1.1 Quotient stacks for the fpqc-topology

Let S be an algebraic space. We denote by $\mathcal {A}\mathrm {ff}/S$ the category of affine schemes endowed with a morphism to S. We will deal systematically with sheaves (of sets or of groupoids) for the fpqc-topology on $\mathcal {A}\mathrm {ff}/S$ (or equivalently on the category of all algebraic spaces over S) since we will deal with general flat group schemes (not necessarily of finite presentation).

The category $\mathcal {A}\mathrm {ff}/S$ is not small. Moreover, for any given non-empty affine scheme T over S, the category of fpqc-covering sieves of T does not have a small cofinal subcategory. Hence categories of sheaves in the naive sense fail to be Grothendieck topoi and presheaves do not admit a sheafification in general. To overcome this difficulty, we proceed by first limiting cardinalities and then forming the colimit, following [Reference Hesselholt and Pstrągowski23, Section 2.1 and App. A], where this is explained in the more complicated case of the fpqc-topology on the category of Dirac rings, see also [Reference Gregoric18, §2].

We will consider presheaves and sheaves with values in arbitrary anima (= spaces = $\infty $ -groupoids) even though in this paper we need only (pre)sheaves with values in 1-truncated anima (= groupoids). We denote the $\infty $ -category of anima by ${\mathcal {A}\mathrm {ni}}$ .

Let $\kappa $ be an uncountable strong limit cardinal and let R be a $\kappa $ -small ring. Let $\mathcal {A}\mathrm {ff}_{R,\kappa }$ be the opposite category of $\kappa $ -small R-algebras, which we always endow with the fpqc-topology. Let $\mathcal {S}\mathrm {h}(\mathcal {A}\mathrm {ff}_{R,\kappa })$ be the $\infty $ -category of fpqc-sheaves on $\mathcal {A}\mathrm {ff}_{R,\kappa }$ with values in ${\mathcal {A}\mathrm {ni}}$ , see, for instance, [Reference Hesselholt and Pstrągowski23, Proposition 2.5].

If $\kappa '> \kappa $ are strongly inaccessible cardinals, then left Kan extension along $\mathcal {A}\mathrm {ff}_{R,\kappa } \hookrightarrow \mathcal {A}\mathrm {ff}_{R,\kappa '}$ induces fully faithful functors $\mathcal {S}\mathrm {h}(\mathcal {A}\mathrm {ff}_{R,\kappa }) \hookrightarrow \mathcal {S}\mathrm {h}(\mathcal {A}\mathrm {ff}_{R,\kappa '})$ that commute with all colimits and with $\kappa $ -small limits [Reference Hesselholt and Pstrągowski23, Theorem 2.15, Lemma A.8]. For any ring R we set

$$\begin{align*}\mathcal{S}\mathrm{h}(\mathcal{A}\mathrm{ff}_{R}) := \mathop{\mathrm{colim}}_{\kappa}\mathcal{S}\mathrm{h}(\mathcal{A}\mathrm{ff}_{R,\kappa}), \end{align*}$$

where the colimit is indexed by the large partially ordered set of uncountable strongly inaccessible cardinals $\kappa $ such that R is $\kappa $ -small.

Then $\mathcal {S}\mathrm {h}(\mathcal {A}\mathrm {ff}_{R})$ is equivalent to the $\infty $ -category of fpqc-sheaves on $\mathcal {A}\mathrm {ff}_R$ which are accessible as presheaves [Reference Hesselholt and Pstrągowski23, Remark 2.16] or, equivalently by [Reference Hesselholt and Pstrągowski23, Proposition A.2], which, as presheaves, are a colimit of a small diagram of representable functors. We denote the $\infty $ -category of these presheaves by $\mathcal {P}\mathrm {Sh}(\mathcal {A}\mathrm {ff}_{R})$ .

By [Reference Hesselholt and Pstrągowski23, Corollary 2.17] we obtain:

Now let G be an fpqc-sheaf of groups on the category of S-schemes. For instance, G could be any group algebraic space over S by a theorem of Gabber [30, 0APL]. Let X be an fpqc-sheaf over S with a left G-action $a\colon G \times _S X \to X$ . We define $[G\backslash X]$ as the fpqc-stack in groupoids on ${\mathcal C}$ attached to the prestack

$$\begin{align*}[G\,{}_p\!\backslash X]\colon T \mapsto [G(T)\backslash X(T)] \end{align*}$$

where $[G(T)\backslash X(T)]$ is the groupoid whose objects are the elements of $X(T)$ and whose maps $x \to x'$ are the elements $g \in G(T)$ with $gx = x'$ . Composition is defined via the group law in $G(T)$ . The stackification (i.e., sheafification of presheaves with values in 1-truncated anima) of $[G\,{}_p\!\backslash X]$ always exists by Proposition 1.1 since we can write $[G\,{}_p\!\backslash X]$ as the (homotopy) colimit of the bar construction

(1.1.1)

As usual, the quotient stack can also be described via G-bundles, which also shows its existence. Here we mean by a G-bundle an fpqc-sheaf P on $\mathcal {A}\mathrm {ff}/S$ endowed with a G-action that is fpqc-locally isomorphic to G endowed with the G-action by left multiplication. Then $[G\backslash X]$ is identified with the stack that sends an object T of $\mathcal {A}\mathrm {ff}/S$ to the groupoid of pairs $(P,\varphi )$ , where P is a G-bundle over T and $\varphi \colon P \to X \times _S T$ is a G-equivariant map of sheaves over T. A morphism $(P,\varphi ) \to (P',\varphi ')$ in $[G\backslash X](T)$ is a G-equivariant isomorphism $u\colon P \overset {\sim }{\to } P'$ of sheaves such that $\varphi = \varphi ' \circ u$ .

In particular, we have the classifying stack $\mathbb {B}G := [G\backslash S]$ , where G acts trivially on S, which classifies G-bundles for the fpqc-topology.

In the sequel we will also use the following notion. Recall from [30, 0AP5] that a morphism $f\colon X \to Y$ of schemes is called ind-quasi-affine if for every open affine subscheme V of Y every open quasi-compact subscheme of $f^{-1}(V)$ is quasi-affine. The property to be ind-quasi-affine is stable under composition and base change and stable under fpqc-descent. Every immersion and every quasi-affine morphism is ind-quasi-affine. By the usual argument (e.g., [Reference Görtz and Wedhorn16, Remark 9.11]) this implies that if a composition $g \circ f$ of scheme morphisms is ind-quasi-affine, then f is ind-quasi-affine.

We call a morphism ${\mathcal X} \to {\mathcal Y}$ of prestacks ind-quasi-affine if it is representable by schemes and for every morphism of schemes $T \to {\mathcal Y}$ , the base change ${\mathcal X} \times _{{\mathcal Y}} T \to T$ is ind-quasi-affine.

Proof. Consider the following diagram

(1.1.2)

where the square is 2-cartesian. Here $\sigma $ is the map given by $(g,x) \mapsto (gx,x)$ on points. Since $G \to S$ is ind-quasi-affine, so is $p_2 \circ \sigma $ . Therefore $\sigma $ is ind-quasi-affine by cancellation, which implies that $\Delta _{[G\backslash X]}$ is ind-quasi-affine by fpqc-descent.

For ‘affine’ the argument is the same where we use that if a composition $g \circ f$ is affine and g has affine diagonal, then f is affine.

1.2 G-bundles over stacks

We continue to assume that S is an algebraic space.

Definition and Remark 1.5. Let ${\mathcal X}$ be a prestack over S and let G be a group algebraic space over S. We define the groupoid $\operatorname {\mathrm {Bun}}_G({\mathcal X})$ of G-bundles over ${\mathcal X}$ as the groupoid of S-morphisms ${\mathcal X} \to \mathbb {B}G$ , i.e., we set

$$\begin{align*}\operatorname{\mathrm{Bun}}_G({\mathcal X}) := \operatorname{\mathrm{Hom}}_S({\mathcal X},\mathbb{B}G). \end{align*}$$

We denote by $H^1({\mathcal X},G)$ the set of isomorphism classes of G-bundles on ${\mathcal X}$ .

The canonical map $S \to \mathbb {B}G$ corresponds to the trivial G-bundle on S. More generally, the morphism ${\mathcal X} \to S \to \mathbb {B}G$ is called the trivial G-bundle on ${\mathcal X}$ . By distinguishing it, $H^1({\mathcal X},G)$ is made into a pointed set.

We also denote by $\underline {\mathcal {B}\mathrm {un}}_G({\mathcal X})$ the prestack over S given by

$$\begin{align*}\underline{\mathcal{B}\mathrm{un}}_G({\mathcal X})(T) = \operatorname{\mathrm{Hom}}_S({\mathcal X} \times_S T,\mathbb{B}G). \end{align*}$$

If $u\colon {\mathcal Y} \to {\mathcal X}$ is a morphism of prestacks over S and ${\mathscr E}$ is a G-bundle on ${\mathcal X}$ given by a map ${\mathcal X} \to \mathbb {B}G$ , then its pullback by u is the G-bundle on ${\mathcal Y}$ given by the composition ${\mathcal Y} \mathrel {\smash {\overset {u}{\longrightarrow }}} {\mathcal X} \to \mathbb {B}G$ . It is denoted by $u^*{\mathscr E}$ .

Suppose that $G \to S$ is ind-quasi-affine and an fpqc-covering. In many cases, it is possible to describe G-bundles via their representations. We set

$$\begin{align*}\operatorname{\mathrm{Rep}}_S(G) := \operatorname{\mathrm{Rep}}(G) := \mathrm{Vec}(\mathbb{B}G), \end{align*}$$

which we consider as an exact symmetric monoidal category. Since $S \to \mathbb {B}G$ is an fpqc-covering and fpqc-descent is effective for vector bundles, this is the category of vector bundles ${\mathscr E}$ over S endowed with a homomorphism of S-group algebraic spaces $G \to \underline {\operatorname {\mathrm {Aut}}}_{{\mathscr O}_S}({\mathscr E})$ , where $\underline {\operatorname {\mathrm {Aut}}}_{{\mathscr O}_S}({\mathscr E})$ is the reductive group scheme over S whose T-valued points for an affine S-scheme $f\colon T \to S$ are given by the group of ${\mathscr O}_T$ -linear automorphisms of $f^*{\mathscr E}$ .

Then we have the following application of a result of Tonini [Reference Tonini32, Theorem A].

Proposition 1.6. Let S be a quasi-compact scheme and let G be a group scheme over S such that $G \to S$ is flat and quasi-affine. Suppose that $\mathbb {B}G$ has the resolution property, i.e., every quasi-coherent module over $\mathbb {B}G$ is a quotient of a direct sum of vector bundles. Let ${\mathcal X}$ be any prestack over S. Then the canonical morphism of groupoids

(1.2.1) $$ \begin{align} \begin{aligned} \operatorname{\mathrm{Bun}}_G({\mathcal X}) = \operatorname{\mathrm{Hom}}_S({\mathcal X},\mathbb{B}G) &\longrightarrow \left\{\begin{matrix}\text{exact strong monoidal } {\mathscr O}_S\text{-}\text{linear}\\ \text{functors } \operatorname{\mathrm{Rep}}(G) \to \text{Vec}({\mathcal X})\end{matrix}\right\},\\ f &\longmapsto f^* \end{aligned} \end{align} $$

is an equivalence.

If ${\mathcal Y}$ is a noetherian algebraic stack, every quasi-coherent ${\mathscr O}_{{\mathcal Y}}$ -module is the colimit of its submodules of finite type by [Reference Laumon and Moret-Bailly25, Proposition 15.4]). Hence ${\mathcal Y}$ has the resolution property if every coherent ${\mathscr O}_{{\mathcal Y}}$ -module is quotient of a vector bundle over ${\mathcal Y}$ . Therefore [Reference Thomason31, §2] yields the following examples of classifying stacks with resolution properties.

For further examples, see, for instance, [Reference Gross19].

1.3 G-bundles on classifying stacks

For ${\mathcal X} = \mathbb {B}G$ , this implies immediately the following more precise assertions.

From the description (1.3.2) we obtain the following result using a result of Cotner.

2.1 Full faithfulness of restriction

Proposition 2.1. Let ${\mathcal X}$ be a stack over S whose diagonal is representable by algebraic spaces and that admits an fpqc-covering by an algebraic space. Let $f\colon {\mathcal U} \to {\mathcal X}$ be a qcqs morphism of stacks that is representable by schemes. Suppose that $f_*{\mathscr O}_{{\mathcal U}} = {\mathscr O}_{{\mathcal X}}$ .

Suppose we are given a 2-commutative diagram of solid arrows of stacks

with $g\colon {\mathcal Y} \to {\mathcal Z}$ a morphism representable by schemes. If g is quasi-affine (resp. affine), then there exists at most one (resp. a unique) morphism ${\mathcal X} \to {\mathcal Y}$ making the above diagram commutative.

The condition that the diagonal of ${\mathcal X}$ is representable by algebraic spaces ensures that every morphism $X \to {\mathcal X}$ , where X is an algebraic space, is representable by algebraic spaces. In particular, we know what it means for such a morphism to be an fpqc-covering.

Proof. Replacing ${\mathcal Y}$ by ${\mathcal Y} \times _{{\mathcal Z}} {\mathcal X}$ and g by its base change ${\mathcal Y} \times _{{\mathcal Z}} {\mathcal X} \to {\mathcal X}$ , it suffices to show that for every representable quasi-affine (resp. affine) morphism $g\colon {\mathcal Y} \to {\mathcal X}$ , every section of g over ${\mathcal U}$ has at most one (resp. a unique) extension to ${\mathcal X}$ . We can work fpqc-locally on ${\mathcal X}$ and hence can assume that ${\mathcal X} = \operatorname {\mathrm {Spec}} A$ is an affine scheme. Then $f\colon {\mathcal U} \to {\mathcal X}$ is a qcqs map of schemes. Since $f_*{\mathscr O}_{{\mathcal U}} = {\mathscr O}_{{\mathcal X}}$ , we have in particular $\Gamma ({\mathcal X},{\mathscr O}_{{\mathcal X}}) = \Gamma ({\mathcal U},{\mathscr O}_{{\mathcal U}})$ .

Suppose that g is affine. Then ${\mathcal Y} = \operatorname {\mathrm {Spec}} B$ is an affine scheme and we find

$$ \begin{align*} \operatorname{\mathrm{Hom}}_{{\mathcal X}}({\mathcal U},{\mathcal Y}) &= \operatorname{\mathrm{Hom}}_A(B,\Gamma({\mathcal U},{\mathscr O}_{{\mathcal U}})) = \operatorname{\mathrm{Hom}}_A(B,A) \\ &= \operatorname{\mathrm{Hom}}_{{\mathcal X}}({\mathcal X},{\mathcal Y}). \end{align*} $$

Now suppose g is quasi-affine and set $B := \Gamma ({\mathcal Y},{\mathscr O}_{{\mathcal Y}})$ . The canonical map ${\mathcal Y} \to \operatorname {\mathrm {Spec}} B$ is an open immersion and we have a commutative diagram

where the vertical maps are injective and the lower horizontal map is bijective. Therefore the upper horizontal map is injective.

Let us make some remarks about the hypotheses of the above proposition.

Corollary 2.3. Let ${\mathcal X}$ be a stack over S whose diagonal is representable by algebraic spaces and that admits an fpqc-covering by an algebraic space, and let $f\colon {\mathcal U} \to {\mathcal X}$ be a morphism of stacks that is representable by schemes and qcqs. Suppose that $f_*{\mathscr O}_{{\mathcal U}} = {\mathscr O}_{{\mathcal X}}$ . Let ${\mathcal Y} \to {\mathcal Z}$ be a morphism of stacks with representable affine diagonal.

Then composition with f yields a fully faithful map of groupoids

$$\begin{align*}\operatorname{\mathrm{Hom}}_{{\mathcal Z}}({\mathcal X},{\mathcal Y}) \longrightarrow \operatorname{\mathrm{Hom}}_{{\mathcal Z}}({\mathcal U},{\mathcal Y}). \end{align*}$$

Proposition 2.4. Let G be a group algebraic space over S such that $G \to S$ is flat and affine. Let ${\mathcal X}$ be a stack over S whose diagonal is representable by algebraic spaces and that admits an fpqc-covering by an algebraic space, and let $f\colon {\mathcal U} \to {\mathcal X}$ be a qcqs morphism of stacks over S that is representable by schemes. Suppose that ${\mathscr O}_{{\mathcal X}} \to f_*{\mathscr O}_{{\mathcal U}}$ is an isomorphism.

Then the functors

(2.1.1) $$ \begin{align}\kern6pt f^*\colon \mathrm{Vec}({\mathcal X}) &\to \mathrm{Vec}({\mathcal U}), \end{align} $$

(2.1.2) $$ \begin{align} f^*\colon \operatorname{\mathrm{Bun}}_G({\mathcal X}) &\to \operatorname{\mathrm{Bun}}_G({\mathcal U}) \end{align} $$

are fully faithful.

2.2 Extension of G-bundles

Let us give criteria for when a vector bundle (resp. a G-bundle) is in the essential image of (2.1.1) (resp. (2.1.2)).

Proof. Assertion (1) follows formally from the fact that $f^*$ is fully faithful and that fpqc-descent is effective for vector bundles and for G-bundles. Assertion (2) can be shown as in [Reference Görtz and Wedhorn17, Lemma 24.67], where the argument is given for a morphism of schemes.

Let us show (3). By (1) we may assume that $\iota $ exists globally. Consider the map ${\bar \iota \colon \mathbb {B}G \to \mathbb {B}H}$ induced by $\iota $ . Since $\iota $ is a monomorphism and $G\backslash H$ is affine, $\bar \iota $ is affine. The condition translates into the existence of a 2-commutative diagram

Now Proposition 2.1 yields a map ${\mathcal X} \to \mathbb {B}G$ commuting with the diagram, i.e., a G-bundle on ${\mathcal X}$ extending P.

In practice, we will usually apply Criterion (3) of Proposition 2.5 with $H = \operatorname {\mathrm {GL}}_m$ to reduce the essential surjectivity to a problem of vector bundles. Thus to apply this criterion we want to give examples of G satisfying the following properties.

1. (1) There exists an embedding $G \to \operatorname {\mathrm {GL}}_m$ fpqc-locally.

2. (2) Given an embedding $G \to \operatorname {\mathrm {GL}}_m$ , the quotient $G\backslash \operatorname {\mathrm {GL}}_m$ is affine.

With respect to (1) there are the following results.

To deal with the question whether $G\backslash \operatorname {\mathrm {GL}}_m$ is affine, recall the following notion, first studied by Franjou and van der Kallen in [Reference Franjou and van der Kallen13] under the notion ‘power-reductive’.

Here we follow the terminology of Alper introduced in [Reference Alper2]. It strengthens the notion of geometric reductivity defined by Seshadri [Reference Seshadri27].

The property of geometric reductivity is stable under base change and under fpqc-descent [Reference Alper2, Proposition 9.3.1].

For us this notion is useful because of the following results.

Thus we can apply Propositions 2.5 and 2.4 as follows; this is used in the classification of G-bundles on the projective line below (Theorem 4.2).

Let us give one further application of the results above.

Proposition 2.10. Let $m \geq 1$ be an integer. Let S be an algebraic space and let G be a geometrically reductive group algebraic space over S that admits locally for the étale topology a closed embedding into $\operatorname {\mathrm {GL}}({\mathscr E})$ for some vector bundle ${\mathscr E}$ on S (for instance, if G is a reductive group scheme over S; see Remark 2.6 (2)).

Let ${\mathcal X}$ be a stack over S whose diagonal is representable by algebraic spaces and that admits an fpqc-covering by an algebraic space. Let $f\colon {\mathcal U} \to {\mathcal X}$ be a morphism of stacks that is representable by schemes and that is proper, flat and of finite presentation with geometrically connected and geometrically reduced fibres. Then

(*) $$\begin{align} f^*\colon \operatorname{\mathrm{Bun}}_G({\mathcal X}) \to \operatorname{\mathrm{Bun}}_G({\mathcal U}) \end{align}$$

is fully faithful.

Let P be a G-bundle on ${\mathcal U}$ and suppose there exist a reduced scheme Y and an fpqc-covering $Y \to {\mathcal X}$ such that the pullback of P to $V := {\mathcal U} \times _{{\mathcal X}} Y$ is trivial on the fibres of $V \to Y$ . Then P is in the essential image of (*).

3.1 Connected components and clopen subsets

For a topological space X, we denote by $\pi _0(X)$ its space of connected components; i.e., the underlying set is the set of connected components of X and $\pi _0(X)$ is endowed with the quotient topology of X. This is a totally disconnected topological space. If ${f\colon X \to Y}$ is a continuous map of topological spaces, sending a connected component $C \subseteq X$ to the unique connected component of Y that contains $f(C)$ yields a continuous map $\pi _0(f)\colon \pi _0(X) \to \pi _0(Y)$ . We obtain a functor $\pi _0$ from the category of topological spaces to the category of totally disconnected spaces. This functor is left adjoint to the inclusion functor.

Let X be a spectral topological space. We call a subset Z of X pro-clopen if it is an intersection of subsets of X that are both open and closed. By [30, 04PL, 0906], the following assertions hold.

From these results one easily deduces the following lemma.

Lemma 3.2. Let X be a spectral space. Let $p\colon X \to \pi _0(X)$ be the canonical quotient map. Then sending $Y \subseteq \pi _0(X)$ to $p^{-1}(Y) \subseteq X$ yields inclusion-preserving bijections

$$ \begin{align*} \{Y \subseteq \pi_0(X) \text{ closed}\} &\overset{\sim}{\to} \{Z \subseteq X \text{ pro-clopen}\}, \\ \{Y = \{C\} \subseteq \pi_0(X) \text{ singleton}\} &\overset{\sim}{\to} \{\emptyset \ne Z \subseteq X \text{ minimal pro-clopen}\}, \\ \{Y \subseteq \pi_0(X) \text{ open and closed}\} &\overset{\sim}{\to} \{Z \subseteq X \text{ open and closed}\}. \end{align*} $$

The inverse map is given by $Z \mapsto p(Z)$ .

It follows that the minimal non-empty pro-clopen subsets of X are precisely the connected components of X.

For a topological space X, we denote by $\operatorname {\mathrm {OC}}(X)$ the set of open and closed subsets of X.

These conditions are, for instance, satisfied if f has connected fibres (so that, in particular, f is surjective) and if f endows Y with the quotient topology of X because then $\pi _0(Y)$ satisfies the universal property of $\pi _0(X)$ .

3.2 Henselian pairs of stacks

We call an fpqc-stack ${\mathcal X}$ algebraic if its diagonal is representable by algebraic spaces and if there exists a surjective and smooth morphism $U \to {\mathcal X}$ , where U is a scheme.

In the Stacks Project it is only assumed that ${\mathcal X}$ is an fppf-stack. Since our main results use only algebraic stacks that are quotient stacks by affine, flat, finitely presented group schemes G and since all G-bundles for the fpqc-topology are automatically fppf-locally trivial, this difference in terminology will not trouble us.

We denote by $|{\mathcal X}|$ the underlying topological space of an algebraic stack ${\mathcal X}$ [30, 04Y8].

For any algebraic stack ${\mathcal X}$ , we denote by $\operatorname {\mathrm {OC}}({\mathcal X})$ the set of open and closed substacks of ${\mathcal X}$ . We denote by $\pi _0({\mathcal X})$ the space of connected components of the underlying topological space of ${\mathcal X}$ . If ${\mathcal X}$ is qcqs, the underlying topological space of ${\mathcal X}$ is spectral [30, 0DQP] and hence $\pi _0({\mathcal X})$ is pro-finite [30, 0906].

Recall the notion of a henselian pair for rings.

In [Reference Bouthier and Česnavičius7, Theorem 2.1.6] it is shown that for an affine henselian pair $(X,Z)$ as above and a quasi-affine smooth group scheme G over X, restriction of G-bundles defines a bijective map

$$\begin{align*}H^1(X,G) \overset{\sim}{\longrightarrow} H^1(Z,G). \end{align*}$$

We want to generalize this result to certain pairs $({\mathcal X},{\mathcal Z})$ consisting of an algebraic stack ${\mathcal X}$ and a closed substack ${\mathcal Z}$ .

We have the following result, which follows from the purely topological statement in Proposition 3.3.

As a special case one has the notion of a henselian pair of algebraic stacks:

The only sources of henselian pairs of algebraic stacks that we will need in this paper are algebraic stacks that admit moduli spaces that form a henselian pair as in the following remark.

Remark 3.8. Let S be an algebraic space. Let G be a geometrically reductive group scheme over S (see Definition 2.7). Let $p\colon \tilde {X} \to S$ be an affine morphism of algebraic spaces and let $\tilde {X}$ be endowed with an action of G over S. By [Reference Alper2, Theorem 9.1.4] and [Reference Alper, Hall and Rydh3, Theorem 3.12], the map

$$\begin{align*}\pi\colon [G\backslash \tilde{X}] \longrightarrow \operatorname{\mathrm{Spec}}(p_*{\mathscr O}_{\tilde{X}})^G \end{align*}$$

is initial in the category of all maps from $[G\backslash \tilde {X}]$ to algebraic spaces. The morphism $\pi $ is surjective and universally closed by [Reference Alper2, Theorem 5.3.1]. The same reference also shows that the underlying topological space of each geometric fibre of $\pi $ contains at most one closed point. Moreover, since $\pi $ is qcqs, every geometric fibre is spectral. In particular, it is a quasi-compact Kolmogorov space and therefore always has at least one closed point [30, 005E]. Therefore every geometric fibre is spectral with a unique closed point and hence homeomorphic to the spectrum of a local ring. In particular, $\pi $ has geometrically connected fibres.

Let ${\mathcal Z}$ be a closed substack of ${\mathcal X} := [G\backslash X]$ and let Z be a closed subscheme of ${X := \operatorname {\mathrm {Spec}}(p_*{\mathscr O}_{\tilde {X}})^G}$ such that $\pi ({\mathcal Z}) = Z$ (set-theoretically). Then $({\mathcal X},{\mathcal Z})$ is henselian if and only if $(X,Z)$ is henselian by [Reference Alper, Hall and Rydh3, Theorem 3.10].

Below, we will also use the following remark.

Remark 3.9. Let G be a group algebraic space over S such that $G \to S$ is an fpqc-covering. Suppose G acts on an algebraic space X over S. Let $x\colon T \to [G\backslash X]$ be a T-valued point of the fpqc-quotient, where T is an algebraic space, and let $G_x := \underline {\operatorname {\mathrm {Aut}}}_{{\mathcal X}(T)}(x)$ be the fpqc-sheaf of automorphisms of x.

By definition of $[G\backslash X]$ as the fpqc-stackification of $[G\,{}_p\!\backslash X]$ , there exist an fpqc-covering $\pi \colon T' \to T$ and a morphism $\tilde {x} \colon T' \to X$ such that

commutes. Hence

$$\begin{align*}G_x \times_T T' = G_{x \circ \pi} = G_{p \circ \tilde{x}} = \operatorname{\mathrm{Stab}}_{G_{T'}}(\tilde{x}), \end{align*}$$

where $\operatorname {\mathrm {Stab}}_{G_{T'}}(\tilde {x})$ denotes the stabilizer of $\tilde {x} \in X(T')$ in $G_{T'}$ . In particular, $G_x$ is always fpqc-locally on T isomorphic to a subgroup algebraic space of $G_T$ .

The main result of this section is an application of [Reference Alper, Hall and Rydh3, Proposition 7.10], which we use in the following situation.

The condition on H is, for instance, satisfied if H is an arbitrary group scheme of multiplicative type of finite type over A and A is a normal domain: in that case H even becomes diagonalizable after a finite étale surjective base change [Reference Conrad10, Corollary B.3.6].

In particular, we see that for every smooth quasi-affine group scheme G over A the restriction functor

$$\begin{align*}\operatorname{\mathrm{Bun}}_G({\mathcal X}) \longrightarrow \operatorname{\mathrm{Bun}}_G({\mathcal Z}) \end{align*}$$

is essentially surjective and full. Therefore one has a bijection

(*) $$ \begin{align} H^1({\mathcal X},G) \overset{\sim}{\longrightarrow} H^1({\mathcal Z},G). \end{align} $$

If G is flat, quasi-affine and finitely presented over A, then (*) is still surjective.

3.3 Reformulation of the lifting theorem in terms of graded algebras

Let M be an abelian group and let $H := \underline {\operatorname {\mathrm {Hom}}}(\underline {M},\mathbb {G}_{m})$ be the corresponding diagonalizable group scheme over $\mathbb {Z}$ . Recall that H is a flat and affine group scheme over $\mathbb {Z}$ [Reference Grothendieck.28, VIII, Proposition 2.1]. In this case, we can make Situation 3.10 more concrete as follows.

Let A be a ring. By [Reference Grothendieck.29, Exp. I, Corollaire 4.7.3.1] the functor $B \mapsto \operatorname {\mathrm {Spec}} B$ induces a contravariant equivalence of the category of M-graded A-algebras and the category of affine A-schemes with an $H_A$ -action. Hence in the sequel we will identify affine A-schemes with an $H_A$ -action and M-graded A-algebras.

For an M-graded A-algebra B we form the quotient stack ${\mathcal X} := [H_A\backslash \operatorname {\mathrm {Spec}} B]$ . Let ${I = \bigoplus _{m}I_m \subseteq B}$ be a graded ideal and let ${\mathcal Z} := [H_A\backslash (\operatorname {\mathrm {Spec}} B/I)]$ be the corresponding closed substack of ${\mathcal X}$ .

We obtain the following special case of Theorem 3.11.

Corollary 3.14. With the above notation, suppose that M is finitely generated and that $(B_0,I_0)$ is a henselian pair. Let G be a flat, quasi-affine, finitely presented (resp. smooth quasi-affine) group scheme over A.

Then the restriction functor $\operatorname {\mathrm {Bun}}_G({\mathcal X}) \to \operatorname {\mathrm {Bun}}_G({\mathcal Z})$ is essentially surjective (resp. essentially surjective and full). In particular,

$$\begin{align*}H^1({\mathcal X},G) \longrightarrow H^1({\mathcal Z},G) \end{align*}$$

is surjective (resp. bijective).

Example 3.15. As a special case suppose that $M = \mathbb {Z}$ and that B is concentrated in degrees $\geq 0$ . We choose $I := \bigoplus _{m>0}B_m$ . Then $I_0 = 0$ and $(B_0,0)$ is trivially henselian. Hence one has for any smooth quasi-affine group scheme over A a bijection

(3.3.1) $$ \begin{align} H^1([\mathbb{G}_m\backslash (\operatorname{\mathrm{Spec}} B)],G) \overset{\sim}{\to} H^1(\mathbb{B}\mathbb{G}_{m,B_0},G). \end{align} $$

In this case, the maps $B_0 \to B \to B/I = B_0$ induce maps

$$\begin{align*}H^1(\mathbb{B}\mathbb{G}_{m,B_0},G) \to H^1([(\operatorname{\mathrm{Spec}} B)/\mathbb{G}_m],G) \overset{\sim}{\to} H^1(\mathbb{B}\mathbb{G}_{m,B_0},G) \end{align*}$$

whose composition is the identity. Here the first map is given by pullback along $\pi \colon [(\operatorname {\mathrm {Spec}} B)/\mathbb {G}_m] \to \mathbb {B}\mathbb {G}_{m,B_0}$ induced by the structure map of $\operatorname {\mathrm {Spec}} B$ as a $B_0$ -scheme. This shows that the inverse of (3.3.1) is given by $\pi ^*$ .

The same assertions also hold if B is concentrated in degrees $\leq 0$ .

The above example is in fact a special case of considering the fixed point locus:

Remark 3.16. Suppose that A, B, X and ${\mathcal X}$ are defined as above.

We claim that the fixed point locus of the $H_A$ -scheme X is given by the closed subscheme $\operatorname {\mathrm {Spec}} (B/I^0)$ , where $I^0$ is the graded ideal of B generated by the homogeneous elements in non-zero degree. Indeed, if R is any A-algebra, then $X^{H_A}(R)$ is the subset of M-graded A-algebra homomorphisms $B \to R$ with R trivially M-graded, which shows the claim.

The inclusion $B_0 \to B$ induces an isomorphism of trivially M-graded A-algebras

$$\begin{align*}B^0 := B_0/I_0^0 \overset{\sim}{\to} B/I^0, \end{align*}$$

where $I^0_0 = \sum _{m\ne 0}B_{-m}B_m$ is the degree $0$ part of $I^0$ .

The closed immersion $\operatorname {\mathrm {Spec}} B^0 = (\operatorname {\mathrm {Spec}} B)^{H_A} \hookrightarrow \operatorname {\mathrm {Spec}} B$ induces a closed immersion

$$\begin{align*}i\colon \mathbb{B}H_{B^0} = [H_A\backslash \operatorname{\mathrm{Spec}} B^0] \hookrightarrow [H_A\backslash \operatorname{\mathrm{Spec}} B]. \end{align*}$$

If H is of finite type over $\mathbb {Z}$ , the pair $([H_A\backslash \operatorname {\mathrm {Spec}} B], \mathbb {B}H_{B^0})$ is henselian if and only if $(B_0,I^0_0)$ is henselian. If this is the case, then $i^*$ yields a bijection

(3.3.2) $$ \begin{align} H^1({\mathcal X},G) \overset{\sim}{\to} H^1(\mathbb{B}H_{B^0},G) \end{align} $$

for every smooth quasi-affine group scheme G over A. We refer to Remark 1.9 for a description of the right-hand side of (3.3.2).

3.4 Lifting of vector bundles over quotient stacks by diagonalizable groups

In this section we give an elementary proof of (a generalization of) Theorem 3.11 in the case of vector bundles and that H is diagonalizable (but not necessarily of finite type) with character group M.

Let us describe quasi-coherent modules and vector bundles on quotient stacks by H.

Example 3.18. Let A be a ring. Then the category of quasi-coherent modules over $[\mathbb {G}_{m,A}\backslash \mathbb {A}_A^1]$ is equivalent to the category of $\mathbb {Z}$ -graded A-modules $E = \bigoplus _i E_i$ together with a graded A-linear endomorphism $T\colon E \to E$ of degree $1$ given by the indeterminate $T \in A[T] = {\mathscr O}_{\mathbb {A}^1_A}(\mathbb {A}^1_A)$ .

Such a quasi-coherent module $(E,T)$ corresponds to a vector bundle over $[\mathbb {G}_{m,A}\backslash \mathbb {A}_A^1]$ if and only if

1. (a) $E_i$ is a finite projective A-module for all i,

2. (b) $T\colon E_i \to E_{i+1}$ is injective with projective cokernel,

3. (c) $E_i = 0$ for $i \ll 0$ and $T\colon E_i \to E_{i+1}$ is an isomorphism for $i \gg 0$ .

In other words, setting

$$\begin{align*}E_{\infty} := \mathop{\mathrm{colim}} (\cdots \mathrel{\smash{\overset{T}{\longrightarrow}}} E_i \mathrel{\smash{\overset{T}{\longrightarrow}}} E_{i+1} \mathrel{\smash{\overset{T}{\longrightarrow}}} \cdots ), \end{align*}$$

a vector bundle over $[\mathbb {G}_{m,A}\backslash \mathbb {A}_A^1]$ is the same as a finite projective A-module $E_{\infty }$ together with a filtration by direct summands.

We continue to denote by M an abelian group. Let B be an M-graded ring and let $I \subseteq B$ be a graded ideal. Recall the following graded version of Nakayama.

If B is $\mathbb {Z}$ -graded with $B_n = 0$ for $n < 0$ and $I = \bigoplus _{n>0}B_n$ , then $B/I = B_0$ and the proposition is well known; see, for instance, [Reference Eisenbud12]. In general, we argue as in [Reference Lau24, Lemma 3.1.1, Corollary 3.1.2], where the result is proved in a special case of the $\mathbb {Z}$ -graded case. For the convenience of the reader, we reproduce Lau’s proof.

We endow $B_0$ with the trivial M-grading. Then the maps $B_0 \to B \to B/IB$ of M-graded rings, by pullback via

$$\begin{align*}[H_A\backslash (\operatorname{\mathrm{Spec}} B/IB)] \to [H_A\backslash (\operatorname{\mathrm{Spec}} B)] \to [H_A\backslash (\operatorname{\mathrm{Spec}} B_0)], \end{align*}$$

induce functors

$$\begin{align*}\mathrm{Vec}([H_A\backslash(\operatorname{\mathrm{Spec}} B_0)]) \mathrel{\smash{\overset{\alpha}{\longrightarrow}}} \mathrm{Vec}([H_A\backslash(\operatorname{\mathrm{Spec}} B)]) \mathrel{\smash{\overset{\beta}{\longrightarrow}}} \mathrm{Vec}([H_A\backslash(\operatorname{\mathrm{Spec}} B/IB)]). \end{align*}$$

4.1 G-bundles on the classifying stack of $\mathbb {G}_m$

Let S be an algebraic space and let G be a flat quasi-compact group scheme over S. By Remark 1.9 the groupoid of G-bundles on $\mathbb {B}\mathbb {G}_{m,S}$ is the groupoid of pairs $(P, \mu )$ , where P is a G-bundle on S and $\mu \colon \mathbb {G}_{m,S} \to \underline {\operatorname {\mathrm {Aut}}}_G(P)$ is a cocharacter of the strong inner form of G attached to P. A morphism $({\mathscr E},\mu ) \to ({\mathscr E}',\mu ')$ is a morphism of G-bundles intertwining $\mu $ and $\mu '$ .

If G is affine and smooth over S, (1.3.2) and Proposition 1.10 show that

$$\begin{align*}\underline{\mathcal{B}\mathrm{un}}_G(\mathbb{B}\mathbb{G}_{m,S}) = [G\backslash \underline{\operatorname{\mathrm{Hom}}}_{S-\mathrm{Grp}}(\mathbb{G}_{m,S},G)] \end{align*}$$

is an algebraic stack locally of finite presentation over S.

We also have the following Tannakian description of $\operatorname {\mathrm {Bun}}_G(\mathbb {B}\mathbb {G}_m)$ .

4.2 Classification of G-bundles over the projective line

As an application we obtain a new proof of the classification of G-bundles over $\mathbb {P}^1_k$ in terms of G-bundles over $\mathbb {B}\mathbb {G}_m$ :

Theorem 4.2. Let k be a field and let G be a smooth affine group scheme over k whose identity component $G^0$ is reductive. Then one has a bijection

(4.2.1) $$ \begin{align} H^1(\mathbb{B}\mathbb{G}_{m,k},G) \cong H^1(\mathbb{P}^1_k,G). \end{align} $$

Proof. As G is defined over a field, $G/G^0$ is automatically finite. By Remark 2.8 (2) we see that G is geometrically reductive. Consider $X = \mathbb {A}^2_k$ with its standard $\mathbb {G}_m$ -action. Then one has bijective maps

Here the first map is a special case of (3.3.1) and one has the second bijection by Proposition 2.9 (2) using that $\mathbb {P}^1_k = [(\mathbb {A}^2_k \setminus \{0\})/\mathbb {G}_m]$ .

Remark 4.3. One can make (4.2.1) more explicit as follows. As recalled above, a G-bundle over $\mathbb {B}\mathbb {G}_{m,k}$ is given by a G-bundle P and by a map of group schemes ${\mu \colon \mathbb {G}_{m,k} \to \underline {\operatorname {\mathrm {Aut}}}_G(P)}$ .

Let $f\colon \mathbb {A}_k^2 \setminus \{0\} \to \operatorname {\mathrm {Spec}} k$ be the structure morphism. By Example 3.15, the corresponding G-bundle on $[\mathbb {G}_{m,k}\backslash (\mathbb {A}_k^2 \setminus \{0\})]$ is given by the G-bundle $\tilde {P} := f^*P$ over $\mathbb {A}_k^2 \setminus \{0\}$ and the $\mathbb {G}_m$ -equivariant structure defined by $\tilde \mu \colon \mathbb {G}_{m,k} \to \underline {\operatorname {\mathrm {Aut}}}_G(\tilde {P})$ obtained by base change from $\mu $ . Since the geometric line bundle $\mathbb {A}^2_k \setminus \{0\} \to \mathbb {P}^1_k$ is ${\mathscr O}(-1)$ , the G-bundle over $\mathbb {P}^1_k$ corresponding to $(P,\mu )$ is given by the image of the isomorphism class of ${\mathscr O}(-1)$ under the composition

where $\Theta ^P$ is the twist by P.

Since G is smooth over k, every G-bundle is étale locally trivial. Moreover, we can also characterize all G-bundles on $\mathbb {P}^1_k$ that are Zariski locally trivial:

As a special case one also obtains the following more classical version (a slight generalization of [Reference Biswas and Nagaraj6]) of the classification of Zariski locally trivial G-bundles over $\mathbb {P}^1_k$ whose formulation avoids G-bundles over an algebraic stack.

Remark 4.6. One can also apply Theorem 4.2 to finite étale group schemes G over k. Then every cocharacter $\mathbb {G}_m$ of any strong inner form of G is trivial and pullback along $\operatorname {\mathrm {Spec}} k \to \mathbb {B}\mathbb {G}_{m,k}$ yields an equivalence $\operatorname {\mathrm {Bun}}_G(\mathbb {B}\mathbb {G}_{m,k}) \overset {\sim }{\to } \operatorname {\mathrm {Bun}}_G(k)$ . Therefore, Theorem 4.2 yields in this case a bijection

$$\begin{align*}H^1(k,G) \cong H^1(\mathbb{P}^1_k,G). \end{align*}$$

Hence every connected finite étale Galois covering of $\mathbb {P}^1_k$ arises by base change from a finite Galois extension of k, which implies the well-known isomorphism

$$\begin{align*}\pi_1(\mathbb{P}^1_k) \overset{\sim}{\to} \operatorname{\mathrm{Gal}}(k^{\mathrm{sep}}/k) \end{align*}$$

of étale fundamental groups.

4.3 Splitting of filtered fiber functors

Let $S_0$ be a qcqs scheme and let $G_0 \to S_0$ be a flat quasi-affine group scheme such that $\mathbb {B}G_0$ satisfies the resolution property (see Example 1.7 for examples). Let S be an $S_0$ -scheme and set $G := S \times _{S_0} G_0$ . We now consider G-bundles on ${\mathcal X} := [\mathbb {G}_{m,S}\backslash \mathbb {A}_S^1]$ . By Proposition 1.6, sending a morphism f to the pullback functor $f^*$ yields an equivalence of the groupoid

$$\begin{align*}\operatorname{\mathrm{Bun}}_G({\mathcal X}) = \operatorname{\mathrm{Hom}}_S({\mathcal X},\mathbb{B}G) = \operatorname{\mathrm{Hom}}_{S_0}({\mathcal X},\mathbb{B}G_0) \end{align*}$$

and the groupoid of exact ${\mathscr O}_{S_0}$ -linear strong monoidal functors $\operatorname {\mathrm {Rep}}(G_0) \to \mathrm {Vec}({\mathcal X})$ . Since $\mathrm {Vec}({\mathcal X})$ can be identified with the category of vector bundles on S endowed with a separated and exhaustive increasing $\mathbb {Z}$ -filtration by subbundles (Example 3.18), it follows that $\operatorname {\mathrm {Bun}}_G({\mathcal X})$ can be identified with the groupoid of filtered fibre functors of $\operatorname {\mathrm {Rep}}(G_0)$ over S.

The zero section of $S \to \mathbb {A}^1_S$ yields a closed immersion

$$\begin{align*}\mathbb{B}\mathbb{G}_{m,S} = [\mathbb{G}_{m,S}\backslash S] \mathrel{\smash{\overset{i}{\longrightarrow}}} [\mathbb{G}_{m,S}\backslash \mathbb{A}^1_S]. \end{align*}$$

Pullback of G-bundles via i defines a functor

(4.3.1) $$ \begin{align} i^*\colon \operatorname{\mathrm{Bun}}_G([\mathbb{G}_{m,S}\backslash \mathbb{A}^1_S]) \longrightarrow \operatorname{\mathrm{Bun}}_G(\mathbb{B}\mathbb{G}_{m,S}). \end{align} $$

It attaches to every filtered fibre functor of $\operatorname {\mathrm {Rep}}(G_0)$ over S its associated graded fibre functor (Remark 4.1) induced by the exact strong monoidal functor that sends a filtered vector bundle $({\mathscr E}, (\operatorname {\mathrm {Fil}}^i{\mathscr E})_{i \in \mathbb {Z}})$ over S to the $\mathbb {Z}$ -graded vector bundle $\bigoplus _i \operatorname {\mathrm {Fil}}^i{\mathscr E}/\operatorname {\mathrm {Fil}}^{i-1}{\mathscr E}$ .

The structure map $\mathbb {A}^1_S \to S$ induces a map $p\colon [\mathbb {G}_{m,S}\backslash \mathbb {A}^1_S] \to \mathbb {B}\mathbb {G}_{m,S}$ and hence via pullback an exact strong monoidal functor

(4.3.2) $$ \begin{align} p^*\colon \operatorname{\mathrm{Bun}}_G(\mathbb{B}G) \longrightarrow \operatorname{\mathrm{Bun}}_G([\mathbb{G}_{m,S}\backslash \mathbb{A}^1_S]) \end{align} $$

induced by the exact strong monoidal functor that sends a graded vector bundle $({\mathscr F} = \bigoplus _i {\mathscr F}_i)$ over S to the filtered vector bundle $({\mathscr F}, \operatorname {\mathrm {Fil}}^i{\mathscr F} := \bigoplus _{j\leq i}{\mathscr F}_j)$ . Clearly one has that $i^* \circ p^* \cong \operatorname {\mathrm {id}}$ .

A filtered fibre functor of $\operatorname {\mathrm {Rep}}(G_0)$ over S is then called splittable if it lies in the essential image of $p^*$ .

We obtain as another direct application of Theorem 3.11 the following result which for smooth affine group schemes over fields is a special case of [Reference Ziegler33, Theorem 4.16].

Note that the bijection $H^1([\mathbb {G}_{m,S}\backslash \mathbb {A}^1_S],G) \overset {\sim }{\to } H^1(\mathbb {B}\mathbb {G}_{m,S},G)$ holds for arbitrary smooth quasi-affine group schemes G over an affine scheme S. The resolution property is needed only to identify both sides with filtered, respectively graded, fibre functors.