mathcal{T}\) assume there exist morphisms \(r_i : T_i \to R\) such that \(g \circ a_i = s \circ r_i\), \(g \circ a_i' = t \circ r_i\). Since by construction \(R' = R \times_{U \times_S U} (U' \times_S U')\) we see that \((r_i, (a_i, a_i')) \in R'(T_i)\) and this shows that \(a_i\) and \(a_i'\) define the same section of \(U'/R'\) over \(T_i\). By the sheaf condition this implies \(\xi = \xi'\).

If \(h_{U'} \to h_U\) is a surjection of sheaves, then of course \(U'/R' \to U/R\) is surjective also. If \(\{g : U' \to U\}\) is a \(\tau\)-covering, then the map of sheaves \(h_{U'} \to h_U\) is surjective, see Sites, Lemma 00WT. Hence \(U'/R' \to U/R\) is surjective also in this case.

Lemma

Let \(\tau \in \{Zariski, \etale, fppf, smooth, syntomic\}\). Let \(S\) be a scheme. Let \((U, R, s, t, c)\) be a groupoid scheme over \(S\). Let \(g : U' \to U\) be a morphism of schemes over \(S\). Let \((U', R', s', t', c')\) be the restriction of \((U, R, s, t, c)\) to \(U'\). Assume \(U, U', R, S\) are objects of a \(\tau\)-site \(\Sch_\tau\). The map of quotient sheaves \[U'/R' \longrightarrow U/R\] is injective. If the composition \[\xymatrix{ U' \times_{g, U, t} R \ar[r]_-{\text{pr}_1} \ar@/^3ex/[rr]^h & R \ar[r]_s & U }\] defines a surjection of sheaves in the \(\tau\)-topology then the map is bijective. This holds for example if \(\{h : U' \times_{g, U, t} R \to U\}\) is a \(\tau\)-covering, or if \(U' \to U\) defines a surjection of sheaves in the \(\tau\)-topology, or if \(\{g : U' \to U\}\) is a covering in the \(\tau\)-topology.

Proof

Injectivity follows on combining Lemmas 0232 and 045Z. To see surjectivity (see Sites, Section 00WL for a characterization of surjective maps of sheaves) we argue as follows. Suppose that \(T\) is a scheme and \(\sigma \in U/R(T)\). There exists a covering \(\{T_i \to T\}\) such that \(\sigma|_{T_i}\) is the image of some element \(f_i \in U(T_i)\). Hence we may assume that \(\sigma\) is the image of \(f \in U(T)\). By the assumption that \(h\) is a surjection of sheaves, we can find a \(\tau\)-covering \(\{\varphi_i : T_i \to T\}\) and morphisms \(f_i : T_i \to U' \times_{g, U, t} R\) such that \(f \circ \varphi_i = h \circ f_i\). Denote \(f'_i = \text{pr}_0 \circ f_i : T_i \to U'\). Then we see that \(f'_i \in U'(T_i)\) maps to \(g \circ f'_i \in U(T_i)\) and that \(g \circ f'_i \sim_{T_i} h \circ f_i = f \circ \varphi_i\) notation as in (02VF). Namely, the element of \(R(T_i)\) giving the relation is \(\text{pr}_1 \circ f_i\). This means that the restriction of \(\sigma\) to \(T_i\) is in the image of \(U'/R'(T_i) \to U/R(T_i)\) as desired.

If \(\{h\}\) is a \(\tau\)-covering, then it induces a surjection of sheaves, see Sites, Lemma 00WT. If \(U' \to U\) is surjective, then also \(h\) is surjective as \(s\) has a section (namely the neutral element \(e\) of the groupoid scheme).

Lemma

Let \(S\) be a scheme. Let \(f : (U, R, j) \to (U', R', j')\) be a morphism between equivalence relations over \(S\). Assume that \[\xymatrix{ R \ar[d]_s \ar[r]_f & R' \ar[d]^{s'} \\ U \ar[r]^f & U' }\] is cartesian. For any \(\tau \in \{Zariski, \etale, fppf, smooth, syntomic\}\) the diagram \[\xymatrix{ U \ar[d] \ar[r] & U/R \ar[d]^f \\ U' \ar[r] & U'/R' }\] is a fibre product square of \(\tau\)-sheaves.

Proof

By Lemma 045Y the quotient sheaves have a simple description which we will use below without further mention. We first show that \[U \longrightarrow U' \times_{U'/R'} U/R\] is injective. Namely, assume \(a, b \in U(T)\) map to the same element on the right hand side. Then \(f(a) = f(b)\). After replacing \(T\) by the members of a \(\tau\)-covering we may assume that there exists an \(r \in R(T)\) such that \(a = s(r)\) and \(b = t(r)\). Then \(r' = f(r)\) is a \(T\)-valued point of \(R'\) with \(s'(r') = t'(r')\). Hence \(r' = e'(f(a))\) (where \(e'\) is the identity of the groupoid scheme associated to \(j'\), see Lemma 0233). Because the first diagram of the lemma is cartesian this implies that \(r\) has to equal \(e(a)\). Thus \(a = b\).

Finally, we show that the displayed arrow is surjective. Let \(T\) be a scheme over \(S\) and let \((a', \overline{b})\) be a section of the sheaf \(U' \times_{U'/R'} U/R\) over \(T\). After replacing \(T\) by the members of a \(\tau\)-covering we may assume that \(\overline{b}\) is the class of an element \(b \in U(T)\). After replacing \(T\) by the members of a \(\tau\)-covering we may assume that there exists an \(r' \in R'(T)\) such that \(a' = t(r')\) and \(s'(r') = f(b)\). Because the first diagram of the lemma is cartesian we can find \(r \in R(T)\) such that \(s(r) = b\) and \(f(r) = r'\). Then it is clear that \(a = t(r) \in U(T)\) is a section which maps to \((a', \overline{b})\).

Descent in terms of groupoids

Cartesian morphisms are defined as follows.

Definition

Let \(S\) be a scheme. Let \(f : (U', R', s', t', c') \to (U, R, s, t, c)\) be a morphism of groupoid schemes over \(S\). We say \(f\) is cartesian, or that \((U', R', s', t', c')\) is cartesian over \((U, R, s, t, c)\), if the diagram \[\xymatrix{ R' \ar[r]_f \ar[d]_{s'} & R \ar[d]^s \\ U' \ar[r]^f & U }\] is a fibre square in the category of schemes. A morphism of groupoid schemes cartesian over \((U, R, s, t, c)\) is a morphism of groupoid schemes compatible with the structure morphisms towards \((U, R, s, t, c)\).

Cartesian morphisms are related to descent data. First we prove a general lemma describing the category of cartesian groupoid schemes over a fixed groupoid scheme.

Lemma

Let \(S\) be a scheme. Let \((U, R, s, t, c)\) be a groupoid scheme over \(S\). The category of groupoid schemes cartesian over \((U, R, s, t, c)\) is equivalent to the category of pairs \((V, \varphi)\) where \(V\) is a scheme over \(U\) and \[\varphi : V \times_{U, t} R \longrightarrow R \times_{s, U} V\] is an isomorphism over \(R\) such that \(e^*\varphi = \text{id}_V\) and such that \[c^*\varphi = \text{pr}_1^*\varphi \circ \text{pr}_0^*\varphi\] as morphisms of schemes over \(R \times_{s, U, t} R\).

Proof

The pullback notation in the lemma signifies base change. The displayed formula makes sense because \[(R \times_{s, U, t} R) \times_{\text{pr}_1, R, \text{pr}_1} (V \times_{U, t} R) = (R \times_{s, U, t} R) \times_{\text{pr}_0, R, \text{pr}_0} (R \times_{s, U} V)\] as schemes over \(R \times_{s, U, t} R\).

Given \((V, \varphi)\) we set \(U' = V\) and \(R' = V \times_{U, t} R\). We set \(t' : R' \to U'\) equal to the projection \(V \times_{U, t} R \to V\). We set \(s'\) equal to \(\varphi\) followed by the projection \(R \times_{s, U} V \to V\). We set \(c'\) equal to the composition \[\begin{align*} R' \times_{s', U', t'} R' & \xrightarrow{\varphi, 1} (R \times_{s, U} V) \times_V (V \times_{U, t} R) \\ & \xrightarrow{} R \times_{s, U} V \times_{U, t} R \\ & \xrightarrow{\varphi^{-1}, 1} V \times_{U, t} (R \times_{s, U, t} R) \\ & \xrightarrow{1, c} V \times_{U, t} R = R' \end{align*}\] A computation, which we omit, shows that we obtain a groupoid scheme over \((U, R, s, t, c)\). It is clear that this groupoid scheme is cartesian over \((U, R, s, t, c)\).

Conversely, given \(f : (U', R', s', t', c') \to (U, R, s, t, c)\) cartesian then the morphisms \[U' \times_{U, t} R \xleftarrow{t', f} R' \xrightarrow{f, s'} R \times_{s, U} U'\] are isomorphisms and we can set \(V = U'\) and \(\varphi\) equal to the composition \((f, s') \circ (t', f)^{-1}\). We omit the proof that \(\varphi\) satisfies the conditions in the lemma. We omit the proof that these constructions are mutually inverse.

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of schemes over \(S\). Then we obtain a groupoid scheme \((X, X \times_Y X, \text{pr}_1, \text{pr}_0, c)\) over \(S\). Namely, \(j : X \times_Y X \to X \times_S X\) is an equivalence relation and we can take the associated groupoid, see Lemma 0233.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of schemes over \(S\). The construction of Lemma 0APE determines an equivalence \[\begin{matrix} \text{category of groupoid schemes} \\ \text{cartesian over } (X, X \times_Y X, \ldots) \end{matrix} \longrightarrow \begin{matrix} \text{ category of descent data} \\ \text{ relative to } X/Y \end{matrix}\]

Proof

This is clear from Lemma 0APE and the definition of descent data on schemes in Descent, Definition 023V.

Separation conditions

This really means conditions on the morphism \(j : R \to U \times_S U\) when given a groupoid \((U, R, s, t, c)\) over \(S\). As in the previous section we first formulate the corresponding diagram.

Lemma

Let \(S\) be a scheme. Let \((U, R, s, t, c)\) be a groupoid over \(S\). Let \(G \to U\) be the stabilizer group scheme. The commutative diagram \[\xymatrix{ R \ar[d]^{\Delta_{R/U \times_S U}} \ar[rrr]_{f \mapsto (f, s(f))} & & & R \times_{s, U} U \ar[d] \ar[r] & U \ar[d] \\ R \times_{(U \times_S U)} R \ar[rrr]^{(f, g) \mapsto (f, f^{-1} \circ g)} & & & R \times_{s, U} G \ar[r] & G }\] the two left horizontal arrows are isomorphisms and the right square is a fibre product square.

Proof

Omitted. Exercise in the definitions and the functorial point of view in algebraic geometry.

Lemma

Let \(S\) be a scheme. Let \((U, R, s, t, c)\) be a groupoid over \(S\). Let \(G \to U\) be the stabilizer group scheme.

1. The following are equivalent

   1. \(j : R \to U \times_S U\) is separated,

   2. \(G \to U\) is separated, and

   3. \(e : U \to G\) is a closed immersion.

2. The following are equivalent

   1. \(j : R \to U \times_S U\) is quasi-separated,

   2. \(G \to U\) is quasi-separated, and

   3. \(e : U \to G\) is quasi-compact.

Proof

The group scheme \(G \to U\) is the base change of \(R \to U \times_S U\) by the diagonal morphism \(U \to U \times_S U\), see Lemma 0235. Hence if \(j\) is separated (resp. quasi-separated), then \(G \to U\) is separated (resp. quasi-separated). (See Schemes, Lemma 01KU). Thus (a) \(\Rightarrow\) (b) in both (1) and (2).

If \(G \to U\) is separated (resp. quasi-separated), then the morphism \(U \to G\), as a section of the structure morphism \(G \to U\) is a closed immersion (resp. quasi-compact), see Schemes, Lemma 01KT. Thus (b) \(\Rightarrow\) (c) in both (1) and (2).

By the result of Lemma 02YH (and Schemes, Lemmas 01JY and 01K5) we see that if \(e\) is a closed immersion (resp. quasi-compact) \(\Delta_{R/U \times_S U}\) is a closed immersion (resp. quasi-compact). Thus (c) \(\Rightarrow\) (a) in both (1) and (2).

Finite flat groupoids, affine case

Let \(S\) be a scheme. Let \((U, R, s, t, c)\) be a groupoid scheme over \(S\). Assume \(U = \Spec(A)\), and \(R = \Spec(B)\) are affine. In this case we get two ring maps \(s^\sharp, t^\sharp : A \longrightarrow B\). Let \(C\) be the equalizer of \(s^\sharp\) and \(t^\sharp\). In a formula [03BF]\[\begin{equation} C = \{a \in A \mid t^\sharp(a) = s^\sharp(a) \}. \end{equation}\] We will sometimes call this the ring of \(R\)-invariant functions on \(U\). What properties does \(M = \Spec(C)\) have? The first observation is that the diagram \[\xymatrix{ R \ar[r]_s \ar[d]_t & U \ar[d] \\ U \ar[r] & M }\] is commutative, i.e., the morphism \(U \to M\) equalizes \(s, t\). Moreover, if \(T\) is any affine scheme, and if \(U \to T\) is a morphism which equalizes \(s, t\), then \(U \to T\) factors through \(U \to M\). In other words, \(U \to M\) is a coequalizer in the category of affine schemes.

We would like to find conditions that guarantee the morphism \(U \to M\) is really a “quotient” in the category of schemes. We will discuss this at length elsewhere; here we just discuss some special cases. Namely, we will focus on the case where \(s, t\) are finite locally free.

Example

Let \(k\) be a field. Let \(U = \text{GL}_{2, k}\). Let \(B \subset \text{GL}_2\) be the closed subgroup scheme of upper triangular matrices. Then the quotient sheaf \(\text{GL}_{2, k}/B\) (in the Zariski, étale or fppf topology, see Definition 02VG) is representable by the projective line: \(\mathbf{P}^1 = \text{GL}_{2, k}/B\). (Details omitted.) On the other hand, the ring of invariant functions in this case is just \(k\). Note that in this case the morphisms \(s, t : R = \text{GL}_{2, k} \times_k B \to \text{GL}_{2, k} = U\) are smooth of relative dimension \(3\).

Recall that in Exercises, Exercises 02DU and 02DV we have defined the determinant and the norm for finitely locally free modules and finite locally free ring extensions. If \(\varphi : A \to B\) is a finite locally free ring map, then we will denote \(\text{Norm}_\varphi(b) \in A\) the norm of \(b \in B\). In the case of a finite locally free morphism of schemes, the norm was constructed in Divisors, Lemma 0BD2.

Lemma

Let \(S\) be a scheme. Let \((U, R, s, t, c)\) be a groupoid scheme over \(S\). Assume \(U = \Spec(A)\) and \(R = \Spec(B)\) are affine and \(s, t : R \to U\) finite locally free. Let \(C\) be as in (03BF). Let \(f \in A\). Then \(\text{Norm}_{s^\sharp}(t^\sharp(f)) \in C\).

Proof

Consider the commutative diagram \[\xymatrix{ & U & \\ R \ar[d]_s \ar[ru]^t & R \times_{s, U, t} R \ar[l]^-{\text{pr}_0} \ar[d]^{\text{pr}_1} \ar[r]_-c & R \ar[d]^s \ar[lu]_t \\ U & R \ar[l]_t \ar[r]^s & U }\] of Lemma 02YE. Think of \(f \in \Gamma(U, \mathcal{O}_U)\). The commutativity of the top part of the diagram shows that \(\text{pr}_0^\sharp(t^\sharp(f)) = c^\sharp(t^\sharp(f))\) as elements of \(\Gamma(R \times_{s, U, t} R, \mathcal{O})\). Looking at the right lower cartesian square the compatibility of the norm construction with base change shows that \(s^\sharp(\text{Norm}_{s^\sharp}(t^\sharp(f))) = \text{Norm}_{\text{pr}_1^\sharp}(c^\sharp(t^\sharp(f)))\). Similarly we get \(t^\sharp(\text{Norm}_{s^\sharp}(t^\sharp(f))) = \text{Norm}_{\text{pr}_1^\sharp}(\text{pr}_0^\sharp(t^\sharp(f)))\). Hence by the first equality of this proof we see that \(s^\sharp(\text{Norm}_{s^\sharp}(t^\sharp(f))) = t^\sharp(\text{Norm}_{s^\sharp}(t^\sharp(f)))\) as desired.

Lemma

Let \(S\) be a scheme. Let \((U, R, s, t, c)\) be a groupoid scheme over \(S\). Assume \(s, t : R \to U\) finite locally free. Then \[U = \coprod\nolimits_{r \geq 1} U_r\] is a disjoint union of \(R\)-invariant opens such that the restriction \(R_r\) of \(R\) to \(U_r\) has the property that \(s, t : R_r \to U_r\) are finite locally free of rank \(r\).

Proof

By Morphisms, Lemma 04MH there exists a decomposition \(U = \coprod\nolimits_{r \geq 0} U_r\) such that \(s : s^{-1}(U_r) \to U_r\) is finite locally free of rank \(r\). As \(s\) is surjective we see that \(U_0 = \emptyset\). Note that \(u \in U_r \Leftrightarrow\) the scheme theoretic fibre \(s^{-1}(u)\) has degree \(r\) over \(\kappa(u)\). Now, if \(z \in R\) with \(s(z) = u\) and \(t(z) = u'\) then using notation as in Lemma 02YE \[\text{pr}_1^{-1}(z) \to \Spec(\kappa(z))\] is the base change of both \(s^{-1}(u) \to \Spec(\kappa(u))\) and \(s^{-1}(u') \to \Spec(\kappa(u'))\) by the lemma cited. Hence \(u \in U_r \Leftrightarrow u' \in U_r\), in other words, the open subsets \(U_r\) are \(R\)-invariant. In particular the restriction of \(R\) to \(U_r\) is just \(s^{-1}(U_r)\) and \(s : R_r \to U_r\) is finite locally free of rank \(r\). As \(t : R_r \to U_r\) is isomorphic to \(s\) by the inverse of \(R_r\) we see that it has also rank \(r\).

Lemma

Let \(S\) be a scheme. Let \((U, R, s, t, c)\) be a groupoid scheme over \(S\). Assume \(U = \Spec(A)\) and \(R = \Spec(B)\) are affine and \(s, t : R \to U\) finite locally free. Let \(C \subset A\) be as in (03BF). Then \(A\) is integral over \(C\).

Proof

First, by Lemma 03BI we know that \((U, R, s, t, c)\) is a disjoint union of groupoid schemes \((U_r, R_r, s, t, c)\) such that each \(s, t : R_r \to U_r\) has constant rank \(r\). As \(U\) is quasi-compact, we have \(U_r = \emptyset\) for almost all \(r\). It suffices to prove the lemma for each \((U_r, R_r, s, t, c)\) and hence we may assume that \(s, t\) are finite locally free of rank \(r\).

Assume that \(s, t\) are finite locally free of rank \(r\). Let \(f \in A\). Consider the element \(x - f \in A[x]\), where we think of \(x\) as the coordinate on \(\mathbf{A}^1\). Since \[(U \times \mathbf{A}^1, R \times \mathbf{A}^1, s \times \text{id}_{\mathbf{A}^1}, t \times \text{id}_{\mathbf{A}^1}, c \times \text{id}_{\mathbf{A}^1})\] is also a groupoid scheme with finite source and target, we may apply Lemma 03BH to it and we see that \(P(x) = \text{Norm}_{s^\sharp}(t^\sharp(x - f))\) is an element of \(C[x]\). Because \(s^\sharp : A \to B\) is finite locally free of rank \(r\) we see that \(P\) is monic of degree \(r\). Moreover \(P(f) = 0\) by Cayley-Hamilton (Algebra, Lemma 00DX). More precisely, Cayley-Hamilton says the polynomial \(s^\sharp(P) \in B[x]\) annihilates the element \(t^\sharp(f)\). Since the coefficients of \(P\) are in \(C\) we get that \(t^\sharp(P(f)) = 0\) in \(B\). Since \(t\) is faithfully flat, we get \(P(f) = 0\).

Lemma

Let \(S\) be a scheme. Let \((U, R, s, t, c)\) be a groupoid scheme over \(S\). Assume \(U = \Spec(A)\) and \(R = \Spec(B)\) are affine and \(s, t : R \to U\) finite locally free. Let \(C \subset A\) be as in (03BF). Let \(C \to C'\) be a ring map, and set \(U' = \Spec(A \otimes_C C')\), \(R' = \Spec(B \otimes_C C')\). Then

1. The maps \(s, t, c\) induce maps \(s', t', c'\) such that \((U', R', s', t', c')\) is a groupoid scheme. Let \(C^1 \subset A'\) be the \(R'\)-invariant functions on \(U'\).

2. The canonical map \(\varphi : C' \to C^1\) satisfies

   1. for every \(f \in C^1\) there exists an \(n > 0\) and a polynomial \(P \in C'[x]\) whose image in \(C^1[x]\) is \((x - f)^n\), and

   2. for every \(f \in \Ker(\varphi)\) there exists an \(n > 0\) such that \(f^n = 0\).

3. If \(C \to C'\) is flat then \(\varphi\) is an isomorphism.

Proof

The proof of part (1) is omitted. Let us denote \(A' = A \otimes_C C'\) and \(B' = B \otimes_C C'\). Then we have \[C^1 = \{a \in A' \mid (t')^\sharp(a) = (s')^\sharp(a) \} = \{a \in A \otimes_C C' \mid t^\sharp \otimes 1(a) = s^\sharp \otimes 1(a) \}.\] In other words, \(C^1\) is the kernel of the difference map \((t^\sharp - s^\sharp) \otimes 1\) which is just the base change of the \(C\)-linear map \(t^\sharp - s^\sharp : A \to B\) by \(C \to C'\). Hence (3) follows.

Proof of part (2)(b). Since \(C \to A\) is integral (Lemma 03BJ) and injective we see that \(\Spec(A) \to \Spec(C)\) is surjective, see Algebra, Lemma 00GQ. Thus also \(\Spec(A') \to \Spec(C')\) is surjective as a base change of a surjective morphism (Morphisms, Lemma 01S1). Hence \(\Spec(C^1) \to \Spec(C')\) is surjective also. This implies (2)(b) holds for example by Algebra, Lemma 00FL.

Proof of part (2)(a). By Lemma 03BI our groupoid scheme \((U, R, s, t, c)\) decomposes as a finite disjoint union of groupoid schemes \((U_r, R_r, s, t, c)\) such that \(s, t : R_r \to U_r\) are finite locally free of rank \(r\). Pulling back by \(U' = \Spec(A') \to U\) we obtain a similar decomposition of \(U'\) and \(U^1 = \Spec(C^1)\). We will show in the next paragraph that (2)(a) holds for the corresponding system of rings \(A_r, B_r, C_r, C'_r, C^1_r\) with \(n = r\). Then given \(f \in C^1\) let \(P_r \in C'_r[x]\) be the polynomial whose image in \(C^1_r[x]\) is the image of \((x - f)^r\). Choosing a sufficiently divisible integer \(n\) we see that there is a polynomial \(P \in C'[x]\) whose image in \(C^1[x]\) is \((x - f)^n\); namely, we take \(P\) to be the unique element of \(C'[x]\) whose image in \(C'_r[x]\) is \(P_r^{n/r}\).

In this paragraph we prove (2)(a) in case the ring maps \(s^\sharp, t^\sharp : A \to B\) are finite locally free of a fixed rank \(r\). Let \(f \in C^1 \subset A' = A \otimes_C C'\). Choose a flat \(C\)-algebra \(D\) and a surjection \(D \to C'\). Choose a lift \(g \in A \otimes_C D\) of \(f\). Consider the polynomial \[P = \text{Norm}_{s^\sharp \otimes 1}(t^\sharp \otimes 1(x - g))\] in \((A \otimes_C D)[x]\). By Lemma 03BH and part (3) of the current lemma the coefficients of \(P\) are in \(D\) (compare with the proof of Lemma 03BJ). On the other hand, the image of \(P\) in \((A \otimes_C C')[x]\) is \((x - f)^r\) because \(t^\sharp \otimes 1(x - f) = s^\sharp(x - f)\) and \(s^\sharp\) is finite locally free of rank \(r\). This proves what we want with \(P\) as in the statement (2)(a) given by the image of our \(P\) under the map \(D[x] \to C'[x]\).

Lemma

Let \(S\) be a scheme. Let \((U, R, s, t, c)\) be a groupoid scheme over \(S\). Assume \(U = \Spec(A)\) and \(R = \Spec(B)\) are affine and \(s, t : R \to U\) finite locally free. Let \(C \subset A\) be as in (03BF). Then \(U \to M = \Spec(C)\) has the following properties:

1. the map on points \(|U| \to |M|\) is surjective and \(u_0, u_1 \in |U|\) map to the same point if and only if there exists a \(r \in |R|\) with \(t(r) = u_0\) and \(s(r) = u_1\), in a formula \[|M| = |U|/|R|\]

2. for any algebraically closed field \(k\) we have \[M(k) = U(k)/R(k)\]

3. the canonical morphism \[(t, s) : R \longrightarrow U \times_M U\] is surjective.

Proof

Since \(C \to A\) is integral (Lemma 03BJ) and injective we see that \(\Spec(A) \to \Spec(C)\) is surjective, see Algebra, Lemma 00GQ. Thus \(|U| \to |M|\) is surjective.

Let \(k\) be an algebraically closed field and let \(C \to k\) be a ring map. Since surjective morphisms are preserved under base change (Morphisms, Lemma 01S1) we see that \(A \otimes_C k\) is not zero. Now \(k \subset A \otimes_C k\) is a nonzero integral extension. Hence any residue field of \(A \otimes_C k\) is an algebraic extension of \(k\), hence equal to \(k\). Thus we see that \(U(k) \to M(k)\) is surjective.

Let \(a_0, a_1 : A \to k\) be two ring maps. If there exists a ring map \(b : B \to k\) such that \(a_0 = b \circ t^\sharp\) and \(a_1 = b \circ s^\sharp\) then we see that \(a_0|_C = a_1|_C\) by definition. Thus the map \(U(k) \to M(k)\) equalizes the two maps \(R(k) \to U(k)\). Conversely, suppose that \(a_0|_C = a_1|_C\). Let us name this algebra map \(c : C \to k\). Consider the diagram \[\xymatrix{ & & B \ar@{-->}[lld] \\ k & & A \ar@<0.5ex>[ll]^{a_0} \ar@<-0.5ex>[ll]_{a_1} \ar@<1ex>[u] \ar@<-1ex>[u] \\ & & C \ar[u] \ar[llu]^c }\] If we can construct a dotted arrow making the diagram commute, then the proof of part (2) of the lemma is complete. Since \(s : A \to B\) is finite there exist finitely many ring maps \(b_1, \ldots, b_n : B \to k\) such that \(b_i \circ s^\sharp = a_1\). If the dotted arrow does not exist, then we see that none of the \(a'_i = b_i \circ t^\sharp\), \(i = 1, \ldots, n\) is equal to \(a_0\). Hence the maximal ideals \[\mathfrak m'_i = \Ker(a_i' \otimes 1 : A \otimes_C k \to k)\] of \(A \otimes_C k\) are distinct from \(\mathfrak m = \Ker(a_0 \otimes 1 : A \otimes_C k \to k)\). By Algebra, Lemma 00DS we would get an element \(f \in A \otimes_C k\) with \(f \in \mathfrak m\), but \(f \not \in \mathfrak m_i'\) for \(i = 1, \ldots, n\). Consider the norm \[g = \text{Norm}_{s^\sharp \otimes 1}(t^\sharp \otimes 1(f)) \in A \otimes_C k\] By Lemma 03BH this lies in the invariants \(C^1 \subset A \otimes_C k\) of the base change groupoid (base change via the map \(c : C \to k\)). On the one hand, \(a_1(g) \in k^*\) since the value of \(t^\sharp(f)\) at all the points (which correspond to \(b_1, \ldots, b_n\)) lying over \(a_1\) is invertible. On the other hand, since \(f \in \mathfrak m\), we see that \(f\) is not a unit, hence \(t^\sharp(f)\) is not a unit (as \(t^\sharp \otimes 1\) is faithfully flat), hence its norm is not a unit. We conclude that \(C^1\) contains an element which is not nilpotent and not a unit. We will now show that this leads to a contradiction. Namely, apply Lemma 03BK to the map \(c : C \to C' = k\), then we see that the map of \(k\) into the invariants \(C^1\) is injective and moreover, that for any element \(x \in C^1\) there exists an integer \(n > 0\) such that \(x^n \in k\). Hence every element of \(C^1\) is either a unit or nilpotent.

We still have to finish the proof of (1). We already know that \(|U| \to |M|\) is surjective. It is clear that \(|U| \to |M|\) is \(|R|\)-invariant. Finally, suppose \(u_0, u_1 \in U\) maps to the same point \(m \in M\). Then the induced field extensions \(\kappa(u_0)/\kappa(m)\) and \(\kappa(u_1)/\kappa(m)\) are algebraic (as \(A\) is integral over \(C\) as used above). Hence if \(k\) is an algebraic closure of \(\kappa(m)\) then we can find \(\kappa(m)\)-embeddings \(\overline{u}_0 : \kappa(u_0) \to k\) and \(\overline{u}_1 : \kappa(u_1) \to k\). These determine \(k\)-valued points \(\overline{u}_0, \overline{u}_1 \in U(k)\) mapping to the same point of \(M(k)\). By part (2) we see that there exists a point \(\overline{r} \in R(k)\) with \(s(\overline{r}) = \overline{u}_0\) and \(t(\overline{r}) = \overline{u}_1\). The image \(r \in R\) of \(\overline{r}\) is a point with \(s(r) = u_0\) and \(t(r) = u_1\) as desired.

Finally, let \(z \in U \times_M U\) be a point and let \(k\) be an algebraic closure of \(\kappa(z)\). The corresponding geometric point of \(U \times_M U\) is a pair of points \(u_0, u_1 \in U(k)\) with the same image in \(M(k)\). By (2), there is an \(r \in R(k)\) mapping to \((u_0, u_1)\). Consequently \(z\) is in the image of \((t, s)\), which proves (3).

Lemma

Let \(S\) be a scheme. Let \(f : (U', R', s', t', c') \to (U, R, s, t, c)\) be a morphism of groupoid schemes over \(S\).

1. \(U\), \(R\), \(U'\), \(R'\) are affine,

2. \(s, t, s', t'\) are finite locally free,

3. the diagrams \[\xymatrix{ R' \ar[d]_{s'} \ar[r]_f & R \ar[d]^s \\ U' \ar[r]^f & U } \quad \quad \xymatrix{ R' \ar[d]_{t'} \ar[r]_f & R \ar[d]^t \\ U' \ar[r]^f & U } \quad \quad \xymatrix{ G' \ar[d] \ar[r]_f & G \ar[d] \\ U' \ar[r]^f & U }\] are cartesian where \(G\) and \(G'\) are the stabilizer group schemes, and

4. \(f : U' \to U\) is étale.

Then the map \(C \to C'\) from the \(R\)-invariant functions on \(U\) to the \(R'\)-invariant functions on \(U'\) is étale and \(U' = \Spec(C') \times_{\Spec(C)} U\).

Proof

Set \(M = \Spec(C)\) and \(M' = \Spec(C')\). Write \(U = \Spec(A)\), \(U' = \Spec(A')\), \(R = \Spec(B)\), and \(R' = \Spec(B')\). We will use the results of Lemmas 03BJ, 03BK, and 03BL without further mention.

Assume \(C\) is a strictly henselian local ring. Let \(p \in M\) be the closed point and let \(p' \in M'\) map to \(p\). Claim: in this case there is a disjoint union decomposition \((U', R', s', t', c') = (U, R, s, t, c) \amalg (U'', R'', s'', t'', c'')\) over \((U, R, s, t, c)\) such that for the corresponding disjoint union decomposition \(M' = M \amalg M''\) over \(M\) the point \(p'\) corresponds to \(p \in M\).

The claim implies the lemma. Suppose that \(M_1 \to M\) is a flat morphism of affine schemes. Then we can base change everything to \(M_1\) without affecting the hypotheses (1) – (4). From Lemma 03BK we see \(M_1\), resp. \(M_1'\) is the spectrum of the \(R_1\)-invariant functions on \(U_1\), resp. the \(R'_1\)-invariant functions on \(U'_1\). Suppose that \(p' \in M'\) maps to \(p \in M\). Let \(M_1\) be the spectrum of the strict henselization of \(\mathcal{O}_{M, p}\) with closed point \(p_1 \in M_1\). Choose a point \(p'_1 \in M'_1\) mapping to \(p_1\) and \(p'\). From the claim we get \[(U'_1, R'_1, s'_1, t'_1, c'_1) = (U_1, R_1, s_1, t_1, c_1) \amalg (U''_1, R''_1, s''_1, t''_1, c''_1)\] and correspondingly \(M'_1 = M_1 \amalg M''_1\) as a scheme over \(M_1\). Write \(M_1 = \Spec(C_1)\) and write \(C_1 = \colim C_i\) as a filtered colimit of étale \(C\)-algebras. Set \(M_i = \Spec(C_i)\). Then \(M_1 = \lim M_i\) and similarly for the other schemes. By Limits, Lemmas 01Z4 and 081E we can find an \(i\) such that \[(U'_i, R'_i, s'_i, t'_i, c'_i) = (U_i, R_i, s_i, t_i, c_i) \amalg (U''_i, R''_i, s''_i, t''_i, c''_i)\] We conclude that \(M'_i = M_i \amalg M''_i\). In particular \(M' \to M\) becomes étale at a point over \(p'\) after an étale base change. This implies that \(M' \to M\) is étale at \(p'\) (for example by Morphisms, Lemma 0476). We will prove \(U' \cong M' \times_M U\) after we prove the claim.

Proof of the claim. Observe that \(U_p\) and \(U'_{p'}\) have finitely many points. For \(u \in U_p\) we have \(\kappa(u)/\kappa(p)\) is algebraic, hence \(\kappa(u)\) is separably closed. As \(U' \to U\) is étale, we conclude the morphism \(U'_{p'} \to U_p\) induces isomorphisms on residue field extensions. Let \(u' \in U'_{p'}\) with image \(u \in U_p\). By assumption (3) the morphism of scheme theoretic fibres \((s')^{-1}(u') \to s^{-1}(u)\), \((t')^{-1}(u') \to t^{-1}(u)\), and \(G'_{u'} \to G_u\) are isomorphisms. Observing that \(U_p = t(s^{-1}(u))\) (set theoretically) we conclude that the points of \(U'_{p'}\) surject onto the points of \(U_p\). Suppose that \(u'_1\) and \(u'_2\) are points of \(U'_{p'}\) mapping to the same point \(u\) of \(U_p\). Then there exists a point \(r' \in R'_{p'}\) with \(s'(r') = u'_1\) and \(t'(r') = u'_2\). Consider the two towers of fields \[\kappa(r')/\kappa(u'_1)/\kappa(u)/\kappa(p) \quad \kappa(r')/\kappa(u'_2)/\kappa(u)/\kappa(p)\] whose “ends” are the same as the two “ends” of the two towers \[\kappa(r')/\kappa(u'_1)/\kappa(p')/\kappa(p) \quad \kappa(r')/\kappa(u'_2)/\kappa(p')/\kappa(p)\] These two induce the same maps \(\kappa(p') \to \kappa(r')\) as \((U'_{p'}, R'_{p'}, s', t', c')\) is a groupoid over \(p'\). Since \(\kappa(u)/\kappa(p)\) is purely inseparable, we conclude that the two induced maps \(\kappa(u) \to \kappa(r')\) are the same. Therefore \(r'\) maps to a point of the fibre \(G_u\). By assumption (3) we conclude that \(r' \in (G')_{u'_1}\). Namely, we may think of \(G\) as a closed subscheme of \(R\) viewed as a scheme over \(U\) via \(s\) and use that the base change to \(U'\) gives \(G' \subset R'\). In particular we have \(u'_1 = u'_2\). We conclude that \(U'_{p'} \to U_p\) is a bijective map on points inducing isomorphisms on residue fields. It follows that \(U'_{p'}\) is a finite set of closed points (Algebra, Lemma 00GA) and hence \(U'_{p'}\) is closed in \(U'\). Let \(J' \subset A'\) be the radical ideal cutting out \(U'_{p'}\) set theoretically.

Second part proof of the claim. Let \(\mathfrak m \subset C\) be the maximal ideal. Observe that \((A, \mathfrak m A)\) is a henselian pair by More on Algebra, Lemma 09XK. Let \(J = \sqrt{\mathfrak m A}\). Then \((A, J)\) is a henselian pair (More on Algebra, Lemma 09XJ) and the étale ring map \(A \to A'\) induces an isomorphism \(A/J \to A'/J'\) by our deliberations above. We conclude that \(A' = A \times A''\) by More on Algebra, Lemma 09XI. Consider the corresponding disjoint union decomposition \(U' = U \amalg U''\). The open \((s')^{-1}(U)\) is the set of points of \(R'\) specializing to a point of \(R'_{p'}\). Similarly for \((t')^{-1}(U)\). Similarly we have \((s')^{-1}(U'') = (t')^{-1}(U'')\) as this is the set of points which do not specialize to \(R'_{p'}\). Hence we obtain a disjoint union decomposition \[(U', R', s', t', c') = (U, R, s, t, c) \amalg (U'', R'', s'', t'', c'')\] This immediately gives \(M' = M \amalg M''\) and the proof of the claim is complete.

We still have to prove that the canonical map \(U' \to M' \times_M U\) is an isomorphism. It is an étale morphism (Morphisms, Lemma 02GW). On the other hand, by base changing to strictly henselian local rings (as in the third paragraph of the proof) and using the bijectivity \(U'_{p'} \to U_p\) established in the course of the proof of the claim, we see that \(U' \to M' \times_M U\) is universally bijective (some details omitted). However, a universally bijective étale morphism is an isomorphism (Descent, Lemma 02LC) and the proof is complete.

Lemma

Let \(S\) be a scheme. Let \((U, R, s, t, c)\) be a groupoid scheme over \(S\). Assume

1. \(U = \Spec(A)\), and \(R = \Spec(B)\) are affine, and

2. there exist elements \(x_i \in A\), \(i \in I\) such that \(B = \bigoplus_{i \in I} s^\sharp(A)t^\sharp(x_i)\).

Then \(A = \bigoplus_{i\in I} Cx_i\), and \(B \cong A \otimes_C A\) where \(C \subset A\) is the \(R\)-invariant functions on \(U\) as in (03BF).

Proof

During this proof we will write \(s, t : A \to B\) instead of \(s^\sharp, t^\sharp\), and similarly \(c : B \to B \otimes_{s, A, t} B\). We write \(p_0 : B \to B \otimes_{s, A, t} B\), \(b \mapsto b \otimes 1\) and \(p_1 : B \to B \otimes_{s, A, t} B\), \(b \mapsto 1 \otimes b\). By Lemma 03C6 and the definition of \(C\) we have the following commutative diagram \[\xymatrix{ B \otimes_{s, A, t} B & B \ar@<-1ex>[l]_-c \ar@<1ex>[l]^-{p_0} & A \ar[l]^t \\ B \ar[u]^{p_1} & A \ar@<-1ex>[l]_s \ar@<1ex>[l]^t \ar[u]_s & C \ar[u] \ar[l] }\] Moreover the two left squares are cocartesian in the category of rings, and the top row is isomorphic to the diagram \[\xymatrix{ B \otimes_{t, A, t} B & B \ar@<-1ex>[l]_-{p_1} \ar@<1ex>[l]^-{p_0} & A \ar[l]^t }\] which is an equalizer diagram according to Descent, Lemma 023M because condition (2) implies in particular that \(s\) (and hence also the isomorphic arrow \(t\)) is faithfully flat. The lower row is an equalizer diagram by definition of \(C\). We can use the \(x_i\) and get a commutative diagram \[\xymatrix{ B \otimes_{s, A, t} B & B \ar@<-1ex>[l]_-c \ar@<1ex>[l]^-{p_0} & A \ar[l]^t \\ \bigoplus_{i \in I} B x_i \ar[u]^{p_1} & \bigoplus_{i \in I} A x_i \ar@<-1ex>[l]_s \ar@<1ex>[l]^t \ar[u]_s & \bigoplus_{i \in I} C x_i \ar[u] \ar[l] }\] where in the right vertical arrow we map \(x_i\) to \(x_i\), in the middle vertical arrow we map \(x_i\) to \(t(x_i)\) and in the left vertical arrow we map \(x_i\) to \(c(t(x_i)) = t(x_i) \otimes 1 = p_0(t(x_i))\) (equality by the commutativity of the top part of the diagram in Lemma 02YE). Then the diagram commutes. Moreover the middle vertical arrow is an isomorphism by assumption. Since the left two squares are cocartesian we conclude that also the left vertical arrow is an isomorphism. On the other hand, the horizontal rows are exact (i.e., they are equalizers). Hence we conclude that also the right vertical arrow is an isomorphism.

Proposition

Let \(S\) be a scheme. Let \((U, R, s, t, c)\) be a groupoid scheme over \(S\). Assume

1. \(U = \Spec(A)\), and \(R = \Spec(B)\) are affine,

2. \(s, t : R \to U\) finite locally free, and

3. \(j = (t, s)\) is an equivalence relation.

In this case, let \(C \subset A\) be as in (03BF). Then \(U \to M = \Spec(C)\) is finite locally free and \(R = U \times_M U\). Moreover, \(M\) represents the quotient sheaf \(U/R\) in the fppf topology (see Definition 02VG).

Proof

During this proof we use the notation \(s, t : A \to B\) instead of the notation \(s^\sharp, t^\sharp\). By Lemma 03C5 it suffices to show that \(C \to A\) is finite locally free and that the map \[t \otimes s : A \otimes_C A \longrightarrow B\] is an isomorphism. First, note that \(j\) is a monomorphism, and also finite (since already \(s\) and \(t\) are finite). Hence we see that \(j\) is a closed immersion by Morphisms, Lemma 03BB. Hence \(A \otimes_C A \to B\) is surjective.

We will perform base change by flat ring maps \(C \to C'\) as in Lemma 03BK, and we will use that formation of invariants commutes with flat base change, see part (3) of the lemma cited. We will show below that for every prime \(\mathfrak p \subset C\), there exists a local flat ring map \(C_{\mathfrak p} \to C_{\mathfrak p}'\) such that the result holds after a base change to \(C_{\mathfrak p}'\). This implies immediately that \(A \otimes_C A \to B\) is injective (use Algebra, Lemma 00HN). It also implies that \(C \to A\) is flat, by combining Algebra, Lemmas 00HR, 00HT, and 00HJ. Then since \(U \to \Spec(C)\) is surjective also (Lemma 03BL) we conclude that \(C \to A\) is faithfully flat. Then the isomorphism \(B \cong A \otimes_C A\) implies that \(A\) is a finitely presented \(C\)-module, see Algebra, Lemma 03C4. Hence \(A\) is finite locally free over \(C\), see Algebra, Lemma 00NX.

By Lemma 03BI we know that \(A\) is a finite product of rings \(A_r\) and \(B\) is a finite product of rings \(B_r\) such that the groupoid scheme decomposes accordingly (see the proof of Lemma 03BJ). Then also \(C\) is a product of rings \(C_r\) and correspondingly \(C'\) decomposes as a product. Hence we may and do assume that the ring maps \(s, t : A \to B\) are finite locally free of a fixed rank \(r\).

The local ring maps \(C_{\mathfrak p} \to C_{\mathfrak p}'\) we are going to use are any local flat ring maps such that the residue field of \(C_{\mathfrak p}'\) is infinite. By Algebra, Lemma 03C3 such local ring maps exist.

Assume \(C\) is a local ring with maximal ideal \(\mathfrak m\) and infinite residue field, and assume that \(s, t : A \to B\) is finite locally free of constant rank \(r > 0\). Since \(C \subset A\) is integral (Lemma 03BJ) all primes lying over \(\mathfrak m\) are maximal, and all maximal ideals of \(A\) lie over \(\mathfrak m\). Similarly for \(C \subset B\). Pick a maximal ideal \(\mathfrak m'\) of \(A\) lying over \(\mathfrak m\) (exists by Lemma 03BL). Since \(t : A \to B\) is finite locally free there exist at most finitely many maximal ideals of \(B\) lying over \(\mathfrak m'\). Hence we conclude (by Lemma 03BL again) that \(A\) has finitely many maximal ideals, i.e., \(A\) is semi-local. This in turn implies that \(B\) is semi-local as well. OK, and now, because \(t \otimes s : A \otimes_C A \to B\) is surjective, we can apply Algebra, Lemma 03C1 to the ring map \(C \to A\), the \(A\)-module \(M = B\) (seen as an \(A\)-module via \(t\)) and the \(C\)-submodule \(s(A) \subset B\). This lemma implies that there exist \(x_1, \ldots, x_r \in A\) such that \(M\) is free over \(A\) on the basis \(s(x_1), \ldots, s(x_r)\). Hence we conclude that \(C \to A\) is finite free and \(B \cong A \otimes_C A\) by applying Lemma 03C8.

Finite flat groupoids

In this section we prove a lemma that will help to show that the quotient of a scheme by a finite flat equivalence relation is a scheme, provided that each equivalence class is contained in an affine. See Properties of Spaces, Proposition 07S6.

Lemma

Let \(S\) be a scheme. Let \((U, R, s, t, c)\) be a groupoid scheme over \(S\). Assume \(s\), \(t\) are finite locally free. Let \(u \in U\) be a point such that \(t(s^{-1}(\{u\}))\) is contained in an affine open of \(U\). Then there exists an \(R\)-invariant affine open neighbourhood of \(u\) in \(U\).

Proof

Since \(s\) is finite locally free it has finite fibres. Hence \(t(s^{-1}(\{u\})) = \{u_1, \ldots, u_n\}\) is a finite set. Note that \(u \in \{u_1, \ldots, u_n\}\). Let \(W \subset U\) be an affine open containing \(\{u_1, \ldots, u_n\}\), in particular \(u \in W\). Consider \(Z = R \setminus (s^{-1}(W) \cap t^{-1}(W))\). This is a closed subset of \(R\). The image \(t(Z)\) is a closed subset of \(U\) which can be loosely described as the set of points of \(U\) which are \(R\)-equivalent to a point of \(U \setminus W\). Hence \(W' = U \setminus t(Z)\) is an \(R\)-invariant, open subscheme of \(U\) contained in \(W\), and \(\{u_1, \ldots, u_n\} \subset W'\). Picture \[\{u_1, \ldots, u_n\} \subset W' \subset W \subset U.\] Let \(f \in \Gamma(W, \mathcal{O}_W)\) be an element such that \(\{u_1, \ldots, u_n\} \subset D(f) \subset W'\). Such an \(f\) exists by Algebra, Lemma 00DS. By our choice of \(W'\) we have \(s^{-1}(W') \subset t^{-1}(W)\), and hence we get a diagram \[\xymatrix{ s^{-1}(W') \ar[d]_s \ar[r]_-t & W \\ W' }\] The vertical arrow is finite locally free by assumption. Set \[g = \text{Norm}_s(t^\sharp f) \in \Gamma(W', \mathcal{O}_{W'})\] By construction \(g\) is a function on \(W'\) which is nonzero in \(u\), as \(t^\sharp(f)\) is nonzero in each of the points of \(R\) lying over \(u\), since \(f\) is nonzero in \(u_1, \ldots, u_n\). Similarly, \(D(g) \subset W'\) is equal to the set of points \(w\) such that \(f\) is not zero in any of the points equivalent to \(w\). This means that \(D(g)\) is an \(R\)-invariant affine open of \(W'\). The final picture is \[\{u_1, \ldots, u_n\} \subset D(g) \subset D(f) \subset W' \subset W \subset U\] and hence we win.

Descending quasi-projective schemes

We can use Lemma 03JE to show that a certain type of descent datum is effective.

Lemma

Let \(X \to Y\) be a surjective finite locally free morphism. Let \(d\) be a positive integer. Assume that for every geometric point \(\bar{y} : \mathrm{Spec}(k) \to Y\) the fiber \(X_{\bar{y}}\) has at most \(d\) points. Let \(V\) be a scheme over \(X\) such that for all \((y, v_1, \ldots, v_d)\) where \(y \in Y\) and \(v_1, \ldots, v_d \in V_y\) there exists an affine open \(U \subset V\) with \(v_1, \ldots, v_d \in U\). Then any descent datum on \(V/X/Y\) is effective.

Proof

Let \(\varphi\) be a descent datum as in Descent, Definition 023V. Recall that the functor from schemes over \(Y\) to descent data relative to \(\{X \to Y\}\) is fully faithful, see Descent, Lemma 02W0. Thus using Constructions, Lemma 01LH it suffices to prove the lemma in the case that \(Y\) is affine. Some details omitted (this argument can be avoided if \(Y\) is separated or has affine diagonal, because then every morphism from an affine scheme to \(X\) is affine).

Assume \(Y\) is affine. If \(V\) is also affine, then we have effectivity by Descent, Lemma 0245. Hence by Descent, Lemma 0AP4 it suffices to prove that every point \(v\) of \(V\) has a \(\varphi\)-invariant affine open neighbourhood. Consider the groupoid \((X, X \times_Y X, \text{pr}_1, \text{pr}_0, \text{pr}_{02})\). By Lemma 0APF the descent datum \(\varphi\) determines and is determined by a cartesian morphism of groupoid schemes \[(V, R, s, t, c) \longrightarrow (X, X \times_Y X, \text{pr}_1, \text{pr}_0, \text{pr}_{02})\] over \(\Spec(\mathbf{Z})\). Since \(X \to Y\) is finite locally free, we see that \(\text{pr}_i : X \times_Y X \to X\) and hence \(s\) and \(t\) are finite locally free, and their geometric fibers have at most \(d\) points. In particular the \(R\)-orbit \(t(s^{-1}(\{v\}))\) of our point \(v \in V\) has at most \(d\) points. Using the equivalence of categories of Lemma 0APF once more we see that \(\varphi\)-invariant opens of \(V\) are the same thing as \(R\)-invariant opens of \(V\). Our assumption shows there exists an affine open of \(V\) containing the orbit \(t(s^{-1}(\{v\}))\) as all the points in this orbit map to the same point of \(Y\). Thus Lemma 03JE provides an \(R\)-invariant affine open containing \(v\).

Lemma

Let \(X \to Y\) be a surjective finite locally free morphism. Let \(V\) be a scheme over \(X\) such that for every positive integer \(d\) and all \((y, v_1, \ldots, v_d)\) where \(y \in Y\) and \(v_1, \ldots, v_d \in V_y\) there exists an affine open \(U \subset V\) with \(v_1, \ldots, v_d \in U\). Then any descent datum on \(V/X/Y\) is effective.

Proof

As in the proof of Lemma 0H8K, we may assume that \(Y\) is affine, in particular quasi-compact. Thus there exists an integer \(d\) such that all geometric fibers of \(X \to Y\) have at most \(d\) points, see the discussion in Morphisms, Section 03J3. Our assumption on \(V\) trivially implies that we can apply Lemma 0H8K, which completes the proof.

Lemma

Let \(X \to Y\) be a surjective finite locally free morphism. Let \(V\) be a scheme over \(X\) such that one of the following holds

1. \(V \to X\) is projective,

2. \(V \to X\) is quasi-projective,

3. there exists an ample invertible sheaf on \(V\),

4. there exists an \(X\)-ample invertible sheaf on \(V\),

5. there exists an \(X\)-very ample invertible sheaf on \(V\).

Then any descent datum on \(V/X/Y\) is effective.

Proof

We check the condition in Lemma 0CCI. Let \(y \in Y\) and \(v_1, \ldots, v_d \in V\) points over \(y\). Case (1) is a special case of (2), see Morphisms, Lemma 07RL. Case (2) is a special case of (4), see Morphisms, Definition 01VW. If there exists an ample invertible sheaf on \(V\), then there exists an affine open containing \(v_1, \ldots, v_d\) by Properties, Lemma 01ZY. Thus (3) is true. In cases (4) and (5) it is harmless to replace \(Y\) by an affine open neighbourhood of \(y\). Then \(X\) is affine too. In case (4) we see that \(V\) has an ample invertible sheaf by Morphisms, Definition 01VH and the result follows from case (3). In case (5) we can replace \(V\) by a quasi-compact open containing \(v_1, \ldots, v_d\) and we reduce to case (4) by Morphisms, Lemma 01VN.

Lemma

Let \(S' \to S\) be a surjective finite locally free morphism of constant degree. Let \(X \to S\) be a morphism. If \(X' = S' \times_S X \to S'\) is projective, then \(X \to S\) is projective.

Proof

The morphism \(X \to S\) is proper by Descent, Lemma 02L1. Choose an \(S'\)-ample invertible sheaf \(\mathcal{L}'\) on \(X'\); this is possible by Morphisms, Lemma 07RL. The projection \(\pi : X' \to X\) is surjective finite locally free. The finite locally free norm and the norm of an invertible sheaf (Divisors, Lemmas 0BD2 and 0BCY) give an invertible sheaf \[\mathcal{N} = \mathop{\rm Norm}\nolimits_\pi(\mathcal{L}')\] on \(X\).

We claim that \(\mathcal{N}\) is \(S\)-ample. This may be checked over affine opens \(U \subset S\). Then \(S'_U\) is affine and \(\mathcal{L}'|_{X'_U}\) is ample. The norm construction commutes with restriction to open subschemes. Hence \(\mathcal{N}|_{X_U}\) is ample by Divisors, Lemma 0BD0. Thus \(X \to S\) is quasi-projective. Since it is also proper, it is projective by Morphisms, Lemma 0BCL.

Lemma

Let \(X \to Y\) be a surjective, quasi-compact, flat, and radicial morphism. Let \(V\) be a scheme over \(X\). Then any descent datum on \(V/X/Y\) is effective.

Proof

Let \(\varphi\) be a descent datum as in Descent, Definition 023V. Since \(X \to Y\) is radicial, the diagonal \(X \to X \times_Y X\) is surjective by Morphisms, Lemma 01S4.

Let \(v \in V\) and choose an affine open \(A \subset X\) containing the image of \(v\). The inverse images of \(A\) by the two projections \(X \times_Y X \to X\) agree: their inverse images by the surjective diagonal agree, and open subsets are determined by their underlying sets. By Descent, Lemma 03N0, there is an open \(U \subset Y\) such that \(A = X \times_Y U\). After replacing everything by its restriction over \(U\), we may assume that \(X\) is affine.

Choose an affine open neighbourhood \(W \subset V\) of \(v\). The two opens \[W \times_Y X \quad\text{and}\quad \varphi^{-1}(X \times_Y W)\] of \(V \times_Y X\) have the same inverse image by the map \(V \to V \times_Y X\) induced by the diagonal. This map is surjective on underlying topological spaces, and the pullback of \(\varphi\) by the diagonal is the identity. Hence the two opens agree. Thus \(W\) is \(\varphi\)-invariant. The morphism \(W \to X\) is affine, so the restricted descent datum is effective by Descent, Lemma 0245. We have found such a neighbourhood for every \(v\), and the result follows from Descent, Lemma 0AP4.

Lemma

Let \(X \to Y\) be a surjective finite locally free morphism which is radicial. Let \(V\) be a scheme over \(X\). Then any descent datum on \(V/X/Y\) is effective.

Proof

Apply Lemma groupoids-lemma-descend-along-fpqc-radicial.