aa r X i v : . [ m a t h . AG ] F e b BRAUER SPACES OF SPECTRAL ALGEBRAIC STACKS
CHANG-YEON CHOUGH
Abstract.
We study the question of whether the Brauer group is isomorphic to the co-homological one in spectral algebraic geometry. For this, we prove the compact generationof the derived category of twisted sheaves for quasi-compact spectral algebraic stacks withquasi-affine diagonal, which admit a quasi-finite presentation; in particular, we obtain thecompact generation of the unbounded derived category of quasi-coherent sheaves and theexistence of compact perfect complexes with prescribed support for such stacks. We alsostudy the relationship between derived and spectral algebraic stacks, so that our results canbe extended to the setting of derived algebraic geometry.
Contents
1. Introduction 12. Spectral Algebraic Stacks 53. Points of Quasi-geometric Spectral Algebraic Stacks 104. Excision Squares 135. Twisted Compact Generations 196. Brauer Spaces and Azumaya Algebras 28References 301.
Introduction
The purpose of this paper is to formulate and prove the following question addressed byGrothendieck in the setting of spectral algebraic geometry: given a scheme X , is the canonicalmap δ : Br( X ) → H ( X, G m ) an isomorphism of abelian groups? Recall from [6, 1.2] that anAzumaya algebra is a sheaf of O X -algebra which is locally isomorphic to a matrix algebra andthat the Brauer group Br( X ) classifies Azumaya algebras up to Morita equivalence. The map δ is injective, and its image lies in the torsion subgroup if X is quasi-compact by virtue of [6,1.4, p.205]. Note that δ needs not be an isomorphism in general. In the positive direction,we have many important results which include that δ is surjective for noetherian schemes X of dimension ≤ X is also regular by Grothendieck [7, 2.2] andthat the image of δ is the torsion subgroup of H ( X, G m ) for quasi-compact and separatedschemes which admit an ample line bundle by Gabber (and the proof of de Jong [3, 1.1]). To¨enestablished a more general result for quasi-compact and quasi-separated (derived) schemes byintroducing the notion of derived Azumaya algebra of [19, 2.1], which is a “dg-enhancement of Azumaya algebras”; see [19, 5.1]. In fact, To¨en addressed the question regarding the map δ by investigating the compact generation of α -twisted derived dg-category L α ( X ) (see [19,4.1]) for each element α ∈ H ( X, G m ): the endomorphism algebra of a compact generator ofL α ( X ) is a derived Azumaya algebra whose associated element of H ( X, G m ) is α ; see [19,4.6]. Extending this idea to the spectral setting, Antieau–Gepner obtained a similar resultfor quasi-compact quasi-separated spectral schemes [1, 7.2]. This paper originated from thedesire to extend these results to algebraic stacks in the derived and spectral settings.The main result of this paper is the following: Theorem 1.2.
Let X be a quasi-geometric spectral algebraic stack which admits a quasi-finitepresentation. Then each element of Br † ( X ) has the form [ A ] for some Azumaya algebra A on X . Remark 1.3.
Here the quasi-geometric spectral algebraic stacks of 2.6 are a formulation ofquasi-compact algebraic stacks with quasi-affine diagonal in spectral algebraic geometry. Wewill say that a quasi-geometric spectral algebraic stack X admits a quasi-finite presentation ifthere exist an E ∞ -ring A and a morphism Spec A → X which is locally quasi-finite, faithfullyflat, and locally almost of finite presentation (see [14, 4.2.0.1]). The extended Brauer group Br † ( X ) is the set of connected components of the extended Brauer space B r † ( X ) of [14,11.5.2.1]. Given an Azumaya algebra A of [14, 11.5.3.7], the extended Brauer class [ A ] isdefined as in [14, 11.5.3.9]. Remark 1.4. R be a commutative ring. Then 1.2 can be applied to the underlying quasi-geometricspectral algebraic stacks of quasi-geometric derived algebraic stacks over R which admita quasi-finite presentation (see 2.31). In particular, we obtain the cases of ordinaryquasi-compact algebraic stacks with quasi-finite and separated diagonal, and of quasi-compact derived Deligne-Mumford stacks with quasi-affine diagonal by Hall–Rydh [8,9.3, 9.4] (hence of quasi-compact quasi-separated derived schemes by To¨en [19, 5.1]). Example 1.5.
Let p be a prime number. Let F p denote a finite field of order p , and let µ p denote the ordinary group scheme of p -th roots of unity over Spec F p . Let X be a quasi-affinespectral Deligne-Mumford stack over F p . Then 1.2 can be applied to the classifying stack of µ p over X , in which case our result is new. Our approach to 1.2 is based on Lurie’s reformulation of the work of Antieau–Gepner interms of the theory of quasi-coherent stacks developed in [14, 10.1.1]. Given a quasi-geometricspectral algebraic stack X and an object C of the ∞ -category QStk PSt ( X ) of prestable quasi-coherent stacks (see [14, 10.1.2.4]), the ∞ -category QCoh( X ; C ) of global sections (see [14,10.4.1.1]) is a spectral analogue of the twisted derived dg-category of To¨en. This perspectiveleads to the following central definition of interest: Definition 1.7.
A quasi-geometric spectral algebraic stack X is of twisted compact generation if it satisfies the following condition: RAUER SPACES OF SPECTRAL ALGEBRAIC STACKS 3 ( ∗ ) For each compactly generated stable quasi-coherent stack C on X , the stable ∞ -category QCoh( X ; C ) is compactly generated. Remark 1.8.
Our definition of twisted compact generation 1.7 is related to the β -Thomasoncondition on ordinary algebraic stacks of [8, 8.1]. Let β be a regular cardinal and considerthe following straightforward generalization which we refer to as the β -Thomason conditionon a quasi-geometric spectral algebraic stack X :(i) The ∞ -category QCoh( X ) is compactly generated by a set of cardinality at most β .(ii) For every quasi-compact open subset U ⊆ | X | , there exists a compact object F ofQCoh( X ) with support | X | − U .In the special case of ordinary quasi-compact quasi-separated algebraic stacks, the β -Thomasoncondition is equivalent to the following condition: for each quasi-compact open immersion U → X , the full subcategory QCoh X − U ( X ) ⊆ QCoh( X ) spanned by those objects which aresupported on | X | − | U | (see 5.12) is compactly generated by a set of cardinality at most β .Although its spectral analogue is not evident (the author does not know if [8, 4.10] is true inthe spectral setting), we will see in 5.16 and 5.18 that if a quasi-geometric spectral algebraicstack X is of twisted compact generation, then it not only satisfies the β -Thomason conditionfor some β , but also the aforementioned condition on QCoh X − U ( X ). The main ingredient in our proof of 1.2 is the following result, which is closely relatedto [19, 4.8] (which asserts that for each quasi-compact quasi-separated derived scheme X andeach element α ∈ H ( X, G m ), the α -twisted derived dg-category L α ( X ) admits a compactgenerator): Theorem 1.10.
Let X be a quasi-geometric spectral algebraic stack which admits a quasi-finite presentation. Then X is of twisted compact generation. In particular, for each quasi-compact open subset U ⊆ | X | , there exists a compact object F of QCoh( X ) with support | X | − U . Remark 1.11. (i) According to [14, 10.3.2.1], if X is a quasi-compact quasi-separated spectral algebraicspace and C is a compactly generated prestable quasi-coherent stack on X , then the ∞ -category QCoh( X ; C ) is compactly generated. In the case where C is stable, we candeduce the statement from 1.10.(ii) Let X be an ordinary quasi-compact algebraic stack with quasi-finite and separateddiagonal. Then [8, Theorem A] shows that for each quasi-compact open subset U ⊆ | X | ,there exists a compact complex F ∈ D qc ( X ) with support | X | − U (cf. 5.3). This canbe viewed as a special case of 1.10.(iii) If X is a quasi-compact quasi-separated spectral algebraic space and U ⊆ | X | is a quasi-compact open subset, then [14, 11.1.2.1] guarantees that there exists a perfect object F ∈ QCoh( X ) with support | X | − U . This can be obtained from 1.10. The main difficulty in the proof of 1.10 is that we do not know if the compact generationof R -linear ∞ -categories (see [14, D.1.2.1]), where R is an E ∞ -ring, is local for the fpqctopology. On the other hand, in the setting of derived algebraic geometry, [19, 4.13] (whichasserts that the existence of a compact generator of a locally presentable dg-category over asimplicial commutative ring is local for the fppf topology) is essential to the proof of [19, 4.8]. CHANG-YEON CHOUGH
As Antieau–Gepner mentioned in [1, p.1215], To¨en’s proof of [19, 4.13] makes use of quotientsof simplicial commutative rings, and therefore it cannot be carried out in the spectral settingdue to the lack of quotient construction of E ∞ -rings. Nonetheless, Antieau–Gepner showedthat the existence of a compact generator of an R -linear ∞ -category, where R is an E ∞ -ring,is local for the ´etale topology (see [1, 6.16]), and attributed the idea of the proof to Lurie of[16, 6.1] in which the key ingredient is that the compact generation of R -linear ∞ -categorysatisfies descent for the Nisnevich topology. Moreover, the notion of scallop decomposition of[14, 2.5.3.1], which is closely related to the Nisnevich topology, plays a crucial role in the proofof [14, 10.3.2.1]: a scallop decomposition of a spectral Deligne-Mumford stack X consists of asequence of open immersions ∅ ≃ U → U → · · · → U n ≃ X such that for every 1 ≤ i ≤ n ,there exists an excision square of spectral Deligne-Mumford stacks V / / (cid:15) (cid:15) Y (cid:15) (cid:15) U i − / / U i where Y is affine and V is quasi-compact. However, the essential difficulty in extending [14,10.3.2.1] from quasi-compact quasi-separated spectral algebraic spaces to quasi-geometricspectral algebraic stacks is that the notion of scallop decomposition is designed to accom-modate spectral algebraic spaces: more concretely, if a spectral Deligne-Mumford admits ascallop decomposition, then it must be a quasi-compact quasi-separated spectral algebraicspace (see [14, 3.4.2.1]). To address this difficulty, we develop a theory of the underlyingtopological space of quasi-geometric spectral algebraic stacks (see 3.1), so that we can extendthe definitions of an excision square and a scallop decomposition to those stacks, withoutimposing the requirement that Y appearing in the excision square above is affine (see 4.1 and4.4). We will then prove 1.10 by generalizing the “induction principle” for ordinary alge-braic stacks by Hall–Rydh [9, Theorem E] to quasi-geometric spectral algebraic stacks. Moreprecisely, we show the representability of the spectral Hilbert functor to provide a special pre-sentation of quasi-geometric spectral algebraic stacks (which is an analogue of [9, 4.1] in thespectral setting; see 4.12), from which we are reduced to proving that the property of beingof twisted compact generation satisfies descent for finite morphisms and excision squares ofquasi-geometric spectral algebraic stacks (see 5.28 and 5.33). Remark 1.13.
As a consequence of our proof of 1.10, we will see in 5.36 that for stable R -linear ∞ -categories, where R is a connective E ∞ -ring, the property of being compactlygenerated satisfies descent with respect to the maps which are quasi-finite, faithfully flat, andalmost of finite presentation. This is a generalization of [14, D.5.3.1] in the stable case, whichasserts that the property is local for the ´etale topology.1.14. Outline of the paper.
In Section 2, we formulate quasi-compact algebraic stacks withquasi-affine diagonal in the setting of derived and spectral algebraic geometry, and study therelationship between them. In Section 3, we define the underlying topological space of quasi-geometric stacks and establish some of its basic properties. In Section 4, we first introduceexcision squares, stacky scallop decompositions, and Nisnevich coverings of quasi-geometricspectral algebraic stacks. We then show the representability of the Hilbert functor in thespectral setting to provide a special presentation of those stacks. Section 5 is devoted tointroducing the notion of twisted compact generation for quasi-geometric spectral algebraic
RAUER SPACES OF SPECTRAL ALGEBRAIC STACKS 5 stacks and developing some descent results. In Section 6, we study the extended Brauergroups and Azumaya algebras in spectral algebraic geometry.1.15.
Conventions.
We will follow the set-theoretic convention of [13].1.16.
Acknowledgements.
The author is grateful to Benjamin Antieau, Jack Hall, andBertrand To¨en for helpful comments and conversations. This work was supported by IBS-R003-D1. 2.
Spectral Algebraic Stacks
In this section, we introduce the basic objects of study in this paper: quasi-geometricspectral algebraic stacks. We also investigate the relationship between derived algebraicgeometry and spectral algebraic geometry, so that we can incorporate quasi-geometric derivedalgebraic stacks (which are defined similarly) into our study.
For an E ∞ -ring R , let CAlg R denote the ∞ -category E ∞ -algebras over R (see [17,7.1.3.8]). The following big ´etale topology on the opposite ∞ -category CAlg op R (whose ex-istence is evident from the small ´etale topology [14, B.6.2.1]) is ubiquitous in [14], but notmentioned explicitly. We record it here for reference: Lemma 2.2.
Let R be an E ∞ -ring. Then there exits a Grothendieck topology on the ∞ -category CAlg op R which can be characterized as follows: if A is an E ∞ -algebra over R , then asieve C ⊂ (CAlg op R ) /A ≃ CAlg op A is a covering if and only if it contains a finite collection ofmaps { A → A i } ≤ i ≤ n for which the induced map A → Q ≤ i ≤ n A i is faithfully flat and ´etale. Remark 2.3.
In the case where R is connective, the same proof provides an apparent ana-logue for the ∞ -category CAlg cn R of connective E ∞ -algebras over R . Let d S hv ´et (CAlg cn R ) ⊆ Fun(CAlg cn R , b S ) denote the full subcategory spanned by the ´etale sheaves (here b S denotes the ∞ -category of (not necessarily small) spaces; see [13, 1.2.16.4]). Let R be a connective E ∞ -ring. We say that a morphism X → Y in Fun(CAlg cn R , b S )is representable if, for every connective R -algebra R ′ and every morphism Spec R ′ → Y , thefiber product X × Y Spec R ′ is representable by a spectral Deligne-Mumford stack over Spec R (cf. [14, 6.3.2.1]). Let P be a property of morphism of spectral Deligne-Mumford stackswhich is local on the target with respect to the ´etale topology [14, 6.3.1.1] and stable underbase change [14, 6.3.3.1]. We say that a representable morphism X → Y has the propertyP if, for every connective E ∞ -algebra R ′ over R and every morphism Spec R ′ → Y , theprojection X × Y Spec R ′ → Spec R ′ , which can be identified with a morphism of spectralDeligne-Mumford stacks, has the property P (cf. [14, 6.3.3.3]). According to [14, 9.1.0.1], a quasi-geometric stack is a functor X : CAlg cn → b S satisfyingthe following conditions:(i) The functor X is a sheaf for the fpqc topology of [14, B.6.1.3].(ii) The diagonal ∆ : X → X × X is representable and quasi-affine (see [14, 6.3.3.6]).(iii) There exists a faithfully flat morphism Spec A → X , where A is a connective E ∞ -ring;see [14, 6.3.3.7]. CHANG-YEON CHOUGH
We now introduce a special class of quasi-geometric stacks, called quasi-geometric spectralalgebraic stacks . This collection of quasi-geometric stacks admits a “smooth covering”. Thereare at least two different ways to construct a suitable “smoothness” in the setting of spectralalgebraic geometry: for example, fiber smoothness and differentially smoothness (see [14,11.2.5.5]). Fiber smooth morphisms are closely related to smooth morphisms in the classicalalgebraic geometry (see 2.10), and we adopt the fiber smoothness in our definition of quasi-geometric spectral algebraic stacks (the terminology is not standard):
Definition 2.6.
Let R be a connective E ∞ -ring. A quasi-geometric spectral algebraic stackover R is a functor X : CAlg cn R → b S which satisfies the following conditions:(i) The functor X is a sheaf for the fpqc topology.(ii) The diagonal morphism ∆ : X → X × X is representable and quasi-affine.(iii) There exist a connective E ∞ -algebra A over R and a morphism Spec A → X which isfiber smooth and surjective.In the special case where R is the sphere spectrum (that is, an initial object of CAlg cn ),we simply say that X is a quasi-geometric spectral algebraic stack . In other words, a quasi-geometric spectral algebraic stack is a quasi-geometric stack which satisfies condition (iii). Remark 2.7.
Let Fun(CAlg cn , b S ) /R denote the slice ∞ -category Fun(CAlg cn , b S ) / Spec R . Let X ′ : CAlg cn R → b S be the image of an object ( X → Spec R ) ∈ Fun(CAlg cn , b S ) /R underthe equivalence of ∞ -categories Fun(CAlg cn , b S ) /R ≃ Fun(CAlg cn R , b S ). Then X ′ is a quasi-geometric spectral algebraic stack over R if and only if the functor X is a quasi-geometricspectral algebraic stack (over the sphere spectrum). Example 2.8.
Quasi-geometric spectral algebraic stacks exist in abundance:(i) Every quasi-geometric spectral Deligne-Mumford stack is a quasi-geometric spectralalgebraic stack because every ´etale morphism is fiber smooth [14, 11.2.3.2].(ii) Let R be a commutative ring. We will see in 2.30 that for each quasi-geometric derivedalgebraic stack over R , there is an underlying quasi-geometric spectral algebraic stackover R ; in particular, each ordinary quasi-compact algebraic stack over R with quasi-affine diagonal can be regarded as a quasi-geometric spectral algebraic stack over R . Our choice of “smooth coverings” in the definition of quasi-geometric spectral algebraicstacks 2.6 is motivated by the following characterization of fiber smoothness in terms of the0-truncations of [14, 1.4.6.5]:
Lemma 2.10.
Let f : X → Y be a morphism of spectral algebraic spaces. Then f is fibersmooth if and only if it is flat and the underlying morphism of ordinary algebraic spaces τ ≤ ( f ) is smooth.Proof. The assertion is ´etale-local on X and Y , so we may assume that X and Y are affine. Inthis case, the desired result is an immediate consequence of [14, 11.2.3.5] and [14, 11.2.4.1]. (cid:3) In classical algebraic geometry, the big smooth topology does not play as significant roleas the big ´etale topology. This is in part due to the fact that a smooth surjection of ordinaryschemes ´etale-locally admits a section (see [5, 17.16.3]), and therefore the topoi inducedby these topologies are equivalent. In the spectral setting, an analogous statement holds for
RAUER SPACES OF SPECTRAL ALGEBRAIC STACKS 7 differentially smooth morphisms by virtue of [1, 4.47]. The following topology defined by fibersmooth maps (whose existence can be proven in the same way as [14, B.6.1.3]) is neverthelessof interest to us in this paper:
Lemma 2.12.
Let R be a connective E ∞ -ring. Then there exists a Grothendieck topology onthe ∞ -category (CAlg cn R ) op which can be described as follows: if A is a connective E ∞ -algebraover R , then a sieve C ⊂ (CAlg op R ) /A ≃ CAlg op A is a covering if and only if it contains a finitecollection of maps { A → A i } ≤ i ≤ n for which the induced map A → Q ≤ i ≤ n A i is faithfully flatand fiber smooth. Remark 2.13.
We refer to the Grothendieck topology of 2.12 as the fiber smooth topologyon (CAlg cn R ) op . We will see later in 5.8 that it has the virtue of connecting quasi-coherentstacks in derived algebraic geometry and spectral algebraic geometry. Let d S hv fsm (CAlg cn R ) ⊆ Fun(CAlg cn R , b S ) denote the full subcategory spanned by the fiber smooth sheaves. Remark 2.14.
In the situation of 2.6, we can replace (i) by the apparently weaker conditionthat X is a sheaf for the fiber smooth topology. Indeed, if X satisfies this condition alongwith conditions (ii) and (iii) of 2.6, then X is a (hypercomplete) sheaf with respect to thefpqc topology; this can be established by mimicking the proof of [14, 9.1.4.3]. For the rest of this section, we study how to deal with derived stacks in the contextof spectral algebraic geometry. In particular, we will see in 2.30 that one can associatea quasi-geometric spectral algebraic stack to each quasi-geometric derived algebraic stack.This connection has the virtue of allowing us to apply our main theorems 1.2 and 1.10—which are described in terms of quasi-geometric spectral algebraic stacks—to quasi-geometricderived algebraic stacks as well.
Let R be a commutative ring. Let CAlg ∆ R denote the ∞ -category of simplicial commu-tative R -algebras (see, for example, [14, 25.1.1.1]). It follows from [14, 25.1.2.1] that there isa forgetful functor Θ R : CAlg ∆ R → CAlg cn R . We denote the image of A ∈ CAlg ∆ R under Θ R by A ◦ and refer to it as the underlying E ∞ -algebra of A . By virtue of [13, 4.3.3.7], the restriction functor Θ ∗ R : Fun(CAlg cn R , b S ) → Fun(CAlg ∆ R , b S ) admits a left adjoint Θ R ! which carries each functor X : CAlg ∆ R → b S to its leftKan extension along Θ R . Remark 2.17.
There are evident analogues of the ´etale and fpqc topologies (see 2.2 and [14,B.6.1.3]) for the ∞ -category (CAlg ∆ R ) op . Let ˚Θ : CAlg ∆ → CAlg cn denote the composition of Θ Z with the forgetful functorCAlg cn Z → CAlg cn . To every derived Deligne-Mumford stack X = ( X , O X ) (which can bedefined as in [14, 1.4.4.2], using CAlg ∆ in place of CAlg cn ), one can associate a spectralDeligne-Mumford stack ( X , ˚Θ ◦ O X ), which we denote by X ◦ and refer to as the underlyingspectral Deligne-Mumford stack of X . We can regard this construction as a functor from the ∞ -category DerDM of derived Deligne-Mumford stacks to the ∞ -category SpDM of spectralDeligne-Mumford stacks; it carries the affine spectrum of a simplicial commutative ring A tothe affine spectrum of the underlying E ∞ -ring A ◦ . Let DerDM /R and SpDM /R denote theslice ∞ -categories DerDM / Spec R and SpDM / Spec R , respectively. We then have a functorDerDM /R → SpDM /R CHANG-YEON CHOUGH which carries a derived Deligne-Mumford stack X over R to its underlying spectral Deligne-Mumford stack X ◦ over R . Let L ´et : Fun(CAlg cn R , b S ) → d S hv ´et (CAlg cn R ) denote a left adjoint tothe inclusion (see 2.3). To extend the construction X X ◦ to derived (algebraic) stacks, weneed the following “functor of points” perspective (cf. [15, 9.27]): Lemma 2.19.
Let R be a commutative ring. Then the composite functor L ´et ◦ Θ R ! : Fun(CAlg ∆ R , b S ) → Fun(CAlg cn R , b S ) → d S hv ´et (CAlg cn R ) restricts to the functor DerDM /R → SpDM /R .Proof. Suppose we are given a derived Deligne-Mumford stack X = ( X , O X ) over R . Let h X : CAlg ∆ R → b S denote the functor represented by X (given by the formula h X ( A ) =Map DerDM /R (Spec A, X )), and define h X ◦ similarly. We wish to show that the natural mor-phism of functors ( ∗ X ) : L ´et (Θ R ! h X ) → h X ◦ is an equivalence (note that h X ◦ is a sheaf for the ´etale topology). Let X be the full sub-category of X spanned by those objects U ∈ X for which ( ∗ X U ) is an equivalence, where X U : CAlg ∆ R → b S denotes the functor represented by the derived Deligne-Mumford stack X U = ( X /U , O X | U ) over R . It follows immediately that X contains all affine objects U ∈ X .By virtue of [14, 1.4.7.9], it will suffice to show that X is closed under small colimits in X .To prove this, suppose we are given a small diagram { U α } in X having a colimit U ∈ X . Wethen have a commutative diagram in d S hv ´et (CAlg cn R )colim L ´et Θ R ! h X Uα / / (cid:15) (cid:15) colim h X ◦ Uα (cid:15) (cid:15) L ´et Θ R ! h X U / / h X ◦ U . Since the transition morphisms in the diagram { X ◦ U α } are ´etale, the right vertical arrow is anequivalence. Combing the analogous equivalence for the diagram { X U α } of derived Deligne-Mumford stacks with the fact that the composition L ´et ◦ Θ R ! commutes with small colimits,we see that the left vertical arrow is also an equivalence, thereby completing the proof. (cid:3) Remark 2.20.
Let us say that a derived Deligne-Mumford stack X is quasi-geometric if it isquasi-compact and the diagonal ∆ X : X → X × X is quasi-affine. In this case, the underlyingspectral Deligne-Mumford stack X ◦ is quasi-geometric (see [14, 9.1.4.1]), so that the functor h X ◦ that it represents is a (hypercomplete) sheaf with respect to the fpqc topology by virtueof [14, 9.1.4.3]; in particular, it satisfies descent for the fiber smooth topology.Let L fsm : Fun(CAlg cn R , b S ) → d S hv fsm (CAlg cn R ) denote a left adjoint to the inclusion functor(see 2.13). Arguing as in the proof of 2.19 (using L fsm h X ◦ Uα and L fsm h X ◦ U in place of h X ◦ Uα and h X ◦ U , respectively), we deduce that the composite functor L fsm ◦ Θ R ! : Fun(CAlg ∆ R , b S ) → Fun(CAlg cn R , b S ) → d S hv fsm (CAlg cn R )carries a quasi-geometric derived Deligne-Mumford stack X over R to its underlying quasi-geometric spectral Deligne-Mumford stack X ◦ over R . RAUER SPACES OF SPECTRAL ALGEBRAIC STACKS 9
According to [12, 3.4.7], a morphism A → B in CAlg ∆ is smooth if the relative algebraiccotangent complex L alg B/A (see [12, 3.2.14] and [14, 25.3.2.1]) is a dual of connective and perfectobject of Mod B and almost of finite presentation (see [12, 3.1.5]). The following observationshows a close relationship between smooth maps in the derived setting and fiber smooth mapsin the spectral setting: Lemma 2.22.
Let f : A → B be a morphism of simplicial commutative rings. Then f issmooth if and only if the underlying morphism of E ∞ -rings f ◦ : A ◦ → B ◦ is fiber smooth.Proof. By virtue of [12, 3.4.9], f is smooth if and only if it is flat and π ( f ) is a smooth mapof commutative rings, so the desired result follows from 2.10. (cid:3) Remark 2.23.
There is an analogue of the fiber smooth topology of 2.12 for the ∞ -category(CAlg ∆ R ) op by replacing “fiber smooth” with “smooth” in 2.12. We refer to this Grothendiecktopology as the smooth topology on (CAlg ∆ R ) op . Let d S hv sm (CAlg ∆ R ) ⊆ Fun(CAlg ∆ R , b S ) denotethe full subcategory spanned by the smooth sheaves. Using an evident analogue of 2.4, we introduce a variant of 2.6 in the derived setting:
Definition 2.25.
Let R be a commutative ring. A quasi-geometric derived algebraic stackover R is a functor X : CAlg ∆ R → b S which satisfies the following conditions:(i) The functor X is a sheaf for the fpqc topology of 2.17.(ii) The diagonal morphism ∆ : X → X × X is representable and quasi-affine.(iii) There exists a simplicial commutative algebra A over R and a morphism Spec A → X which is smooth and surjective. Remark 2.26.
Arguing as in the proof of [1, 4.47], we see that a smooth morphism ofderived Deligne-Mumford stacks ´etale-locally admits a section. Using this observation, aminor modification of the proof of [14, 9.1.4.3] guarantees that an ´etale sheaf satisfyingconditions (ii) and (iii) of 2.25 is a sheaf for the fpqc topology. In the situation of 2.25, onecan therefore replace (i) by the weaker condition that X is a sheaf for the ´etale topology. In order to extend 2.19 from derived Deligne-Mumford stacks to (quasi-geometric)derived algebraic stacks, we need to understand if the functor L ´et ◦ Θ R ! carries representablemorphisms in Fun(CAlg ∆ R , b S ) to representable morphisms in d S hv ´et (CAlg cn R ); however, this isnot straightforward at all. We will circumvent this difficulty by using the following lemmas: Lemma 2.28.
Let f : X → Y be a representable morphism in the ∞ -category of ´etalesheaves on (CAlg ∆ R ) op . If f is a smooth surjection, then f is an effective epimorphism and ( L fsm ◦ Θ R ! )( f ) is an effective epimorphism of fiber smooth sheaves on (CAlg cn R ) op .Proof. The forgetful functor Θ R : CAlg ∆ R → CAlg cn R is left exact (see [14, 25.1.2.2]) and carriessmooth coverings to fiber smooth coverings (see 2.22). In particular, the restriction functorΘ ∗ R restricts to a morphism of ∞ -topoi d S hv fsm (CAlg cn R ) → d S hv sm (CAlg ∆ R ) (see 2.23), whoseleft adjoint is given by the composition of the inclusion d S hv sm (CAlg ∆ R ) ⊆ Fun(CAlg ∆ R , b S ) with L fsm ◦ Θ R ! . Since a left adjoint of a geometric morphism of ∞ -topoi preserves effective epimor-phisms [13, 6.2.3.6], it will suffice to show that f is an effective epimorphism in d S hv sm (CAlg ∆ R ).By virtue of [12, 3.4.4], a smooth surjective morphism f : X → Y of derived Deligne-Mumford stacks satisfies an infinitesimal lifting criterion. Using the argument of [1, 4.47], we deducethat f is an effective epimorphism of ´etale sheaves, which implies the desired result. (cid:3) We are now ready to prove the main result of this section:
Proposition 2.30.
Let R be a commutative ring. Let X be a quasi-geometric derived al-gebraic stack over R . Then the functor ( L fsm ◦ Θ R ! )( X ) : CAlg cn R → b S is a quasi-geometricspectral algebraic stack over R .Proof. Choose a smooth surjection p : Spec A → X where A is a simplicial commutative R -algebra. Let Y be a derived Deligne-Mumford stack representing the fiber product Spec A × X Spec A (note that Y is quasi-affine), so that it fits into a pullback square of fpqc sheaves Y q / / (cid:15) (cid:15) Spec A p (cid:15) (cid:15) Spec A p / / X. Using 2.20 and the fact that the functor L fsm ◦ Θ R ! is left exact (see the proof of 2.28), theabove diagram induces a pullback square of fiber smooth sheaves on (CAlg cn R ) op Y ◦ q ◦ / / (cid:15) (cid:15) Spec A ◦ p ◦ (cid:15) (cid:15) Spec A ◦ p ◦ / / L fsm (Θ R ! ( X )) . Since q : Y → Spec A is quasi-affine, so is its underlying morphism q ◦ of spectral Deligne-Mumford stacks. By virtue of 2.28, the map p ◦ : Spec A ◦ → L fsm (Θ R ! ( X )) is an effectiveepimorphism, so that a relative version of [14, 9.1.1.3] with the fiber smooth topology inplace of the fpqc topology (which can be proven by exactly the same argument) guaranteesthat p ◦ is representable quasi-affine. Since q is a smooth surjection, it follows from 2.22 that q ◦ is fiber smooth and surjective. Consequently, the representable morphism p ◦ is also a fibersmooth surjection because the property of being a fiber smooth morphism is local on thetarget with respect to the flat topology (see [14, 11.2.5.9]). Applying a variant of [14, 9.1.1.2]which uses fiber smooth morphisms in place of flat morphisms to L fsm (Θ R ! ( X )) and p ◦ , wededuce that the diagonal of L fsm (Θ R ! ( X )) is representable quasi-affine. Invoking 2.14, weconclude that L fsm (Θ R ! ( X )) is a quasi-geometric spectral algebraic stack over R . (cid:3) Remark 2.31.
Let X be a quasi-geometric derived algebraic stack X over R . Let X ◦ denoteits image under the functor L fsm ◦ Θ R ! ; we refer to X ◦ as the underlying quasi-geometricspectral algebraic stack over R of X .3. Points of Quasi-geometric Spectral Algebraic Stacks
In this section, we define the notion of points of quasi-geometric spectral algebraic stacks insuch a way that the points of ordinary algebraic stacks are defined (see [11, 5.2]) and establishsome of their basic properties.
Definition 3.1.
Let X : CAlg cn → b S be a functor satisfying the following condition: RAUER SPACES OF SPECTRAL ALGEBRAIC STACKS 11 ( ∗ ) There exist a quasi-separated spectral algebraic space X and a relative spectral al-gebraic space π : X → X in Fun(CAlg cn , b S ) which is quasi-separated, faithfully flat,and locally almost of finite presentation.A point of X is a morphism Spec κ → X , where κ is a field. We define an equivalence relationon the (not necessarily small) set of points of X as follows: given two points p : Spec κ → X and p ′ : Spec κ ′ → X , we will write p ∼ p ′ if there exists a field κ ′′ and a commutative diagramSpec κ ′′ / / (cid:15) (cid:15) Spec κ ′ p ′ (cid:15) (cid:15) Spec κ p / / X in Fun(CAlg cn , b S ); let | X | denote the set of equivalence classes. We endow | X | with thetopology generated by the sets | U | , where U ranges over all representable open j : U → X inFun(CAlg cn , b S ) (here we identify | U | with its image under the natural map of sets | j | : | U | →| X | which is injective). Remark 3.2.
In the special case where the functor X is representable by a quasi-separatedspectral algebraic space, it follows from [14, 3.6.3.1] that the topological space | X | definedabove is homeomorphic to the topological space associated to X in the sense of [14, 3.6.1.1]. For later reference, we record some observations whose proofs are immediate:
Lemma 3.4.
Let f : X → Y be a morphism in Fun(CAlg cn , b S ) , where X and Y satisfycondition ( ∗ ) of . Then: ( i ) The induced map of sets | f | : | X | → | Y | is continuous. ( ii ) If f is representable, then it is surjective if and only if the induced map | f | is surjective. ( iii ) Suppose we are given a pullback diagram X ′ / / (cid:15) (cid:15) Y ′ (cid:15) (cid:15) X / / Y of functors satisfying condition ( ∗ ) of . Then the induced map | X ′ | → | X | × | Y | | Y ′ | is a surjection of topological spaces. The rest of this section is devoted to investigating some properties of the “underlyingtopological spaces” of 3.1.
Lemma 3.6.
Let π : X → X be a morphism of functors appearing in condition ( ∗ ) of .Then the induced map of topological spaces | π | : | X | → | X | is open. In particular, | π | is aquotient map.Proof. Let U ⊆ | X | be an open subset; we wish to show that its image under | π | is open.Let X U be the subfunctor of X which carries each object A ∈ CAlg cn to the summand of X ( A ) spanned by those η ∈ X ( A ) for which the induced map | η | : | Spec A | → | X | factors through | π | ( U ). We claim that the inclusion j : X U → X is representable open. For this, let η : Spec A → X be a point and consider a pullback diagram of functorsSpec A × X X η ′ / / π ′ (cid:15) (cid:15) X π (cid:15) (cid:15) Spec A η / / X. Using 3.4, we can identify | η | − ( | π | ( U )) with | π ′ | ( | η ′ | − U ); in particular, it is an open subsetof | Spec A | because | π ′ | is an open map by virtue of a refinement of [14, 4.3.4.3] withoutthe quasi-compact assumption (which can be proven with little additional effort). Then [14,19.2.4.1] guarantees that j is representable open. By construction, we have that | j | ( | X U | ) = | π | ( U ), thereby completing the proof. (cid:3) Lemma 3.7.
Let X : CAlg cn → b S be a functor satisfying condition ( ∗ ) of and let η : Spec A → X be a representable morphism which is flat and locally almost of finite presen-tation. Then | η | ( | Spec A | ) ⊆ | X | is open.Proof. By virtue of 3.6, | π | is a quotient map. Then the desired result follows by combining3.4 with the variant of [14, 4.3.4.3] mentioned in the proof of 3.6. (cid:3) Lemma 3.8.
Let X : CAlg cn → b S be a functor satisfying condition ( ∗ ) of . Then theunderlying topological space | X | has a basis consisting of quasi-compact open subsets of theform | η | ( | Spec A | ) , where A is a connective E ∞ -ring and η : Spec A → X is a relative spectralalgebraic space which is flat and locally almost of finite presentation.Proof. Using 3.6 and 3.7, we can reduce to the case where X is a quasi-separated spectralalgebraic space, in which case the desired result follows from (the proof of) [14, 3.6.3.3]. (cid:3) Combining 3.7 with 3.8, we immediately deduce the following generalization of 3.7:
Lemma 3.10.
Let f : X ′ → X be a morphism in Fun(CAlg cn , b S ) , where X ′ and X satisfycondition ( ∗ ) of . If f is representable flat and locally almost of finite presentation, thenthe induced map of topological spaces | X ′ | → | X | is open. Let X be a functor which satisfies condition ( ∗ ) of 3.1. Under mild hypotheses, givingan open subset of | X | is equivalent to giving an open immersion U → X : Lemma 3.12.
Let X : CAlg cn → b S be a functor which satisfies condition ( ∗ ) of anddescent for the fpqc topology. Let π : X → X be a morphism as in condition ( ∗ ) of .Assume that the diagonal of X is representable quasi-affine. If U ⊆ | X | is a quasi-compactopen subset, then there exist a quasi-geometric stack U and a representable open immersion j : U → X such that | j | ( | U | ) = U .Proof. Let U be the subfunctor of X which carries an object A ∈ CAlg cn to the summandof X ( A ) spanned by those η ∈ X ( A ) for which the induced map of topological spaces | η | : | Spec A | → | X | factors through U . It follows immediately that the inclusion j : U → X isrepresentable open and that | j | ( | U | ) = U (note that U satisfies condition ( ∗ ) of 3.1). Using [14,6.3.3.8], we see that U is a sheaf for the fpqc topology. Since the diagonal of X is quasi-affine,so is the diagonal of U . By virtue of 3.10, | U | is homeomorphic to U , hence quasi-compact. RAUER SPACES OF SPECTRAL ALGEBRAIC STACKS 13
Then 3.8 guarantees that there exists a relative spectral algebraic space Spec A → U whichis faithfully flat (and locally almost of finite presentation), which completes the proof. (cid:3) We now extend the relationship between reduced closed substacks of a (quasi-geometric)spectral Deligne-Mumford stack X and open subsets of | X | to quasi-geometric spectral alge-braic stacks; see [14, 3.1.6.3]. Lemma 3.14.
Let X be a quasi-geometric spectral algebraic stack. The following conditionsare equivalent: ( i ) For every fiber smooth morphism f : Spec A → X , the E ∞ -ring A is discrete andreduced. ( ii ) There exists a fiber smooth surjection
Spec A → X , where the E ∞ -ring A is discreteand reduced.Proof. According to [14, 2.8.3.9], the property of being a 0-truncated spectral Deligne-Mumfordstack is local with respect to the flat topology, so the desired equivalence follows from thefact that for ordinary algebraic spaces, the property of being reduced is local with respect tothe smooth topology (see, for example, [18, Tag 034E]). (cid:3)
Definition 3.15.
Let X be a quasi-geometric spectral algebraic stack. Let us say that X is reduced if it satisfies the equivalent conditions of 3.14. Proposition 3.16.
Let j : U → X be a representable open immersion of quasi-geometricspectral algebraic stacks. Then there exist a reduced quasi-geometric spectral algebraic stack K and a representable closed immersion i : K → X such that | i || K | = | X | − | j || U | .Proof. Choose a fiber smooth surjection f : X → X , where X is affine. Let X • denotethe ˇCech nerve of the morphism f , which is a simplicial object of the ∞ -category SpDM ofspectral Deligne-Mumford stacks. The projections U × X X n → X n are open immersions ofspectral Deligne-Mumford stacks, so that there is a simplicial object K • of SpDM, whereeach K n is the reduced closed substack complementary to U × X X n , and is quasi-geometric.Let K denote the geometric realization of K • in the ∞ -category of fpqc sheaves; it is aquasi-geometric stack by virtue of [14, 9.1.1.5]. By construction, the diagram of fpqc sheaves K / / (cid:15) (cid:15) X (cid:15) (cid:15) K / / X is a pullback square, from which it follows immediately that K is a reduced quasi-geometricspectral algebraic stack. Applying [14, 9.1.1.3] to the diagram above, we deduce that thecanonical morphism K → X is representable quasi-affine, thereby a closed immersion withthe property that | K | is complementary to | U | (regarded as subsets of | X | ). (cid:3) Excision Squares
Our goal in this section is to supply a special presentation of quasi-geometric spectralalgebraic stacks in the spirit of “induction principle” for ordinary algebraic stacks; see [9, 4.1]and [9, Theorem E]. For this, we introduce excision squares and stacky scallop decompositionsof such stacks.
Definition 4.1.
A diagram of quasi-geometric spectral algebraic stacks σ : U ′ / / (cid:15) (cid:15) X ′ f (cid:15) (cid:15) U j / / X is an excision square if it satisfies the following conditions:(i) The diagram σ is a pullback square.(ii) The morphism j is a representable open immersion.(iii) The morphism f is representable ´etale.(iv) The projection K × X X ′ → K is an equivalence (here K denotes the reduced closedsubstack of X complementary to U ; see 3.16). Remark 4.2.
According to [14, p.321], a diagram of spectral Deligne-Mumford stacks U ′ j ′ / / f ′ (cid:15) (cid:15) X ′ (cid:15) (cid:15) U / / X is an excision square if it is a pushout square, j ′ is an open immersion, and f ′ is ´etale. If it isa diagram of quasi-geometric spectral Deligne-Mumford stacks, then it is an excision squarein the sense of [14, p.321] if and only if the associated square of quasi-geometric stacks is anexcision square in the sense of 4.1 (see also [14, 9.1.4.4]). Let X be a spectral Deligne-Mumford stack. According to [14, 2.5.3.1], a scallop decom-position of X consists of a sequence of open immersions ∅ ≃ U → U → · · · → U n ≃ X suchthat for each 1 ≤ i ≤ n , there exists an excision square of spectral Deligne-Mumford stacks V / / (cid:15) (cid:15) Y (cid:15) (cid:15) U i − / / U i , where Y is affine and V is quasi-compact. This is a useful device for proving many basicresults in the theory of spectral algebraic geometry by reducing to the affine case. However,a spectral Deligne-Mumford stack admits a scallop decomposition if and only if it is a quasi-compact quasi-separated spectral algebraic space (see [14, 3.4.2.1]), so that the concept ofa scallop decomposition is not adequate for spectral Deligne-Mumford stacks which are notspectral algebraic spaces. To incorporate a wider class of spectral algebro-geometric objects,we should relax the requirement that Y is affine in the diagram above; we therefore allow Y to be quasi-geometric spectral algebraic stacks, which is sufficient for our needs in this paper: Definition 4.4.
Let X be a quasi-geometric spectral algebraic stack. A stacky scallop de-composition of X consists of a sequence of representable open immersions of quasi-geometricspectral algebraic stacks ∅ ≃ U → U → · · · → U n ≃ X RAUER SPACES OF SPECTRAL ALGEBRAIC STACKS 15 satisfying the following condition: for each 1 ≤ i ≤ n , there exists an excision square V / / (cid:15) (cid:15) W (cid:15) (cid:15) U i − / / U i of quasi-geometric spectral algebraic stacks (see 4.1). The notion of
Nisnevich covering of quasi-compact quasi-separated spectral algebraicspace (see [14, 3.7.1.1]) admits a straightforward extension to quasi-geometric spectral alge-braic stacks:
Definition 4.6.
Let X be a quasi-geometric spectral algebraic stack. Let { p α : W α → X } be a collection of representable ´etale morphisms of quasi-geometric spectral algebraic stacks.We say that { p α } is a Nisnevich covering of X if there exists a sequence of open immersionsof quasi-geometric spectral algebraic stacks ∅ ≃ U n +1 ֒ → · · · ֒ → U ≃ X satisfying the following condition: for each 0 ≤ i ≤ n , let K i denote the reduced closedsubstack of U i which is complementary to U i +1 (see 3.16). Then the composition K i → U i → X factors through some p α . Lemma 4.7.
Let p : W → X be a Nisnevich covering of quasi-geometric spectral algebraicstacks. Then p induces a stacky scallop decomposition of X .Proof. In the situation of 4.6, for each 0 ≤ m ≤ n , consider a subset of | U m × X W | which iscomplementary to | i m | ( | K m × X W |−| s m || K m | ), where i m : K m × X W → U m × X W is the closedimmersion determined by K m → U m and s m is a section of the projection K m × X W → K m (note that p is a Nisnevich covering). Since s m is an open immersion, this subset is open.Moreover, it is quasi-compact because it can be written as a disjoint union of the image ofthe map | i m ◦ s m | and | U m +1 × X W | . According to 3.12, this quasi-compact open subsetdetermines a representable open immersion W m → U m × X W of quasi-geometric spectralalgebraic stacks. Composing this with the projection to U m , we obtain a representable ´etalemorphism W m → U m . Note that the composition i m ◦ s m factors through W m , inducing asection of the projection K m × U m W m → K m . By construction, this section is a surjectiveopen immersion, hence an equivalence. Consequently, the pullback square of quasi-geometricspectral algebraic stacks U m +1 × U m W m / / (cid:15) (cid:15) W m (cid:15) (cid:15) U m +1 / / U m is an excision square, thereby completing the proof. (cid:3) Our primary goal in this section is to produce some presentation of quasi-geometricalgebraic stacks, which allows us to apply some d´evissage method for the study of thosestacks. To obtain such a presentation, we will make use of the Hilbert functors in the settingof spectral algebraic geometry. Note that [12, 8.3.3] shows the representability of the Hilbertfunctors in the derived setting; we will prove a similar result for the spectral Hilbert functors.
We begin by defining the Hilbert functors in the spectral setting. Let p : X → S be arepresentable morphism in Fun(CAlg cn , b S ). Let π : CAlg cn S → CAlg cn be a left fibrationclassified by S (see [13, 3.3.2.2]). Let us identify objects of CAlg cn S with pairs ( A, η ), where A is a connective E ∞ -ring and η ∈ S ( A ) is an A -valued point of S . Note that the oppositeof CAlg cn S can be identified with the fiber product (CAlg cn ) op × Fun(CAlg cn , b S ) Fun(CAlg cn , b S ) /S ,where (CAlg cn ) op → Fun(CAlg cn , S ) is the Yoneda embedding. Consider the composition(CAlg cn S ) op ⊆ Fun(CAlg cn , b S ) /S → Fun(CAlg cn , b S ) /X → Fun(CAlg cn , b S ) , where the middle arrow is the base change functor S ′ S ′ × S X and the last is the forgetfulfunctor. This composition can be described more informally as follows: to each pair ( A, η ),it assigns the fiber product Spec A × S X , where Spec A → S is determined by η . We alsoconsider the composition Fun(∆ , SpDM) → Fun( { } , SpDM) → Fun(CAlg cn , b S ), where thefirst map is an evaluation at { } ⊆ ∆ and the second is the fully faithful embedding. Let C denote the full subcategory of the fiber product(CAlg cn S ) op × Fun(CAlg cn , b S ) Fun(∆ , SpDM)spanned by those morphisms f : Y → Spec A × S X , where Y is a spectral Deligne-Mumfordstack, f is a closed immersion, and the composition of f with the projection Spec A × S X → Spec A is proper, flat, and locally almost of finite presentation. Let d Hilb
X/S : CAlg cn → d C at ∞ denote the functor classifying the Cartesian fibration C → (CAlg cn S ) op π op → (CAlg cn ) op (here d C at ∞ denotes the ∞ -category of (not necessarily small) ∞ -categories; see [13, 3.0.0.5]). LetHilb X/S : CAlg cn → b S be the functor given by the formula Hilb X/S ( A ) = d Hilb
X/S ( A ) ≃ , where d Hilb
X/S ( A ) ≃ denotes the largest Kan complex contained in d Hilb
X/S ( A ). Note that there is acanonical morphism of functors Hilb X/S → S . Theorem 4.9.
Let p : X → S be a morphism in Fun(CAlg cn , b S ) which is representable,separated, and locally almost of finite presentation. Then the canonical morphism Hilb
X/S → S is a relative spectral algebraic space which is locally almost of finite presentation.Proof. We will use the criterion for representability supplied by [14, 18.1.0.2]. The canonicalmorphism Hilb
X/S → S is infinitesimally cohesive and nilcomplete (see [14, 17.3.7.1]) byvirtue of [14, 16.3.0.1, 16.3.2.1] and [14, 19.4.1.2, 19.4.2.3], respectively.We next show that the morphism Hilb X/S → S admits a relative cotangent complex of [14,17.2.4.2]. We will prove this by verifying conditions ( a ) and ( b ) of [14, 17.2.4.3]. Let A bea connective E ∞ -ring and let η ∈ Hilb
X/S ( A ) be a point corresponding to a pair ( ζ , i : Y → X × S Spec A ), where ζ ∈ S ( A ) is a point and i : Y → X × S Spec A is a closed immersion ofspectral Deligne-Mumford stacks for which the composition f : Y → Spec A × S X → Spec A is proper, flat, and locally almost of finite presentation. Let F : Mod cn A → S be the functordefined by the formula F ( M ) = fib(Hilb X/S ( A ⊕ M ) → Hilb
X/S ( A ) × S ( A ) S ( A ⊕ M )) , where the fiber is taken over the point of Hilb X/S ( A ) × S ( A ) S ( A ⊕ M ) determined by η . Wewish to show that F is corepresented by an almost connective A -module. According to [14,19.4.3.1], the fiber is canonically equivalent to Map QCoh( Y ) ( L Y /X × S Spec A , Σ f ∗ M ). By virtueof [14, 6.4.5.3], f ∗ : QCoh(Spec A ) → QCoh( Y ) admits a left adjoint f + , so that F ( M ) is RAUER SPACES OF SPECTRAL ALGEBRAIC STACKS 17 corepresented by an A -module Σ − f + L Y /X × S Spec A . Since i is a closed immersion, it followsfrom [14, 17.1.4.3] that L Y /X × S Spec A is 1-connective, so the A -module Σ − f + L Y /X × S Spec A isconnective as desired (here we use the fact that f is flat). Condition ( b ) is an immediateconsequence of [14, 6.4.5.4]. We note that L Y /X × S Spec A is almost perfect because i is locallyalmost of finite presentation (see [14, 17.1.5.1]), so that Σ − f + L Y /X × S Spec A is almost perfectby virtue of [14, 6.4.5.2] and [17, 7.2.4.11]. We conclude that the relative cotangent complex L Hilb
X/S /S is not only connective, but also almost perfect.We now show that Hilb X/S → S is a relative spectral algebraic space. Since the formationof Hilbert functors is compatible with base change, we may assume that S is an affine spectralDeligne-Mumford stack. We wish to show that Hilb X/S is representable by a spectral algebraicspace. Let CAlg ♥ denote the ∞ -category of discrete E ∞ -rings, which can be identified withthe nerve of the category of commutative rings; see [17, 7.1.0.3]. The restriction of Hilb X/S toCAlg ♥ is equivalent to the ordinary Hilbert functor associated to the morphism of ordinaryalgebraic spaces τ ≤ X → τ ≤ S (here we use the fact that p is a relative spectral algebraicspace; see [14, 3.2.1.1]), which is representable by an ordinary algebraic space; see [2, 6.2].Since S is assumed to be representable, it admits a cotangent complex, infinitesimally cohe-sive, and nilcomplete by virtue of [14, 17.2.5.4, 17.3.1.2, 17.3.2.3]. Combining [14, 17.3.7.3]and [14, 17.3.9.1] with the above discussion, we deduce that Hilb X/S satisfies the hypothesisof [14, 18.1.0.2], and is therefore representable by a spectral algebraic space as desired.It remains to prove that the morphism Hilb
X/S → S is locally almost of finite presentation.We may assume that S is affine. Using [14, 19.4.2.3], we may further assume that S is 0-truncated. We have already seen that Hilb X/S → S is infinitesimally cohesive and admits arelative cotangent complex which is almost perfect. By virtue of [14, 17.4.2.2], it will sufficeto check condition ( ∗ ) of [14, 17.4.2.1]: for every filtered diagram { A α } of commutative ringshaving colimit A , the canonical mapcolim Hilb X/S ( A α ) → colim S ( A α ) × S ( A ) Hilb
X/S ( A )is an equivalence. Since the restrictions of Hilb X/S and Hilb τ ≤ X/S to CAlg ♥ are equivalent, wecan reduce to the case where p : X → S is a morphism of ordinary algebraic spaces, in whichcase the desired result follows from its classical counterpart (see [2, 6.2] and [4, 8.14.2]). (cid:3) Remark 4.10.
Let Hilb ´et
X/S ⊆ Hilb
X/S be the subfunctor which carries an E ∞ -ring A to thesummand of Hilb X/S ( A ) spanned by those pairs ( A, i : Y → Spec A × S X ) for which i is ´etale.Under the additional assumption that the morphism p : X → S is flat, a similar argumentshows that the canonical morphism Hilb ´et X/S → S is a relative spectral algebraic space whichis locally almost of finite presentation; moreover, it is ´etale by construction (here we use thefact that a morphism of spectral Deligne-Mumford stacks which is locally almost of finitepresentation is ´etale if and only if its relative cotangent complex vanishes; see [14, 17.1.5.1])and is separated by reducing to its classical counterpart (see [2, 6.1]). Our proof of 4.12 will make use of the notion of degree of fibers defined as follows:let f : X → Y be a representable flat, quasi-compact, separated, and locally quasi-finitemorphism in Fun(CAlg cn , b S ), where X and Y satisfy condition ( ∗ ) of 3.1. Suppose we are givena point η : Spec κ → Y which represents some y ∈ | Y | . The projection Spec κ × Y X → Spec κ ,which can be identified with a morphism of ordinary schemes, is finite flat of degree d for some d ≥
0; this integer does not depend on the choice of η . We therefore obtain a well-defined map n X/Y : | Y | → Z ≥ which carries y ∈ | Y | to the degree of finite flat morphismSpec κ × Y X → Spec κ determined by any point Spec κ → Y representing y .We are now ready to prove an analogue of [9, 4.1] in spectral algebraic geometry: Theorem 4.12.
Let X : CAlg cn → b S be a quasi-geometric spectral algebraic stack whichadmits a quasi-finite presentation (see ). Then there exist morphisms of quasi-geometricspectral algebraic stacks p : W → X and q : V → W such that p is a separated Nisnevichcovering, V is a quasi-affine spectral Deligne-Mumford stack, and q is representable finite,faithfully flat, and locally almost of finite presentation.Proof. Using our assumption that X admits a quasi-finite presentation, we can choose aconnective E ∞ -ring A and a morphism f : Spec A → X which is locally quasi-finite, faithfullyflat, and locally almost of finite presentation. Choose a fiber smooth surjection g : Y → X ,where Y is an affine spectral Deligne-Mumford stack. Consider a pullback square of quasi-geometric spectral algebraic stacks Y ′ g ′ / / f ′ (cid:15) (cid:15) Spec A f (cid:15) (cid:15) Y g / / X. The underlying morphism τ ≤ f ′ : τ ≤ Y ′ → τ ≤ Y of ordinary of algebraic spaces is quasi-affine, locally quasi-finite, faithfully flat, and locally of finite presentation. Combining [18,Tag 07RZ] with [18, Tag 03JA], we deduce that there exists a sequence of quasi-compact openimmersions of spectral algebraic spaces ∅ ≃ V n +1 ֒ → · · · ֒ → V ≃ Y with the following properties:(i) For every 0 ≤ i ≤ n + 1, we have | V i | = { y ∈ | Y | : n Y ′ /Y ( y ) ≥ i } (see 4.11).(ii) For each 0 ≤ i ≤ n , let K i denote the reduced closed substack of V i complementary to V i +1 (see [14, 3.1.6.3]). Then the projection K i × Y Y ′ → K i is finite flat of degree i .Since g is flat and locally almost of finite presentation, 3.10 guarantees that for each i , theimage of | V i | under | g | is quasi-compact open, and therefore gives rise to an open immersion U i → X of quasi-geometric spectral algebraic stacks (see 3.12). We claim that the sequenceof open immersions of quasi-geometric spectral algebraic stacks ∅ ≃ U n +1 ֒ → · · · ֒ → U ≃ X gives a stacky scallop decomposition of X . Let K ′ i denote the reduced closed substack of U i complementary to U i +1 ; see 3.16. Using the description of | V i | , we see that the canonicalmorphism V i → U i × X Y is an equivalence. In particular, the induced morphism K i → K ′ i canbe identified with a pullback of V i → U i , and therefore is a flat covering of [14, 2.8.3.1]. It thenfollows from [14, 5.2.3.5] that the projection K ′ i × X Spec A → K ′ i is finite flat of degree i , sothat the identity morphism on K ′ i × X Spec A induces a factorization of the immersion K ′ i → X through the subfunctor Hilb ´etSpec A/X ⊆ Hilb
Spec
A/X of 4.10. Using 3.8, we can choose a quasi-compact open subset W ′ ⊆ | Hilb ´etSpec
A/X | which contains the image of | K ′ i | in | Hilb ´etSpec
A/X | RAUER SPACES OF SPECTRAL ALGEBRAIC STACKS 19 for all i . Let j : W ′ → Hilb ´etSpec
A/X be a representable open immersion such that | j || W ′ | = W ′ (see the proof of 3.12). Let p ′ denote the composition W ′ → Hilb ´etSpec
A/X → X . Note that4.10 guarantees that p ′ is representable. Combining this observation with [14, 6.3.3.8], wesee that W ′ is a sheaf for the fpqc topology. Since the projection Y × X W ′ → W ′ is a fibersmooth surjection and | W ′ | is quasi-compact, 3.8 (and its proof) supplies a fiber smoothsurjection π : Spec B → W ′ , where B is a connective E ∞ -ring. Note that π is quasi-affinebecause p ′ is separated and the composition p ′ ◦ π is quasi-affine (here we use the fact thatthe diagonal of X is quasi-affine), so that the diagonal of W ′ is representable quasi-affineby virtue of [14, 9.1.1.2]. Consequently, we conclude that W ′ is a quasi-geometric spectralalgebraic stack. In particular, p ′ is quasi-compact. Combining this observation with the factthat p ′ is ´etale and separated (see 4.10), we deduce that it is quasi-affine by virtue of [14,3.3.0.2]. Let i : V → W ′ × X Spec A denote the clopen immersion of quasi-geometric spectralalgebraic stacks (see [14, 3.1.7.2]) determined by the inclusion W ′ → Hilb ´etSpec
A/X . Let q ′ denote the composition of i with the projection W ′ × X Spec A → W ′ . Since q ′ is flat andlocally almost of finite presentation, 3.10 guarantees that the image of | V | under | q ′ | is quasi-compact open, and therefore induces an open immersion W → W ′ of quasi-geometric spectralalgebraic stacks by virtue of 3.12. Shrinking | W ′ | to the image of | q ′ | , we obtain a surjection q : V → W of quasi-geometric spectral algebraic stacks. Combining the fact that i is clopenimmersion with [14, 21.4.6.4] and [14, 2.5.7.4], we observe that q is flat, locally almost of finitepresentation, and quasi-affine. Since q ′ is proper, q is also proper, hence finite by virtue of [14,5.2.1.1]. Invoking quasi-affineness of p ′ , we conclude that V is representable by a quasi-affineDeligne-Mumford stack. Let p denote the composition of the inclusion W ⊆ W ′ with p ′ . Wewill complete the proof by showing that p is a separated Nisnevich covering. Since each K ′ i factors through W ′ , it will suffice to show that the image of | K ′ i | in | W ′ | is contained in | q ′ || V | .By construction, the projection K ′ i × X Spec A → K ′ i is a pullback of q ′ , so the desired resultfollows by combining this observation with the fact that f is surjective. (cid:3) Twisted Compact Generations
In this section, we prove that quasi-geometric spectral algebraic stacks which admit a quasi-finite presentation are of twisted compact generation. This result will play a central role inour proof of 1.2.
To formulate the main definition of interest to us in this section (that is, 1.7), we recalla bit of terminology. According to [13, 5.5.7.1], an ∞ -category C is compactly generated if itis presentable and ω -accessible, or equivalently if the inclusion Ind( C ω ) → C is an equivalenceof ∞ -categories, where C ω ⊆ C is the full subcategory spanned by the compact objects of C (here Ind( C ω ) denotes the ∞ -category of Ind-objects of C ω ; see [13, 5.3.5.1]).Now let C be a presentable stable ∞ -category, and let { C i } i ∈ I be a collection of compactobjects of C . We say that the collection { C i } is a set of compact generators for C if it satisfiesthe following condition: an object C ∈ C is equivalent to 0 if the graded abelian groupExt ∗ C ( C i , C ) is zero for all i ∈ I . Note that if C is compactly generated, the collection ofcompact objects of C forms a set of compact generators. Suppose we are given an adjunction L : C / / D : R o o between presentable ∞ -categories,where the right adjoint R is conservative and preserves small filtered colimits. It follows from [16, 6.2] that if C is compactly generated, then so is D . Note that in the special case where C and D are presentable stable ∞ -categories, if { C i } i ∈ I is a set of compact generators for C ,then { L ( C i ) } is a set of compact generators for D . Let X : CAlg cn → b S be a functor and let QCoh( X ) denote the ∞ -category of quasi-coherent sheaves on X of [14, 6.2.2.1]. More informally, we can think of an object F ∈ QCoh( X ) as a rule which assigns to each connective E ∞ -ring R and each point η ∈ X ( R ) an R -module F η ∈ Mod R , which depends functorially on R and η (see [14, 6.2.2.7]). According to[14, 6.2.6], the ∞ -category QCoh( X ) can be equipped with a symmetric monoidal structure,where the tensor product is given informally by the formula ( F ⊗ F ′ ) η ≃ F η ⊗ R F ′ η for eachpoint η ∈ X ( R ); let O X denote the unit object of QCoh( X ). In the special case where X is representable by an ordinary Deligne-Mumford stack ( X , O ), let D( X ) denote the derived ∞ -category of the Grothendieck abelian category Mod O of O -modules; see [17, 1.3.5.8]. Itthen follows from [14, 2.2.6.2] that there is a canonical equivalence QCoh( X ) ≃ D qc ( X ), whereD qc ( X ) ⊆ D( X ) is the full subcategory spanned by those chain complexes of O -modules whosehomologies are quasi-coherent. Remark 5.4. If X ′ : CAlg ∆ R → b S is a functor, we define the ∞ -category QCoh( X ′ ) ofquasi-coherent sheaves on X ′ to be the ∞ -category QCoh(Θ R ! X ′ ) (here we regard Θ R ! X ′ asan object of the slice ∞ -category Fun(CAlg cn , b S ) /R ). A slight variant of [14, 6.2.3.1] (usingthe fiber smooth topology in place of the fpqc topology) guarantees that for every quasi-geometric derived algebraic stack X over R , the canonical map QCoh( X ◦ ) → QCoh( X ) is anequivalence of ∞ -categories (here X ◦ is regarded as an object of Fun(CAlg cn , b S ) /R ). In [14], Lurie develops the theory of quasi-coherent stacks, which plays an analogousrole of the categories of twisted sheaves in the setting of spectral algebraic geometry. Inthis analogy, the ∞ -category of global sections of quasi-coherent stacks is an analogue of thederived category of twisted sheaves. We now give a quick review of some basic definitionsand notations. Let LinCat PSt denote the ∞ -category whose objects are pairs ( R, C ), where R is a connective E ∞ -ring and C is a prestable R -linear ∞ -category of [14, D.1.4.1]. LetQStk PSt : Fun(CAlg cn , b S ) op → d C at ∞ denote the functor obtained by applying [14, 6.2.1.11]to the projection q : LinCat PSt → CAlg cn . Let X : CAlg cn → b S be a functor. We referto QStk PSt ( X ) as the ∞ -category of prestable quasi-coherent stacks on X ; see [14, 10.1.2.4].More informally, an object C ∈ QStk
PSt ( X ) is a rule which assigns to each connective E ∞ -ring R and each point η ∈ X ( R ) a prestable R -linear ∞ -category C η , depending functorially onthe pair ( R, η ) (see [14, 10.1.1.3] for more details).
Let QStk
PSt R : Fun(CAlg cn R , b S ) op → d C at ∞ denote the functor obtained by applying [14,6.2.1.11] to the projection q R : CAlg cn R × CAlg cn LinCat
PSt → CAlg cn R . Note that if X is animage of X ′ under the equivalence Fun(CAlg cn R , b S ) → Fun(CAlg cn , b S ) /R of 2.7, the canonicalmap QStk PSt ( X ) → QStk
PSt R ( X ′ ) is an equivalence of ∞ -categories by construction. Remark 5.7.
Let QStk ′ PSt R : Fun(CAlg ∆ R , b S ) op → d C at ∞ denote the functor obtained by apply-ing [14, 6.2.1.11] to the projection q ′ R : CAlg ∆ R × CAlg cn LinCat
PSt → CAlg ∆ R . By construction,the canonical map QStk PSt R ◦ Θ op R ! → QStk ′ PSt R is an equivalence of functors. According to [14,D.4.1.6], the functor χ R : CAlg cn R → d C at ∞ classifying the coCartesian fibration q R is a sheaffor the flat universal descent topology of [14, D.4.1.4], and therefore also a sheaf for the fiber RAUER SPACES OF SPECTRAL ALGEBRAIC STACKS 21 smooth topology of 2.12 by virtue of [14, 11.2.3.3]. Using [14, 1.3.1.7], we deduce that thecanonical map QStk
PSt R ◦ L opfsm → QStk
PSt R is an equivalence of functors from Fun(CAlg cn R , b S ) op to d C at ∞ . We conclude that for each quasi-geometric derived algebraic stack X over R , thecanonical map QStk PSt R ( X ◦ ) → QStk ′ PSt R ( X ) is an equivalence of ∞ -categories. Remark 5.8.
Our choice of the fiber smooth topology over the fpqc topology for the un-derlying quasi-geometric spectral algebraic stacks of quasi-geometric derived algebraic stacks(see 2.30) is motivated by 5.7. Indeed, we do not know if the functor χ R satisfies fpqc descent. Let X be a quasi-geometric stack which satisfies the following condition:( ∗ ) There exists a morphism of quasi-geometric stacks X → X which is faithfully flatand locally almost of finite presentation, where X is affine.Let Groth ∞ denote the ∞ -category of Grothendieck prestable ∞ -categories (see [14, C.3.0.5]).Let S denote the sphere spectrum and let q : X → Spec S be the projection. It follows from[14, 10.4.1.1] that the pullback functor q ∗ : Groth ∞ → QStk
PSt ( X ) induced by q (note thatthere is a canonical equivalence QStk PSt (Spec S ) ≃ Groth ∞ ) admits a right adjointQCoh( X ; • ) : QStk PSt ( X ) → Groth ∞ which we refer to as the global section functor on X . For each prestable quasi-coherent stack C on X , we refer to QCoh( X ; C ) as the ∞ -category of global sections of C . Remark 5.10.
Let f : Y → X be a morphism between quasi-geometric stacks satisfyingcondition ( ∗ ) of 5.9. Assume that f is representable faithfully flat and locally almost of finitepresentation. Applying the argument of [14, 10.4.1.4] to f , we deduce that the canonicalmorphism QCoh( X ; C ) → lim [ n ] ∈ ∆ QCoh( Y n ; f ∗ n C ) is an equivalence in Groth ∞ (here Y • denotesthe ˇCech nerve of f and each f n : Y n → X denotes the projection). Remark 5.11.
Let ∆ s denote the subcategory of ∆ having the same objects but the mor-phisms are given by injective order-preserving maps (see [13, 6.5.3.6]). Let Groth lex ∞ denotethe subcategory of d C at ∞ whose objects are Grothendieck prestable ∞ -categories and whosemorphisms are functors preserving small colimits and finite limits; see [14, C.3.2.3]. By virtueof [14, C.3.2.4], Groth lex ∞ admits small limits and the inclusion Groth lex ∞ ⊆ Groth ∞ preservessmall limits. In the situation of 5.10, the existence of the limit lim [ n ] ∈ ∆ QCoh( Y n ; f ∗ n C ) is suppliedby the right cofinality of the inclusion ∆ s ⊆ ∆ (see [13, 6.5.3.7]) and our assumption that f isflat. Indeed, the limit is given by lim [ n ] ∈ ∆ s QCoh( Y n ; f ∗ n C ), where the flatness assumption guar-antees that the construction [ n ] QCoh( Y n ; f ∗ n C ) determines a functor ∆ s → Groth lex ∞ (cf.[14, 10.1.7.10]). Let P r L denote the subcategory of d C at ∞ whose objects are presentable ∞ -categories and whose morphisms are functors which preserve small colimits; see [13, 5.5.3.1].In the special case where C is stable, combining [14, 10.3.1.8] with [17, 4.8.2.18] (which assertsthat the full subcategory Pr St ⊆ P r L spanned by the presentable stable ∞ -categories is closedunder small limits), we deduce that QCoh( X ; C ) is stable. Let f : X → Y be a representable morphism between quasi-geometric stacks satisfyingcondition ( ∗ ) of 5.9. It follows from [14, 10.1.4.1] that the pullback functor f ∗ : QStk PSt ( Y ) → QStk
PSt ( X ) admits a right adjoint f ∗ . Let C be a prestable quasi-coherent stack on Y . Applying the global section functor on Y (see 5.9) to the unit morphism C → f ∗ f ∗ C , weobtain a functor QCoh( Y ; C ) → QCoh( X ; f ∗ C ) which we refer to as the pullback along f anddenote by f ∗ (see [14, 10.1.7.5]).Let j : U → X be an open immersion of quasi-geometric stacks satisfying condition ( ∗ ) of5.9. For each object C ∈ QStk
PSt ( X ), we let QCoh X − U ( X ; C ) ⊆ QCoh( X ; C ) denote the fullsubcategory spanned by those objects M such that j ∗ M ∈ QCoh( U ; j ∗ C ) is equivalent to 0. Let X be a quasi-geometric stack satisfying condition ( ∗ ) of 5.9. Let QStk St ( X ) ⊆ QStk
PSt ( X ) denote the full subcategory spanned by the stable quasi-coherent stacks (see[14, 10.1.2.1]). Let Q X denote the unit object of QStk St ( X ) (with respect to the symmetricmonoidal structure described in [14, 10.1.6.4]). More informally, it assigns to each point η ∈ X ( R ) the stable R -linear ∞ -category Mod R . Note that Q X is compactly generated(see [17, 7.2.4.2]) and that there is a canonical equivalence of ∞ -categories QCoh( X ; Q X ) ≃ QCoh( X ). More generally, let j : U → X be a representable open immersion. Repeatingthe argument of [14, 10.1.7.3], we obtain a stable quasi-coherent stack Q X − U on X , whichis given informally by the formula ( Q X − U ) η = Mod Nil( I η ) R for each point η ∈ X ( R ) (here I η ⊆ π R is a finitely generated ideal whose vanishing locus is complementary to the opensubset | Spec R × X U | ⊆ | Spec R | and Mod Nil( I η ) R ⊆ Mod R denotes the full subcategory spannedby the I η -nilpotent objects of [14, 7.1.1.6]). For each stable quasi-coherent stack C on X , let C X − U denote the tensor product C ⊗ Q X − U in the symmetric monoidal ∞ -category QStk St ( X ).For an alternative description, let η ∈ X ( R ) be a point and let ( C η ) Nil( I η ) ⊆ C η denote thefull subcategory spanned by the I η -nilpotent objects. It then follows from [14, 7.1.2.11](and its proof) that C X − U is equivalent to the stable quasi-coherent stack determined by theconstruction ( η ∈ X ( R )) ( C η ) Nil( I η ) ; in particular, [14, 7.1.1.12] guarantees that if C iscompactly generated, then so is C X − U . We have the following observation: Lemma 5.14.
Let j : U → X be a representable open immersion of quasi-geometric stackssatisfying condition ( ∗ ) of . Let C be a stable quasi-coherent stack on X and let C X − U be as in . Then the canonical morphism QCoh( X ; C X − U ) → QCoh( X ; C ) induces anequivalence of ∞ -categories QCoh( X ; C X − U ) → QCoh X − U ( X ; C ) (see ).Proof. By virtue of 5.10, we are reduced to the case where X is a quasi-geometric spectralDeligne-Mumford stack. Using the proof of [14, 10.1.4.1], we can reduce further to the casewhere X is affine, in which case the desired result follows immediately from the definition ofthe ∞ -category C Nil( I ) appearing in [14, 7.1.1.6]. (cid:3) The following pair of results asserts that if a quasi-geometric spectral algebraic stackis of twisted compact generation, then it satisfies a spectral analogue of the β -Thomasoncondition of [8, 8.1] for some regular cardinal β (see 1.8): Lemma 5.16.
Let X be a quasi-geometric spectral algebraic stack which is of twisted compactgeneration. Then for each representable open immersion j : U → X of quasi-geometricspectral algebraic stacks, the ∞ -category QCoh X − U ( X ) is compactly generated. In particular, QCoh( X ) is compactly generated.Proof. It follows from [14, 7.1.1.12] that Q X − U is compactly generated and stable, so thedesired result follows immediately by applying 5.14 to C = Q X (cid:3) RAUER SPACES OF SPECTRAL ALGEBRAIC STACKS 23
Before stating our next result, we introduce some terminology. Let X : CAlg cn → b S be a functor which satisfies condition ( ∗ ) of 3.1. Suppose we are given a perfect object F ofQCoh( X ). Let Supp F denote the subset of | X | consisting of those elements x ∈ | X | such thatfor any point η : Spec κ → X representing x , η ∗ F = 0. This set is well-defined; we refer to itas the support of F . In the special case where X is representable by a quasi-separated spectralalgebraic space, it follows from [14, 7.1.5.5] that our definition of support is compatible withthe definition of support in the sense of [14, 7.1.5.4]. Lemma 5.18.
Let X be a quasi-geometric spectral algebraic stack which is of twisted compactgeneration. Then for each quasi-compact open subset U ⊆ | X | , there exists a compact object F of QCoh( X ) with support | X | − U .Proof. U and a rep-resentable open immersion j : U → X such that | j | ( | U | ) = U . Then 5.16 supplies aset of compact generators { F i } i ∈ I for QCoh X − U ( X ). Using [14, 6.3.4.1], we observe that j ∗ : QCoh( X ) → QCoh( U ) admits a fully faithful right adjoint j ∗ . It then follows from [14,7.2.1.7] that the pair of subcategories (QCoh X − U ( X ) , QCoh( U )) determine a semi-orthogonaldecomposition of QCoh( X ) (see [14, 7.2.0.1]). Using [14, 7.2.1.4], we see that the inclusionQCoh X − U ( X ) ⊆ QCoh( X ) admits a right adjoint which we denote by R . According to [14,7.2.0.2], there is a fiber sequence R ( P ) → P → j ∗ j ∗ P for each P ∈ QCoh( X ). Since j ∗ pre-serves small colimits [14, 6.3.4.3] (and QCoh( X ) is stable), R preserves filtered colimits, andtherefore each F i is also compact as an object of QCoh( X ) (see [13, 5.5.7.2]). In particular,every F i ∈ QCoh( X ) is perfect by virtue of [14, 9.1.5.2]. We now proceed as in the proof of[8, 4.10]. We first claim that the | X | − U can be identified with the union of the supports { Supp F i } . For this, it suffices to prove that | X | − U belongs to the union of { Supp F i } . Let x ∈ | X | − U and choose a point η : Spec κ → X representing x . Then [14, 6.3.4.1] guaranteesthat η ∗ O Spec κ ∈ QCoh X − U ( X ). Since { F i } is a set compact generators for QCoh X − U ( X )and η ∗ O Spec κ is nonzero, we conclude that η ∗ F i is not equivalent to 0 for some i , so that x ∈ Supp F i as desired. To complete the proof, it will suffice to show that | X | − U can becovered by finitely many supports Supp F i . Choose a fiber smooth surjection f : X → X ,where X is affine. Replacing X by X , we are reduced to the case where X is affine. In thiscase, [14, 3.6.3.4] guarantees that | X | is a coherent topological space (see also 3.2). Using [14,7.1.5.5], we see that each Supp F i is a closed subset of | X | complementary to a quasi-compactopen subset. Consequently, every Supp F i and | X | − U are constructible, so that the desiredresult follows by using the constructible topology on | X | of [14, 4.3.1.5]. (cid:3) It is a tautology that the class of quasi-geometric spectral algebraic stacks which are oftwisted compact generation includes the basic building blocks of spectral algebraic geometry:
Lemma 5.20.
Let X be an affine spectral Deligne-Mumford stack. Then X is of twistedcompact generation. Our goal in this section is to prove that quasi-geometric spectral algebraic stackswhich admit a quasi-finite presentation are of twisted compact generation 1.10. We begin byestablishing some preliminaries.
Lemma 5.22.
Suppose we are given a pullback diagram of quasi-geometric stacks satisfyingcondition ( ∗ ) of : X ′ f ′ / / g ′ (cid:15) (cid:15) Y ′ g (cid:15) (cid:15) X f / / Y. Let C be a prestable quasi-coherent stack on Y . If f is a relative spectral algebraic space whichis quasi-compact quasi-separated and g is representable flat, then the commutative diagram of ∞ -categories QCoh( Y ; C ) f ∗ / / g ∗ (cid:15) (cid:15) QCoh( X ; f ∗ C ) g ′∗ (cid:15) (cid:15) QCoh( Y ′ ; g ∗ C ) f ′∗ / / QCoh( X ′ ; g ′∗ f ∗ C ) is right adjointable.Proof. This follows by combining [14, 10.1.7.9] with an extension of [14, 10.1.7.13] to quasi-geometric stacks satisfying condition ( ∗ ) of 5.9 (which can be achieved by using [17, 4.7.4.18]). (cid:3) Lemma 5.23.
Let f : X → Y be a relative spectral algebraic space which is quasi-compactquasi-separated morphism of quasi-geometric stacks satisfying condition ( ∗ ) of . Let C be a prestable quasi-coherent stack on Y . Then the pullback functor f ∗ : QCoh( Y ; C ) → QCoh( X ; f ∗ C ) is compact.Proof. By virtue of 5.10 and 5.11, we can reduce to the case where X and Y are representableby spectral Deligne-Mumford stacks, in which case the desired result follows by combining[14, 10.1.7.15] with [14, 10.3.1.13]. (cid:3) As a first step towards the proof of 1.10, we show that the property of being of twistedcompact generation behaves well with respect to open immersions:
Proposition 5.25.
Let j : U → X be an open immersion of quasi-geometric spectral algebraicstacks. If X is of twisted compact generation, then so is U .Proof. Let C be a compactly generated stable quasi-coherent stack on U ; we wish to show thatQCoh( U ; C ) is compactly generated. It follows from 5.22 and 5.23 that we have an adjunction j ∗ : QCoh( X ; j ∗ C ) / / QCoh( U ; C ) : j ∗ , o o where j ∗ is conservative and preserves small filtered colimits. Using [14, 10.3.2.3] and [14,10.3.1.7] (see also [14, 10.1.4.1]), we see that j ∗ C ∈ QStk
PSt ( X ) is compactly generated andstable. Since X is of twisted compact generation, the desired result follows from [16, 6.2]. (cid:3) Our proof of 1.10 will require two descent results about the property of being of twistedcompact generation. Let us begin with the descent along finite morphisms. In what follows,we regard P r L as equipped with the symmetric monoidal structure described in [17, 4.8.1.15].Note that P r St inherits a symmetric monoidal structure from P r L (see [17, 4.8.2.18]). For thefirst descent result, we need the following stable version of [14, 10.2.4.2]: RAUER SPACES OF SPECTRAL ALGEBRAIC STACKS 25
Lemma 5.27.
Let f : X → Y be a representable quasi-affine morphism of quasi-geometricstacks satisfying condition ( ∗ ) of , and let C be a stable quasi-coherent stack on Y . Thenthe canonical morphism QCoh( X ) ⊗ QCoh( Y ) QCoh( Y ; C ) → QCoh( X ; f ∗ C ) is an equivalence of presentable stable ∞ -categories.Proof. We first claim that the functor Mod
QCoh( Y ) ( P r St ) → P r St which carries a presentablestable ∞ -category D equipped with an action of QCoh( Y ) to the presentable stable ∞ -category QCoh( X ) ⊗ QCoh( Y ) D preserves small limits. For this, it will suffice to show thatQCoh( X ) is dualizable when viewed as an object of Mod QCoh( Y ) ( P r St ). Let A ∈ CAlg(QCoh( Y ))denote the pushforward of the structure sheaf of X (that is, the unit object of QCoh( X ))along f . Then [14, 6.3.4.6] supplies an equivalence QCoh( X ) → Mod A (QCoh( Y )), so the de-sired assertion follows from [17, 4.8.4.8]. Combining this observation with 5.10 (see also[14, 6.3.4.7]), we are reduced to the case where X and Y are representable by spectralDeligne-Mumford stacks X and Y , respectively. In this case, it follows from [14, 10.2.1.1] thatLMod A (QCoh( Y ; C )) can be identified with the stabilization of LMod A (QCoh( Y ; C )) ≥ (see[14, 10.2.1.1] for the t-structure). By virtue of the equivalence QCoh( X ) ≃ Mod A (QCoh( Y ))and [17, 4.8.4.6], it can also be identified with QCoh( X ) ⊗ QCoh( Y ) QCoh( Y ; C ). Invokingour assumption that f is quasi-affine, we obtain an equivalence LMod A (QCoh( Y ; C )) ≥ ≃ QCoh( X , f ∗ C ) by combining [14, 10.2.1.3] and [14, 10.2.4.2]; the desired equivalence nowfollows by passing to the stabilization. (cid:3) Proposition 5.28.
Let f : X → Y be a morphism of quasi-geometric spectral algebraic stackswhich is representable, finite, faithfully flat, and locally almost of finite presentation. If X isof twisted compact generation, then so is Y .Proof. Let C be a compactly generated stable quasi-coherent stack on Y ; we wish to showthat QCoh( Y ; C ) is compactly generated. We first show that the pullback functor f ∗ :QCoh( Y ; C ) → QCoh( X ; f ∗ C ) admits a left adjoint. Since it preserves small colimits (see5.22), it will suffice to prove that it preserves small limits by virtue of the adjoint functor the-orem [13, 5.5.2.9]. Let A ∈ CAlg(QCoh( Y )) denote the pushforward f ∗ O X . Combining 5.27with [14, 6.3.4.6] and [17, 4.8.4.6], we can identify QCoh( X ; f ∗ C ) with LMod A (QCoh( Y ; C )),under which f ∗ corresponds to the functor QCoh( Y ; C ) → LMod A (QCoh( Y ; C )) given bytensor product with A . Since the forgetful functor LMod A (QCoh( Y ; C )) → QCoh( Y ; C ) isconservative [17, 4.2.3.2] and preserves small limits [17, 4.2.3.3], it is enough to prove that A is dualizable as an object of QCoh( Y ). Invoking our assumption on f (and using [14,6.3.4.1]), we deduce from [14, 6.1.3.2] that A is perfect, hence dualizable as desired (see [14,6.2.6.2]). Now let f + denote a left adjoint to f ∗ . Since X is of twisted compact generationand f ∗ is conservative (by virtue of 5.10), the desired compact generation follows immediatelyby applying [16, 6.2] to the adjoint pair ( f + , f ∗ ). (cid:3) We next show that the property of being a twisted compact generation satisfies descentfor excision squares. For this purpose, the Thomason–Neeman localization theorem in thesetting of ∞ -categories by Adeel A. Khan (see [10, 2.11]) will play an essential role. Ofgreatest interest to us is the case of semi-orthogonal decompositions [14, 7.2.0.1]: Lemma 5.30.
Let C be a compactly generated presentable stable ∞ -category and let ( C + , C − ) be a semi-orthogonal decomposition of C for which the subcategories C + , C − are compactlygenerated. Suppose that the inclusion functor ι : C − → C preserves small filtered colimits.Let L denote a left adjoint to the inclusion ι (which exists by virtue of [14, 7.2.1.7] ). Let D be a compact object of C − . Then there exists a compact object C of C and an equivalence L ( C ) ≃ D ⊕ D [1] in the ∞ -category C − .Proof. By virtue of [14, 7.2.1.4], the inclusion functor C + → C admits a right adjoint, whichwe denote by R . According to [14, 7.2.0.2], for each object C ∈ C , there is a fiber sequence R ( C ) → C → L ( C ). Combing this with our assumption on ι , we see that R preserves smallfiltered colimits. The desired result now follows immediately from [10, 2.11]. (cid:3) The following “Nisnevich excision” result will be useful in the proof of 5.33:
Lemma 5.32.
Suppose we are given an excision square of quasi-geometric spectral algebraicstacks: U ′ j ′ / / f ′ (cid:15) (cid:15) X ′ f (cid:15) (cid:15) U j / / X. Let C be a prestable quasi-coherent stack on X . If f is quasi-affine, then the commutativediagram of ∞ -categories QCoh( X ; C ) / / (cid:15) (cid:15) QCoh( U ; j ∗ C ) (cid:15) (cid:15) QCoh( X ′ ; f ∗ C ) / / QCoh( U ′ ; j ′∗ f ∗ C ) is a pullback square in Groth ∞ .Proof. We can use 5.10 to reduce to the case where X and Y are quasi-affine spectral Deligne-Mumford stacks. In this case, the desired result follows from [14, 10.2.3.1] and [14, 10.2.4.2]. (cid:3) Proposition 5.33.
Suppose we are given an excision square of quasi-geometric spectral al-gebraic stacks U ′ j ′ / / f ′ (cid:15) (cid:15) X ′ f (cid:15) (cid:15) U j / / X, where f is quasi-affine. If X ′ and U are of twisted compact generation, then so is X .Proof. Let C ∈ QStk
PSt ( X ) be compactly generated stable; we wish to show that QCoh( X ; C )is compactly generated. For this, we proceed as in the proofs of [16, 6.20] and [1, 6.13]. Wefirst claim that the collection of objects of the form j ∗ M , where M is a compact object ofQCoh( X ; C ), is a set of compact generators for QCoh( U ; j ∗ C ) (note that j ∗ M is compactby virtue of 5.23). Using the compact generation of QCoh( U ; j ∗ C ) (because U is of twistedcompact generation), it suffices to prove that for each compact object N ∈ QCoh( U ; j ∗ C ), RAUER SPACES OF SPECTRAL ALGEBRAIC STACKS 27 there exists a compact object M ∈ QCoh( X ; C ) such that j ∗ M ≃ N ⊕ N [1]. By virtueof 5.23 and 5.32, we observe that an object M ∈ QCoh( X ; C ) is compact if and only ifboth j ∗ M and f ∗ M are compact; consequently, in order to lift N ⊕ N [1] to a compactobject of QCoh( X ; C ), it suffices to lift f ′∗ ( N ⊕ N [1]) to a compact object of QCoh( X ′ ; f ∗ C ).Since j ′∗ admits a fully faithful right adjoint j ′∗ X ′ − U ′ ( X ′ ; f ∗ C ) , QCoh( U ′ ; j ′∗ f ∗ C )) determine a semi-orthogonaldecomposition of QCoh( X ′ ; f ∗ C ). Combining the assumption that X ′ is of twisted compactgeneration with 5.14 and 5.25, we deduce that QCoh( X ′ ; f ∗ C ) and the two subcategories arecompactly generated. Since j ′∗ preserves small filtered colimits 5.23, the desired lifting followsfrom the Thomason–Neeman localization theorem [10, 2.11] (see 5.30).On the other hand, 5.32 guarantees that the pullback functor f ∗ induces an equivalence of ∞ -categories QCoh X − U ( X ; C ) → QCoh X ′ − U ′ ( X ′ ; f ∗ C ), so that QCoh X − U ( X ; C ) is also com-pactly generated. Let { M ′ α } α ∈ A be the collection of compact objects of QCoh X − U ( X ; C ). Asin the case of QCoh( X ′ ; f ∗ C ), the pair (QCoh X − U ( X ; C ) , QCoh( U ; j ∗ C )) is a semi-orthogonaldecomposition of QCoh( X ; C ). Since j ∗ preserves small filtered colimits, it follows from theproof of 5.30 that the inclusion QCoh X − U ( X ; C ) ⊆ QCoh( X ; C ) admits a right adjoint R which preserves small filtered colimits, and therefore each M ′ α ∈ QCoh X − U ( X ; C ) is compactas an object of QCoh( X ; C ) as well. To complete the proof, we will show that the collec-tion of objects of the form M ⊕ M ′ α ∈ QCoh( X ; C ), where M ∈ QCoh( X ; C ) is compact,is a set of compact generators for QCoh( X ; C ): let M ′′ ∈ QCoh( X ; C ) and suppose thatExt ∗ QCoh( X ; C ) ( M ⊕ M ′ α , M ′′ ) = 0 for each compact object M ∈ QCoh( X ; C ) and each M ′ α .Since the collection { M ′ α } is a set of compact generators for QCoh X − U ( X ; C ), we see that RM ′′ ≃
0. It then follows from the fiber sequence RM ′′ → M ′′ → j ∗ j ∗ M ′′ (see [14, 7.2.0.2])that the unit map M ′′ → j ∗ j ∗ M ′′ is an equivalence. Combining this with the above discussionof the collection { j ∗ M } , we deduce that j ∗ M ′′ ≃
0, hence M ′′ ≃ (cid:3) We now provide a proof of our main result of this section. Our basic strategy isanalogous to the proof of [8, Theorem A].
Proof of . In view of the special presentation of 4.12 and the stacky scallop decompositioninduced by p (see 4.7), we can use the descent results 5.28 and 5.33 (note that each morphism W m → U m appearing in the proof of 4.7 is quasi-affine by virtue of the separatedness of p and[14, 3.3.0.2]) along with 5.25 to reduce to the case of quasi-affine spectral Deligne-Mumfordstacks, in which case the desired result follows from 5.20 and 5.25 (see also [14, 2.4.2.3]). (cid:3) We close this section by mentioning another consequence of the special presentation of4.12 and the descent results 5.28 and 5.33. According to [14, D.5.3.1], the property of beinga compactly generated prestable R -linear ∞ -category is local for the ´etale topology, where R is a connective E ∞ -ring. In the stable case, we have the following stronger assertion, whoseproof is immediate from the proofs of 5.28 and 5.33 by considering 4.12 and 4.7: Theorem 5.36.
Let X be a quasi-geometric spectral algebraic stack and let C be a stablequasi-coherent stack on X . If there exists a locally quasi-finite, faithfully flat, and locallyalmost of finite presentation of quasi-geometric spectral algebraic stacks f : Spec A → X forwhich f ∗ C is compactly generated, then C is compactly generated. Brauer Spaces and Azumaya Algebras
Our goal in this section is to give a proof of our main result 1.2, and to provide an explicitdescription of the homotopy groups of the extended Brauer sheaf of a quasi-geometric stack.
Let R be a connective E ∞ -ring. According to [14, 11.5.3.1], an E -algebra A over R iscalled the Azumaya algebra over R if it is a compact generator of Mod R and the natural map A ⊗ R A rev → End R ( A ) induced by the left and right actions of A on itself is an equivalence.In the setting of derived algebraic geometry, the property of being a derived Azumaya algebra(see [19, 2.1]) is local for the flat topology [19, 2.3]. We have the following spectral analogue,which can be proven by exactly the same argument: Lemma 6.2.
Let R be a connective E ∞ -ring and let A be an E -algebra over R . Then thecondition that A is an Azumaya algebra is stable under base change [14, 6.2.5.1] and local forthe flat topology [14, 2.8.4.1] . Before giving our proof of the main result 1.2, we need to recall a bit of terminology.Let X : CAlg cn → b S be a functor. Let QStk cg ( X ) denote the subcategory of QStk St ( X )whose objects are compactly generated stable quasi-coherent stacks and whose morphismsare compact morphisms of quasi-coherent stacks of [14, 10.1.3.1]. Note that it inherits asymmetric monoidal structure from QStk St ( X ); see [14, 11.4.0.1]. According to [14, 11.5.2.1],the extended Brauer space B r † ( X ) ⊆ QStk cg ( X ) ≃ of X is defined to be the full subcategoryspanned by the invertible objects of QStk cg ( X ) (here QStk cg ( X ) ≃ denotes the largest Kancomplex contained in QStk cg ( X )), and the extended Brauer group Br † ( X ) of X is defined to bethe set π B r † ( X ). As in [14, 11.5.2.11], we denote the full subcategory of QCoh( X ) ≃ spannedby the invertible objects of QCoh( X ) by P ic † ( X ) and refer to it as the extended Picard space of X . Using the symmetric monoidal structures on QStk St ( X ) and QCoh( X ), we may regard B r † ( X ) and P ic † ( X ) as grouplike commutative monoid objects of the ∞ -category b S .According to [14, 11.5.3.7], an associative algebra object A of QCoh( X ) is called the Azumaya algebra if, for every morphism η : Spec R → X where R is a connective E ∞ -ring, η ∗ A ∈ Alg R is an Azumaya algebra over R . For each Azumaya algebra A ∈ Alg(QCoh( X )),it follows from [14, 11.5.3.9] that the stable quasi-coherent stack on X given by the formula( η : Spec R → X ) (RMod η ∗ A ∈ LinCat St R ) determines an object of the extended Brauerspace B r † ( X ). We refer to the equivalence class of this quasi-coherent stack as the extendedBrauer class of A and denote it by [ A ] ∈ B r † ( X ). We are now ready to prove our main result which extends [14, 11.5.3.10] (which assertsthat if X is a quasi-compact quasi-separated spectral algebraic space, then every object ofBr † ( X ) has the form [ A ] for some Azumaya algebra A ∈ Alg(QCoh( X ))) to quasi-geometricspectral algebraic stacks which admit a quasi-finite presentation (see 1.3). The main ingredi-ent in the proof of [14, 11.5.3.10] is [14, 10.3.2.1] which shows that QCoh( X ; C ) is compactlygenerated for each compactly generated prestable quasi-coherent stack C on X . In our case ofinterest, we will closely follow the proof of [14, 11.5.3.10], using 1.10 in place of [14, 10.3.2.1]. Proof of . Let u ∈ Br † ( X ) be an element, and choose an invertible object C of QStk cg ( X )which represents u . By virtue of 1.10, X is of twisted compact generation, so that QCoh( X ; C )is compactly generated. Choose a set of compact generators { C i } i ∈ I for QCoh( X ; C ). Choose RAUER SPACES OF SPECTRAL ALGEBRAIC STACKS 29 a fiber smooth surjection f : Spec A → X , where A is a connective E ∞ -ring. Since f is quasi-affine, f ∗ : QCoh( X ; C ) → QCoh(Spec A ; f ∗ C ) admits a right adjoint f ∗ (see 5.22), and it fol-lows from the proof of 5.28 that QCoh(Spec A ; f ∗ C ) can be identified with LMod A (QCoh( Y ; C )),under which the pushforward f ∗ corresponds to the forgetful functor LMod A (QCoh( Y ; C )) → QCoh( Y ; C ). In particular, f ∗ is conservative. Combining this observation with the fact that f ∗ is compact (see 5.23), we deduce that { f ∗ C i } i ∈ I is a set of compact generators for f ∗ C (see 5.2). Using [14, 11.5.2.5], we see that f ∗ C is smooth over A , and therefore the proofof [14, 11.3.2.4] guarantees that there exists a finite subset I ⊂ I such that the pullback of C = ⊕ i ∈ I C i along f is a compact generator of f ∗ C . Let E ∈ Alg(QCoh( X )) denote the endo-morphism algebra of C (here we regard QCoh( X ; C ) as tensored over QCoh( X )), and let C ′ bethe stable quasi-coherent stack on X , given by the formula ( η : Spec R → X ) (RMod η ∗ E ∈ LinCat St R ). Consider the morphism of quasi-coherent stacks F : C ′ → C determined by theoperation • ⊗ E C . We will complete the proof by showing that the functor F is an equivalenceand that E is an Azumaya algebra on X . By virtue of 6.2 and [14, D.4.1.6] (see also [14,11.2.3.3]), it will suffice to show the assertion after pulling back along f . Invoking the factthat f ∗ C is a compact generator of f ∗ C , we deduce that f ∗ F is an equivalence of A -linear ∞ -categories (see [17, 7.1.2.1]). Combining this observation with the fact that C is invertibleand [14, 11.5.3.4], we conclude that f ∗ E is an Azumaya algebra over A as desired. (cid:3) The remainder of this section is devoted to describing the homotopy groups of theextended Brauer sheaf B r † ( X ) where X is a quasi-geometric stack. Definition 6.6.
Let X be a quasi-geometric stack. Let CAlg cn X → CAlg cn be a left fibrationclassified by X (so that an object of CAlg cn X can be identified with a pair ( A, η ), where A isa connective E ∞ -ring and η ∈ X ( A ) is an A -valued point of X ). The ∞ -category (CAlg cn X ) op can be equipped with a Grothendieck topology which we refer to as the big ´etale topology : asieve on an object ( A, η ) is a covering if it contains a finite collection of morphisms { ( A, η ) → ( A i , η i ) } ≤ i ≤ n for which the induced map A → Q A i is faithfully flat and ´etale. There is aninduced Grothendieck topology on the opposite of the full subcategory CAlg cn , fpqc X ⊆ CAlg cn X spanned by those objects ( A, η ) for which the corresponding morphism Spec A → X is flat.We let S hv fpqc − ´et X ⊆ Fun(CAlg cn , fpqc X , S ) denote the full subcategory spanned by the sheaveson (CAlg cn , fpqc X ) op and refer to it as the fpqc-´etale ∞ -topos of X . Remark 6.7.
Let B r † X , P ic † X : CAlg cn , fpqc X → S denote the functors given on objects by( A, η ) B r † ( A ) and P ic † ( A ), respectively. They are fpqc-´etale sheaves and factor throughthe ∞ -category CAlg gp ( S ) of grouplike E ∞ -spaces (see [17, 5.2.6.6]). By virtue of [14,11.5.2.11], we have a canonical equivalence Ω B r † X ≃ P ic † X in the ∞ -category of CAlg gp ( S )-valued fpqc-´etale sheaves on X . We refer to B r † X as the extended Brauer sheaf of X . There is an evident forgetful functor QCoh( X ) → Fun(CAlg cn X , S ). More informally, itassigns to each F ∈ QCoh( X ) a functor which carries a pair ( R, η ) ∈ CAlg cn X to the 0-thspace Ω ∞ ( F η ) of the underlying spectrum of the R -module F η . By virtue of [14, 6.2.3.1],the forgetful functor factors through the ∞ -category of big ´etale sheaves on X . For anyinteger n ≥ F ∈ QCoh( X ), let π n F denote the n -th homotopy group ofthe restriction of the underlying big ´etale sheaf of F to S hv fpqc − ´et X . We have the followinganalogue of [14, 11.5.5.3], which can be proven by exactly the same argument: Lemma 6.9.
Let X be a quasi-geometric stack. Then the homotopy groups of B r † X are givenby π n B r † X ≃ if n = 0 Z if n = 1( π O X ) × if n = 2 π n − O X if n ≥ .Here Z denotes the constant sheaf associated to the abelian group Z . In the special case where X is 0-truncated (in the sense of [14, 9.1.6.1]), the restrictionof the underlying big ´etale sheaf of the structure sheaf O X to S hv fpqc − ´et X can be regarded as acommutative ring object of the topos of discrete objects of S hv fpqc − ´et X ; let us denote its groupof units by O × X . Arguing as in [14, 11.5.5.4, 11.5.5.5] (using 6.9 in place of [14, 11.5.5.3]),6.9 supplies an equivalence B r † X ≃ K ( O × X , × K ( Z ,
1) in the ∞ -topos S hv fpqc − ´et X . Since thespace of global sections of B r † X can be identified with B r † ( X ), we have the following: Lemma 6.11.
Let X be a -truncated quasi-geometric stack. Then the homotopy groups of B r † ( X ) are given by π n B r † ( X ) ≃ H − ´et ( X, O × X ) × H − ´et ( X, Z ) if n = 0H − ´et ( X, O × X ) × H − ´et ( X, Z ) if n = 1H − ´et ( X, O × X ) if n = 20 if n ≥ ,where H ∗ fpqc − ´et ( X, • ) denotes of the cohomology group of the fpqc-´etale ∞ -topos of X . Remark 6.12.
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