Sheaf theory for stacks in manifolds and twisted cohomology for S^{1}-gerbes

Ulrich Bunke, Thomas Schick and Markus Spitzweck ^{∗}

March 30, 2006

Contents

1 Introduction 1 1.1 About the motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Statement of the main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Twisted cohomology and gerbes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Sheaf theory for smooth stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Sheaf theory for smooth stacks 11 2.1 Over sites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 The site of a smooth stack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 Presheaves of abelian groups and derived functors . . . . . . . . . . . . . . . . . . . . . . 22 2.4 Simplicial models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.5 Comparison of big and small sites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3 The de Rham complex 32 3.1 The de Rham complex is a flabby resolution . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.2 Calculation for U(1)-gerbes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

1 Introduction

1.1 About the motivation

1.1.1 Given a closed three form λ ∈ Ω^{3}(X) on a smooth manifold X, the usual definition of twisted de Rham cohomology is as the cohomology of the two-periodic complex (Ω_{per}^{•}(X), d_{λ}), where

Ω_{per}^{•}(X) := ⨁_{n∈ℤ} Ω^{•+2n}(X) , and d_{λ} := d_{dR} + λ

is the sum of the de Rham differential and the multiplication operator by the form λ. 1.1.2 Twisted de Rham cohomology is in particular interesting as a target of the Chern character from twisted K-theory. In this case [λ] ∈ H^{3}(X; ℝ) is the real image of an integral class λ_{ℤ}(P) ∈ H^{3}(X; ℤ) which classifies a principal bundle P → X with structure group P U, the projective unitary group of a complex infinite-dimensional separable Hilbert space. The twisted K-theory depends functorially on P in a non-trivial manner.

∗Mathematisches Institut, Universität Göttingen, Bunsenstr. 3-5, 37073 Göttingen, GERMANY, bunke@unimath.gwdg.de, schick@uni-math.gwdg.de, spitz@uni-mah.gwdg.de

The twisted cohomology as defined above depends on the cohomology class [λ] up to (in general) noncanonical isomorphism. The draw-back of this definition of twisted cohomology above is that it is not functorial in the twist P → X of K-theory since there no canonical choice of a three-form λ representing the image of λ_{ℤ}(P) in real cohomology. 1.1.3 The main goal of the present note is to propose an alternative functorial definition of the twisted cohomology as the real cohomology of a stack G_{P} which is canonically associated to the P U-bundle P → X. The stack G_{P} is the stack of U-reductions of P → X, where U is the unitary group of the Hilbert space and U → P U is the canonical projection map. It is also called the lifting gerbe of P. In order to define the cohomology of a stack like G_{P} we develop a sheaf theory set-up for stacks in smooth manifolds. Our main result Theorem 1.1 is the key step in the verification that the cohomology according to the new sheaf-theoretic definition is essentially isomorphic (non-canonically) to the twisted cohomology as defined above. We have chosen to work with stacks in smooth manifolds since we are heading towards a comparison with de Rham cohomology. A parallel theory can be set up in the topological context. Together with applications to T -duality and delocalized cohomology it will be discussed in detail in the subsequent papers [8], [6] and [7]. 1.1.4 The twists for our new cohomology theory are smooth gerbes G → X with band U(1). The lifting gerbe G_{P} → X of a P U-bundle mentioned above is an example. Advantages of our new definition are:

(1) The twisted cohomology depends functorially on the twist.

(2) One can define twisted cohomology with coefficients in an arbitrary abelian group.

(3) The definition can easily be generalized to the topological context.

1.1.5 In Subsection 1.2 we give a complete technical statement of our main result written for a reader familar with the language of stacks, sites, and sheaf theory. The third part of the introduction, Subsection 1.3, is devoted to a detailed motivation with references to the literature and a less technical introduction of the language and the description of the result. Finally, Subsection 1.4 is an introduction to the technical sheaf theoretic part of the present paper.

1.2 Statement of the main result

1.2.1 We consider a stack G on the category of smooth manifolds equipped with the usual topology of open coverings. To G we associate a site G as a subcategory of manifolds over G. The objects of this site are representable smooth maps U → G from smooth manifolds to G. A covering (U_{i} → U)_{i∈I} is a collection of morphisms which are submersions and such that ⊔_{i∈I}U_{i} → U is surjective (see 2.2.3 for a precise definition). 1.2.2 To the site G we associate the categories of presheaves Pr G and sheaves ShG of sets as well as the lower bounded derived categories D^{+}(Pr_{Ab} G) and D^{+}(Sh_{Ab}G) of the abelian categories Pr_{Ab} G and Sh_{Ab}G of presheaves and sheaves of abelian groups. 1.2.3 Let i : ShG → Pr G be the natural inclusion, and let i^{♯} : Pr G → ShG be its left adjoint, the sheafification functor. As a right adjoint the functor i is left exact and admits a right derived functor Ri : D^{+}(Sh_{Ab}G) → D^{+}(Pr_{Ab} G). 1.2.4 If G → X is a morphism of stacks, then we define a functor f_{∗} : Pr G → Pr X. Note that if f is not representable, then this map is not associated to a map of sites. If F ∈ Pr G and (U → X) ∈ X, then we set (see 2.4)

f_{∗}(U) := lim F (V) ,

where the limit is taken over the category of diagrams

\xymatrix{V \ar[r] \ar[d] & G \ar[d]^{f} \\ U \ar@{=>}[ur] \ar[r] & X} .

It turns out that f_{∗} admits a left adjoint. Therefore it is left exact and admits a right derived functor Rf_{∗} : D^{+}(Pr_{Ab} G) → D^{+}(Pr_{Ab} X). 1.2.5 Let f : G → X be a smooth gerbe with band S^{1} over the smooth manifold X. We consider the sheafification i^{♯}ℝ_{G} of the constant presheaf ℝ_{G} on G with value ℝ. Our main result describes

i^{♯} ∘ Rf_{∗} ∘ Ri(i^{♯}ℝ_{G}) ∈ D^{+}(Sh_{Ab}X)

in terms of a deformation of the de Rham complex. The gerbe f : G → X is classified by a Dixmier-Douday class λ_{ℤ} ∈ H^{3}(X; ℤ). Let λ ∈ Ω^{3}(X) be a closed form such that [λ] ∈ H^{3}(X; ℝ) represents the image of λ_{ℤ} under H^{3}(X; ℤ) → H^{3}(X; ℝ). For a manifold X the objects (U, p) of the site X are submersions p : U → X from smooth manifolds U to X. This differs from the usual convention, where the site is the category of open subsets of X. We form the complex of presheaves (U, p) ↦ Ω^{·}[[z]]_{λ}(U, p) on X, which associates to (U, p) ∈ X the complex of formal power series of smooth real differential forms on U with differential

d_{λ} := d_{dR} + T λ ,

where z is a formal variable of degree 2, T := \frac{d}{dz}, d_{dR} is the deRham differential, and λ stands for multiplication by p^{∗}λ. It turns out that this is actually a complex of sheaves (see Lemma 3.1). 1.2.6 The main result of the present paper is the following theorem.

Theorem 1.1 In D^{+}(Sh_{Ab}X) we have an isomorphism i^{♯} ∘ Rf_{∗} ∘ Ri(i^{♯}ℝ_{G}) ≅ Ω^{·}[[z]]_{λ}.

1.2.7 The projection map f : G → X of a gerbe is not representable so that f_{∗} : Pr G → Pr X does not come from an associated map of sites. Therefore, in order to define Rf_{∗} and to verify the theorem we have to develop some standard elements of sheaf theory for stacks in smooth manifolds. This is the contents of Section 2 (see 1.4 for an introduction). In Section 3 we verify Theorem 1.1.

1.3 Twisted cohomology and gerbes

1.3.1 A closed three-form λ ∈ Ω^{3}(X) on a smooth manifold X can be used to perturb the de Rham differential

d_{dR} ⇝ d_{dR} + λ =: d_{λ} .

The cohomology of the two-periodic complex

⋯ →^{d_{λ}} Ω^{even}(X) →^{d_{λ}} Ω^{odd}(X) →^{d_{λ}} Ω^{even}(X) →^{d_{λ}} …

is called the λ-twisted cohomology of X and often denoted by H^{∗}(X; λ). This ad-hoc definition appears in various places in the recent mathematical literature (let us mention just [1], [3], [16], [4]) and in the physics literature. A closely related and essentially equivalent definition [15] uses the complex (Ω^{·}(X)((u)), d_{dR} − uλ), where u is a formal variable of degree −2, and ′′((u))′′ stands for formal Laurent series. 1.3.2 It is known that the isomorphism class of the λ-twisted cohomology group only depends on the cohomology class [λ] ∈ H^{3}(X; ℝ). If f : Y → X is a smooth map, then we have a functorial map f^{∗} : H^{∗}(X; λ) → H^{∗}(Y; f^{∗}λ) which essentially only depends on the homotopy class of f. Furthermore, λ-twisted cohomology has a Mayer-Vietoris sequence and is a module over H^{∗}(X; ℝ). It now appears as a natural question to understand λ-twisted cohomology as a concept of algebraic topology. 1.3.3 One attempt is the approach of [9] in which the complex of smooth differential forms is replaced by similar objects in algebraic topology. The proposal of [1] to use the singular de Rham complex goes into the same direction. Observe that we can use the filtration of Ω^{ev}(X) and Ω^{odd}(X) by degree in order to construct a spectral sequence converging to H^{∗}(X; λ). Its E_{2}-page involves H^{∗}(X; ℝ) (as ℤ/2ℤ-graded vector spaces). The next possibly non-trivial differential of this spectral sequence is the multiplication by the class [λ]. In [1] the higher differentials of this spectral sequence are identified as Massey products.

1.3.4 A natural homotopy theoretic framework for twisted cohomology theories would be some version of parametrized stable homotopy theory as developed e.g. in [17]. In such a theory a twist of a generalized cohomology theory (represented by a spectrum E) is a parametrized spectrum E over X with typical fibre equivalent to E (think of a bundle of spectra). The twisted cohomology groups H^{∗}(X; E) are then given by the homotopy groups of the spectrum of sections of E. In order to interpret λ-twisted cohomology in this manner one would have to relate three-forms on X with parametrized versions of the Eilenberg-MacLane spectrum Hℝ. Let us mention that alternatively to [17] other reasonable versions of a stable homotopy theory over X could be based on presheaves of spectra over X or Ω(X)-equivariant spectra, where Ω(X) denotes the based loops of X. 1.3.5 One motivation for introducing λ-twisted cohomology is based on the fact that it can be used as a target of the Chern character from twisted K-theory. It is known that H^{3}(X; ℤ) classifies a certain subset of isomorphisms classes of parametrized spectra K with fibre equivalent to the complex K-theory spectrum K. This follows from the splitting BGL_{1}(K) ≅ K(ℤ, 3)∧T. Here GL_{1}(K) denotes the grouplike monoid of units of the K-theory spectrum, K(ℤ, 3) denotes an Eilenberg-MacLane space, and T is an auxiliary space. We refer to [18] for more details. Chern characters are constructed in [3], [1], [16], [15]. Note that in these works twisted K-theory is not defined in homotopy theoretic terms but using sections in bundles of Fredholm operators, bundle gerbe modules or K-theory of C^{∗}-algebras. If λ_{ℤ} ∈ H^{3}(X; ℤ) classifies the parametrized K-theory spectrum K, then the Chern character has values in H^{∗}(X; λ), where [λ] is the image of λ_{ℤ} under the map H^{3}(X; ℤ) → H^{3}(X; ℝ). Such a definition can not be natural since in general K has non-trivial automorphisms which are not reflected by H^{3}(X; λ). A completely natural definition of a Chern character with values even in a twisted rational cohomology could be induced from the canonical rationalization map K → K_{ℚ} if we like to define twisted rational cohomology using K_{ℚ}. 1.3.6 Above we have seen that H^{3}(X; ℤ) classifies a subset of the isomorphism classes of parametrized K-theory spectra over X. This can in fact be seen directly. Let U be the unitary group of a separable infinite dimensional complex Hilbert space. Equipped with the topology induced by the operator norm it is a topological group. By Kuipers theorem it is contractibe so that the projective unitary group P U := U/U(1) has the homotopy type of BU(1) ≅ K(ℤ, 2). Taking the classifying space once more we have BP U ≅ K(ℤ, 3). This shows that H^{3}(X; ℤ) classifies isomorphism classes of P U-principal bundles over X. One can now manufacture a P U-equivariant version of a K-theory spectrum K (see e.g. [12]). If P → X is a P U-principal bundle, then one can define the bundle of spectra K := P ×_{P U} K over X. Alternatively one could construct twisted K-theory starting from a bundle of projective Hilbert spaces as in [2]. As a result of this discussion one should consider P U-principal bundles as more primary objects. 1.3.7 The theory of bundle gerbes initiated in [20] and continued in [21] aims at a categorification of H^{3}(X; ℤ) in a similar manner as U(1)-principal bundles categorify H^{2}(X; ℤ). The P U-principal bundles considered above are particularly nice examples of bundle gerbes. Other examples of bundle gerbes are introduced in [11]. In order to simplify we forget the smooth structure of X for the moment and work in the category of topological spaces. Let us represent X as a moduli space of a groupoid A^{1} ⇒ A^{0} → X in topological spaces, i.e. we represent X as the quotient of the space of objects A^{0} by the equivalence relation A^{1}. In addition we shall assume that the range and source maps have local sections. Then a bundle gerbe is the same as a central U(1)-extension

[figure: U(1); Ã^{1}; A^{0}; A^{1}; A^{0}; X]

of topological groupoids.

In order to relate the P U-principal bundle P → X with a bundle gerbe we represent X as the moduli space of the action groupoid P × P U ⇒ P → X. The central U(1)-central extension of this groupoid is given by P × U ⇒ P. 1.3.8 The picture of a gerbe in [11] is obtained by choosing an open covering (U_{i})_{i∈I} of X and forming the representation

⨆_{i,j} U_{i} ∩ U_{j} ⇒ ⨆_{i∈I} U_{i} → X .

The data of a U(1)-central extension of this groupoid is equivalent to transition line bundle data and trivializations over triple intersection considered in [11]. One can build a two-category of topological groupoids by inverting Morita equivalence such that equivalence classes of U(1)-central extensions of groupoids representing X are indeed classified by H^{3}(X; ℤ) (see e.g. [25]). 1.3.9 A more natural view on this category of groupoids is through stacks on topological spaces Top. We consider Top as a Grothendieck site where covering families are given by coverings by families of open subspaces. Note that groupoids form a two-category. A stack G on Top can be viewed as an object which associates to each space U ∈ Top a groupoid G(U), to a morphism U′ → U a homomorphism of groupoids G(U) → G(U′), to a chain of composeable morphisms

\xymatrix{U′′ \ar[dr] \ar[rr] &  & U \\  & U′ \ar[ur]}

a two-isomorphism

[figure: G(U); G(U′′); G(U′)]

satisfying a natural associativity relation, and such that G satisfies descend conditions for the covering families of U. Precise definitions can be found e.g. in [22], [10], [5]. A space V ∈ Top can be viewed as a stack by the Yoneda embedding such that V (U) = Hom_{Top}(U, V) (where we consider sets as groupoids with only identity morphisms). 1.3.10 As an illustration we explain a canonical construction which associates to a P U-principal bundle P → X over a space X a stack G_{P} together with a map G_{P} → X. It will be called the lifting gerbe of P. Observe that U acts on P via the canonical homomorphims U → P U. For a space T ∈ Top the objects of the groupoid G_{P} (T) are the diagrams

\xymatrix{Q \ar[r] \ar[d] & P \ar[d] \\ T \ar[r] & X},

where Q → T is a U-principal bundle, and Q → P is U-equivariant. A morphism between two such objects

\xymatrix{Q \ar[r] \ar[d] & P \ar[d] \\ T \ar[r] & X}, \xymatrix{Q′ \ar[r] \ar[d] & P \ar[d] \\ T \ar[r] & X},

is an isomorphism of U-principal bundles Q → Q′ over T which is compatible with the maps to P.

Finally, for a map T′ → T the functor G_{P} (f) : G_{P} (T) → G_{P} (T′) maps the object

\xymatrix{Q \ar[r] \ar[d] & P \ar[d] \\ T \ar[r] & X}∈ G_{P} (T)

to the induced diagram

\xymatrix{T′ ×_{T} Q \ar[r] \ar[d] & P \ar[d] \\ T′ \ar[r] & X}∈ G_{P} (T′) ,

and a morphism Q → Q′ to the induced morphism T′ ×_{T} Q → T′ ×_{T} Q′. We leave it as an exercise to check that this presheaf of groupoids is a stack. The morphism G_{P} → X maps the object

\xymatrix{Q \ar[r] \ar[d] & P \ar[d] \\ T \ar[r] & X}∈ G_{P} (T)

to the underlying map T → X which is considered as an element of X(T). 1.3.11 A diagram of P U-principal bundles

\xymatrix{P \ar[r] \ar[d] & P′ \ar[d] \\ X \ar[r] & X′}

functorially induces a diagram of stacks

\xymatrix{G_{P} \ar[r] \ar[d] & G_{P′} \ar[d] \\ X \ar[r] & X′}

in the obvious way. 1.3.12 A topological groupoid A : A^{1} ⇒ A^{0} represents a stack [A^{1}/A^{0}] on topological spaces. It associates to each space U the groupoid [A^{0}/A^{1}](U) = Hom(U, [A^{0}/A^{1}]) of A-principal bundles on X and isomorphisms (see [10]). A morphism of groupoids gives rise via an associated bundle construction to a map of stacks. In this way we can embed the two-category of topological groupoids (with Morita equivalence inverted) mentioned at the end of 1.3.7 as a full subcategory of stacks on Top. The image of this embedding consists of topological stacks G, i.e. stacks which admit an atlas A^{0} → G. An atlas is a surjective representable morphism A^{0} → G admitting local sections, where A^{0} is a space. Given an atlas of G we can construct a groupoid A^{1} ⇒ A^{0}. The morphism space of the groupoid is given by A^{1} := A^{0} ×_{G} A^{0}. We then have an equivalence of stacks [A^{0}/A^{1}] ≅ G. A map of stacks G → H is called representable if for any map U → H with U a space U ×_{H} G is equivalent to a space. The representability condition on A^{0} → G ensures that A^{1} := A^{0} ×_{G} A^{0} is a space. 1.3.13 The lifting gerbe G_{P} of a P U-principal bundle 1.3.10 is a topological stack. In order to construct an atlas we choose a covering of X by open subsets on which P is trivial. Let A be the disjoint union of the elements of the convering, and A → X be the canonical map. By choosing local trivializations we obtain the lift in the diagram

\xymatrix{ & P \ar[d] \\ A \ar[r] \ar@{.>}[ur]^{s}^{s} & X}.

We now consider the diagram

\xymatrix{A × U \ar[r]^{ϕ} \ar[d] & P \ar[d] \\ A \ar[r] & X}∈ G_{P} (A) ,

where ϕ(a, u) := s(a)ū and ū denotes the image of u ∈ U under U → P U. We consider this object as a morphism A → G_{P}. We leave it as an exercise to verify that this map is an atlas. 1.3.14 A morphism of stacks G → X with X a space is a topological gerbe with band U(1) if there exists an atlas A → X, a lift

\xymatrix{ & G \ar[d] \\ A \ar[r] \ar@{.>}[ur] & X}  (1.2)

to an atlas of G such that

[figure: U(1); A ×_{G} A; A; A ×_{X} A; A; X]

is a U(1)-extension of topological groupoids. In particular, the bundle gerbes considered in 1.3.7 give rise to topological gerbes with band U(1). For equivalent definitions see [22], [10]. The definition of a gerbe in [5] is slightly more general since the existence of an atlas is not required. 1.3.15 The lifting gerbe G_{P} → X constructed in 1.3.10 is a topological gerbe with band U(1). In fact, the construction 1.3.13 produces the lift (1.2). 1.3.16 In the definitions above the Grothendieck site Top can be replaced by the Grothendieck site of smooth manifolds Mf^{∞}. In this site the covering families are again coverings by families of open submanifolds. Stacks on Mf^{∞} are called stacks in smooth manifolds. If G is a stack in smooth manifolds, then an atlas A → G is a map of stacks which is representable and smooth, i.e. for any map T → G from a smooth manifold T to G the induced map T ×_{G} A → A is a submersion of manifolds. A stack in smooth manifolds which admits an atlas will then be called smooth. 1.3.17 Let Y → X be a map of manifolds. It is representable as a map between stacks in smooth manifolds if for any map Z → X the fibre product Z ×_{X} Y exists as a manifold. Submersions between manifolds are representable maps.^{1} 1.3.18 We come to the conclusion that a basic object classified by λ_{ℤ} ∈ H^{3}(X; ℤ) is the equivalence class of a smooth gerbe f : G → X with band U(1). Instead of going the way through some version of parametrized stable homotopy theory it now seems natural to define a real cohomology twisted by G directly using a suitable sheaf theory on stacks. A natural candidate would be something like H^{∗}(X; G) := H^{∗}(G; ℝ) := H^{∗}(G; i^{♯}ℝ_{G}), where i^{♯}ℝ_{G} is the sheafification of the constant presheaf with value ℝ, and H^{∗}(…, i^{♯}ℝ_{G}) is defined using the derived global sections, or the derived p_{∗}, where p : G → ∗ is the projection to a point. In fact, if G would be a manifold, then the sheaf theoretic H^{∗}(G, i^{♯}ℝ_{G}) would be isomorphic to the de Rham cohomology of the manifold G, and therefore to the topologist‘s H^{∗}(G; ℝ). To proceed in the case of stacks we must clarify what we mean by a sheaf on G, and how we define p_{∗}. The construction of H^{∗}(G; ℝ) will be finalized in Definition 2.31. In order to define sheaves and presheaves on G we associate in 2.2 to G a Grothendieck site G. The notions of presheaves and sheaves on a site are the standard ones. 1.3.19 If f : G → X is a morphism of stacks, then we are interested in functors f_{∗}, f^{∗}. Such operations are usually obtaind from some induced morphisms of sites f^{♯} : X → G. In fact, this works well for representable morphisms. But in the case of a gerbe f : G → X neither f nor p : G → ∗ are representable. We will define f_{∗} and p_{∗} in an ad-hoc way. The same problem with a similar solution also occurs in algebro-geometric set-ups, see e.g. [14]. Because of this ad-hoc definitions we must redevelop some of the basic material of sheaf theory in order to check that the expected properties hold in the present set-up. For details we refer to the introduction 1.4 to sheaf theory part of the present paper. 1.3.20 After the development of elements of sheaf theory on smooth stacks we can define

1 We do not know the converse, i.e. whether a representable map between manifolds is necessarily a submersion.

H^{∗}(X; G) := H^{∗}(G; ℝ) := H^{∗}(ev ∘ Rp_{∗} ∘ Ri(i^{♯}ℝ_{G})) ,

where i : ShG → Pr G is the embedding of sheaves into presheaves, the sheafification functor i^{♯} : Pr G → ShG is the left adjoint of i, and the exact functor

                                  ev : Pr_{Ab}Site(∗) → Ab

evaluates a presheaf of abelian groups on the object (∗ → ∗) ∈ Site(∗). This last evaluation is necessary since our site is the big site of ∗ consisting of all smooth manifolds. As the notation suggests we view this as the cohomology of X twisted by the gerbe G. 1.3.21 This definition is natural in G. If u : G′ → G is a map of stacks, then by Lemma 2.16 we have a functorial map

u^{∗} : H^{∗}(G; ℝ) → H^{∗}(G′; ℝ)

since there is a canonical isomorphism u^{∗}i^{♯}ℝ_{G} ≅ i^{♯}ℝ_{G′}. In particular, H^{∗}(X; G) carries the action of the automorphisms of the gerbe G → X. 1.3.22 The natural question is now how the λ-twisted de Rham cohomology H^{∗}(X; λ) and H^{∗}(X; G) are related. The main step in this relation is provided by Theorem 1.1. Using this result in the isomorphism

≅^{!} and the projection q : X → ∗ we can write

H^{∗}(X; G) = H^{∗}(ev ∘ Rp_{∗} ∘ Ri(i^{♯}R_{G}))
≅ H^{∗}(ev ∘ R(q ∘ f)_{∗} ∘ Ri(i^{♯}R_{G}))
≅^{(∗∗)} H^{∗}(ev ∘ Rq_{∗} ∘ Rf_{∗} ∘ Ri(i^{♯}ℝ_{G}))
≅^{(∗)} H^{∗}(ev ∘ Rq_{∗} ∘ Ri ∘ i^{♯} ∘ Rf_{∗} ∘ Ri(i^{♯}ℝ_{G}))
≅^{!} H^{∗}(ev ∘ Rq_{∗} ∘ Ri(Ω^{·}[[z]]_{λ}))
≅^{(∗∗∗∗)} H^{∗}(ev ∘ R(q_{∗} ∘ i)(Ω^{·}[[z]]_{λ}))
≅^{(∗∗∗)} H^{∗}(ev ∘ q_{∗} ∘ i(Ω^{·}[[z]]_{λ}))
= H^{∗}(Ω^{·}[[z]]_{λ}(X)) .

In order to justify the isomorphism (∗) we use Lemma 2.13 which says that f_{∗} preserves sheaves. The isomorphism (∗∗) follows from Lemma 2.26 since f is smooth. For (∗ ∗ ∗∗) we use Lemma 2.27. Finally, (∗ ∗ ∗) follows from Lemma 2.30 and the fact that Ω^{·}[[z]]_{λ} is a complex of flabby sheaves (see 2.28).

Note that the isomorphism ≅^{!} depends on additional choices.

1.3.23 It remains to relate the cohomology of the complex (Ω^{·}[[z]]_{λ}(X), d_{λ}) (see 1.2.5) with H^{∗}(X; λ). Let

Ω^{·}[[z]]_{λ}(X)^{p} ⊂ Ω^{·}[[z]]_{λ}(X)

be the subset of polynomials ∑_{2n+k=p} z^{n}ω^{k} with ω^{k} ∈ Ω^{k}(X). Then we have d_{λ} : Ω^{·}[[z]]_{λ}(X)^{p} → Ω^{·}[[z]]_{λ}(X)^{p+1}. For p > 0 we construct morphisms ψ_{p} such that the following diagram commutes

\xymatrix{… \ar[r] & Ω^{odd}(X) \ar[d]^{ψ_{2p−1}} \ar[r]^{d_{λ}} & Ω^{even}(X) \ar[d]^{ψ_{2p}} \ar[r]^{d_{λ}} & Ω^{odd}(X) \ar[d]^{ψ_{2p+1}} \ar[r]^{d_{λ}} & … \\ … \ar[r] & Ω^{·}[[z]]_{λ}(X)^{2p−1} \ar[r]^{d_{λ}} & Ω^{·}[[z]]_{λ}(X)^{2p} \ar[r]^{d_{λ}} & Ω^{·}[[z]]_{λ}(X)^{2p+1} \ar[r]^{d_{λ}} & …}

In fact, for e = 0, 1 and ω = ∑_{i=0}^{∞} ω^{e+2i} we define

ψ_{2p+e}(ω) := ∑_{i=0}^{[ \frac{p}{2} ]} \frac{z^{p−i}ω^{e+2i}}{(p − i)!} .

If p > dim(X), then ψ_{p} is an isomorphism. Therefore for large p the isomorphisms ψ_{p} induce embeddings H^{∗}(X; λ) ↪ H^{∗}(X; G). In this way H^{∗}(X; G) is a replacement of H^{∗}(X; λ) with good functorial properties. 1.3.24 The definition of real cohomology of X twisted by a gerbe as

H^{∗}(X; G) := H^{∗}(G; ℝ)

has a couple of additional interesting features.

(1) First of all note that ℝ is a commutative ring. Therefore H^{∗}(X; G) has naturally the structure of a graded commutative ring. In the old picture this structure seems to be partically reflected by the product

H^{∗}(X; aλ) ⊗ H^{∗}(X; bλ) → H^{∗}(X; (a + b)λ) .

(2) One can replace ℝ by any other abelian group. In particular, one can define integral twisted cohomology by H^{∗}(X; G; ℤ) := H(G; ℤ). This definition of an integral twisted cohomology proposes a solution to the question raised in the remark made in [1, Sec. 6]. Using the maps ψ_{p} introduced above we can identify the image of H^{∗}(X; G; ℤ) → H^{∗}(X; G) as a lattice in H^{∗}(X; λ). The result depends on the choice of p, and in view of the denominators in the formula for ψ_{p} the position of lattice is not very obvious.

(3) In the proof of Theorem 1.1 we construct a de Rham model for the cohomology of H^{∗}(G; ℝ). Let Ω_{G}^{<p} be the sheaf of de Rham complexes truncated at p−1 and form the sheaf of Deligne complexes H(p − 1)_{G} := (i^{♯}ℤ_{G} → Ω_{G}^{<p}), where i^{♯}ℤ_{G} sits in degree −1. We can then define the real Deligne cohomology Ĥ^{p}(G; ℤ) of G as the (p − 1)-st hypercohomology of the complex H_{G}(p − 1) (see [5] for a definition of Deligne cohomology for manifolds in a similar fashion).

1.4 Sheaf theory for smooth stacks

1.4.1 This subsection is the introduction to the sheaf theoretic part of the paper. We consider a smooth stack X. In order to define the notion of a sheaf on X we associate to X a Grothendieck site X. A site consists of a category X and the choice of covering families cov_{X}(U) for the objects U ∈ X. Presheaves on X are just contravariant set-valued functors on X. A sheaf on X is a presheaf which satisfies a descend condition with respect to the covering families.

1.4.2 We define the category X as a full subcategory of the category of manifolds U over X such that the structure map U → X is smooth. The covering families of U → X are families of submersions over X whose union maps surjectively to U. Observe that the category of smooth manifolds can be considered as a site with the above mentioned choice of covering families. By the Yoneda embedding it maps to the two-category of smooth stacks. In Subsection 2 we consider this abstract situation. We consider a site S, a two-category C and a functor z : S → C. Furthermore we consider a subcatgeory rC which plays the role of the subcategory of stacks with smooth representable morphisms. In this situation we associate to each object X ∈ C the site X (see 2.1) as the full subcategory of (z(U) → X) ∈ S/X such that the structure map belongs to rC. The covering families are induced from S (see 2.2). 1.4.3 The central topic of Subsection 2 is the adjoint pair (2.5) of functors

f^{∗} : Pr X ⇔ Pr G : f_{∗}

between presheaf categories associated to a morphism f : G → X. Since in general f does not induce a morphism between the sites G and X we define these functors in an ad-hoc manner (see 2.3 and 2.4). For two composeable morphisms f, g we relate (g ∘ f)^{∗}, (g ∘ f)_{∗} with g_{∗} ∘ f_{∗}, f^{∗} ∘ g^{∗} in Lemma 2.6. 1.4.4 In Subsection 2.2 we specialize to smooth stacks. If the morphism f : G → H between smooth stacks is smooth or representable, then it gives rise to a morphism of sites f_{♯} or f^{♯}, respectively (2.2.6 and 2.2.7). We verify that our ad-hoc definitions of f^{∗} or f_{∗} , respectively, coincide with the standard functors induced from the morphism of sites f^{♯} or f_{♯} (see Lemmas 2.7, 2.9, 2.11). 1.4.5 Most of the statements which we formulate for the sheaf theory on stacks are well-known in the usual sheaf theory on sites and for functors associated to morphisms of sites. But for the sheaf theory on stacks we must be very careful about which of these standard facts remain true in general. For other statements we must know and under which additional assumptions they carry over to stacks. 1.4.6 An important point is the observation that for every morphism between smooth stacks the functor f_{∗} preserves sheaves (Lemma 2.13). In the Lemmas 2.16 and 2.22 we study the compatibility of the pull back with the push forward in cartesian squares. In Lemma 2.23 we study under which additional assumptions we have relations like (g ∘ f)^{∗} ≅ f^{∗} ∘ g^{∗}. 1.4.7 In order to define the cohomology of a gerbe we must descend the functors f_{∗} and f^{∗} to the derived categories of presheaves and sheaves of abelian groups. This question is studied in Subsection 2.3. Here the exactness properties of the functors studied in the preceding subsections play an important role. Most of the statements in this subsection are standard for the usual sheaf theory and functors associated to a morphism of sites. Here we study carefully under which additional conditions they remain true for stacks. 1.4.8 The main result (Lemma 2.41) of Subsection 2.4 is that the derived functor Rf_{∗} for a map G → X of smooth stack can be calculated using a simplicial approximation of G → X. In particular, if X is a manifold, then the calculation of Rf_{∗} can be reduced to ordinary sheaf theory on manifolds. We use this simplicial model in the proof of our main theorem, for the explicit calculation of the cohomology of the stack [∗/S^{1}] in Lemma 3.8, but also to verify that pull-back and push-forward commute on the level of derived functors for certain cartesian diagrams in Lemma 2.43. 1.4.9 The covering families of the small site (U) of a manifold are coverings by open subsets. Thus the sheaf theory for (U) is the ordinary one. If (U → X) ∈ X, then a presheaf on X induces a presheaf on (U). In the present paper the sheaf theory on (U) is considered to be well-understood. The main goal of Subsection 2.5 is to compare the sheafification functors on X and (U) (see Lemma 2.47). This result is very useful in explicit calculations since it says that certain questions can be studied for each (U → X) ∈ X separately and with respect to the small site (U). This sort of reasoning will be applied in the proof that the de Rham complex of a stack is a flabby resolution of the constant sheaf with value ℝ, where we use that this fact is well-known on each manifold equipped with the site (U). It is also used in the proof of Lemma 2.50 which says that for a smooth map between smooth stacks the pull back commutes with the sheafification functor.

2 Sheaf theory for smooth stacks

2.1 Over sites

2.1.1 The goal of the present subsection is to develop some elements of sheaf theory in the following situation. Let S be a site (see [24, 1.2.1] for a definition), C a two-category with invertible two-morphisms, and z : S → C a functor (we consider S as a two-category with only identity two-isomorphisms). Finally let rC be a subcategory which we call the category of admissible morphisms. To each object G ∈ C we will associate a site G (sometimes we will write Site(G) := G) and the categories of presheaves Pr G and sheaves ShG of sets on this site. For a morphism f ∈ C(G, H) we will define an adjoint pair of functors

f^{∗} : Pr H ⇔ Pr G : f_{∗} .

In general these functors are not induced by a morphism of sites. 2.1.2 Let G ∈ C. We define the underlying category of G.

Definition 2.1 The objects of G are pairs (U, ϕ), where U ∈ S and ϕ ∈ rC(z(U), G). A morphism (U, ϕ) → (U′, ϕ′) is given by a pair (h, σ), where h ∈ S(U, U′) and σ is a two-isomorphism

[figure: z(U); ϕ; z(h); σ; z(U′); ϕ′; G]

The composition in G is defined in the obvious way.

Sometimes we will abbreviate the notation and write U or (z(U) → G) for (U, ϕ). 2.1.3 Next we define the coverings of an object (U, ϕ) of G.

Definition 2.2 A covering of (U, ϕ) is a collection of morphisms

((U_{i}, ϕ_{i}) →^{(h_{i},σ_{i})} (U, ϕ))_{i∈I}

such that (U_{i} →^{h_{i}} U)_{i∈I} is a covering of U in S.

In fact it is easy to verify the axioms listed in the definition [24, 1.2.1]. The only non-obvious part asserts that given a covering ((U_{i}, ϕ_{i}) → (U, ϕ))_{i∈I} and a morphism (V, ψ) → (U, ϕ), then the fibre products (V_{i}, ψ_{i}) := (V, ψ) ×_{(U,ϕ)} (U_{i}, ϕ_{i}) exist in G and ((V_{i}, ψ_{i}) → (V, ψ))_{i∈I} is a covering of (V, ψ). By a little diagram chase one verifies that (V, ψ) ×_{(U,ϕ)} (U_{i}, ϕ_{i}) ≅ (V ×_{U} U_{i}, ϕ ∘ z(κ)), where V ×_{U} U_{i} is the fibre product in S and κ : V ×_{U} U_{i} → U is the natural map. 2.1.4 Let f: G → H be a morphism in C. Then we can define the functor f^{∗} : Pr H → Pr G as follows. Given (z(V) → G) ∈ G we consider the category V/H (recall that V abbreviates (z(V) → G)) of diagrams

\xymatrix{V \ar[d] & z(V) \ar[r] \ar[d] & G \ar[d]^{f} \\ U & z(U) \ar@{=>}[ur] \ar[r] & H} .

A morphism in this category is given by a morphism (z(U′) → H) → (z(U) → H) in H fitting into

[figure: U; z(V); G; f; V; z(U); H; U′; z(U′); H] .

Let F ∈ Pr H.

Definition 2.3 We define

f^{∗}F (V) := colim_{V/H}F (U) .

A morphism V′ → V in G induces naturally a functor V/H → V′/H. The relevant diagram is

[figure: V′; z(V′); G; V; z(V); G; f; U; z(U); H] .

We therefore get a map f^{∗}F (V) → f^{∗}F (V′), and this makes f^{∗}F a presheaf on G. 2.1.5 Let f : G → H again be a morphism in C. We define a functor f_{∗} : Pr G → Pr H as follows. We consider (z(U) → G) ∈ H. Then we conider the category of diagrams G/U of diagrams

\xymatrix{V \ar[d] & z(V) \ar[d] \ar[r] & G \ar[d]^{f} \\ U & z(U) \ar@{=>}[ur] \ar[r] & H} .

A morphism of such diagrams is given by a morphism V′ → V in G which fits into

[figure: V′; z(V′); G; V; z(V); G; f; U; z(U); H] .

Definition 2.4 We define

f_{∗}F (U) := lim_{G/U} F (V) .

A morphism U → U′ in H induces naturally a functor G/U → G/U′. The relevant diagram is

[figure: V; z(V); G; U; z(U); G; f; U′; z(U′); H] .

We therefore get a map f_{∗}F (U′) → f_{∗}F (U), and this makes f_{∗}F a presheaf on H. 2.1.6 Let f ∈ C(G, H) as before.

Lemma 2.5 The functors f_{∗} and f^{∗} naturally form an adjoint pair

f^{∗} : Pr H ⇔ Pr G : f_{∗} .

Proof. We give the unit and the counit. Let (z(W) → G) ∈ G. Then

f^{∗}f_{∗}F (W) = colim_{U} lim_{V} F (V) ,

where the colimit-limit is taken over a category of diagrams

\xymatrix{V \ar[d] & z(V) \ar[d] \ar[r] & G \ar[d]^{f} \\ U & z(U) \ar[r] & H \\ W \ar[u] & z(W) \ar[u] \ar[r] & G \ar[u]_{f}}

(we leave out the two-isomorphisms). The counit is a natural transformation

f^{∗}f_{∗}F (W) → F (W) .

It is given by the universal property of the colimit and the collection of maps which associates to U the canonical map lim_{V} F (V) → F (W). Furthermore, let (z(U) → H) ∈ H. Then

f_{∗}f^{∗}F (U) = lim_{V} colim_{W} F (W) ,

where the limit-colimit is taken over a category of diagrams

\xymatrix{U & z(U) \ar[r] & H \\ V \ar[u] \ar[d] & z(V) \ar[u] \ar[d] \ar[r] & G \ar[u]_{f} \ar[d]^{f} \\ W & z(W) \ar[r] & H}

(we leave out the two-isomorphisms). The unit is a natural transformation

F (U) → f_{∗}f^{∗}F (U) .

It is given by the universal property of the limit and the collection of maps which associates to V the natural map F (U) → colim_{W} F (W). We leave it to the interested reader to perform the remaining checks. □

2.1.7 Let us consider a pair of composable maps in C

G ⟶^{f} H ⟶^{g} L .

Lemma 2.6 We have natural transformations of functors

(g ∘ f)_{∗} → g_{∗} ∘ f_{∗} , f^{∗} ∘ g^{∗} → (g ∘ f)^{∗} .

Proof. We discuss the transformation f^{∗} ∘ g^{∗} → (g ∘ f)^{∗}. Let F ∈ Pr L and (z(W) → G) ∈ G. Inserting the definitions we have

f^{∗} ∘ g^{∗}(F)(W) ≅ colim_{A}F (U) ,

where A is the category of diagrams

\xymatrix{z(W) \ar[r] \ar[d] & G \ar[d]^{f} \\ z(V) \ar[r] \ar[d] \ar@{=>}[ur] & H \ar[d]^{g} \\ z(U) \ar[r] \ar@{=>}[ur] & L}

with (V → H) ∈ H and (U → L) ∈ L. The vertical composition provides a functor A → W/L, where W/L is the category of diagrams of the form

\xymatrix{z(W) \ar[d] \ar[r] & G \ar[d]^{g∘f} \\ z(U) \ar[r] \ar@{=>}[ur] & L} .

We get an induced map of colimits

f^{∗} ∘ g^{∗}(F)(W) → (g ∘ f)^{∗}F (W) = colim_{W/L}F (U) .

The other transformation (g ∘ f)_{∗} → g_{∗} ∘ f_{∗} is obtained in a similar manner or, equivalently, by adjointness. □

In general, we can not expect that these transformations are isomorphisms. But under additional assumptions they are, see 2.23.

2.2 The site of a smooth stack

2.2.1 We consider the small^{2} site Mf^{∞} of smooth manifolds. Its underlying category is the category of smooth manifolds and smooth maps. A collection of smooth maps (U_{i} → U)_{i∈I} is a covering if and only if this family is isomorphic to the collection of inclusions of the open subsets of U given by an open covering of U. We use the small site Mf^{∞} in order to define stacks in smooth manifolds. We refer to [10], [19], [22] for the language of stacks.

2In the present paper the adjectives small and big refer to the type of covering families. In fact, in order to avoid set-theoretic problems one must require that a site is a small category. In the present paper we will ignore this problem. It can be resolved by either working with universes or replacing Mf^{∞} by an equivalent small category (see e.g. [19]).

2.2.2 We will also consider the big site S on smooth manifolds. In this site a family (U_{i} → U)_{i∈I} of smooth maps is a covering if the maps U_{i} → U are submersions and ⊔_{i∈I}U_{i} → U is surjective. We will use this site in order to define the site of a stack according to 2.1. 2.2.3 In this paragraph we recall the main notions of the theory of smooth stacks.

(1) A morphism of stacks G → H is called representable, if for each manifold U and map U → H the fibre product U ×_{H} G is equivalent to a manifold. A composition of representable maps is representable.

(2) A representable morphism G → H of stacks is called smooth if for each manifold U and map U → H the induced map U ×_{H} G → U (of manifolds) is a submersion.

(3) A map U → G from a manifold to a stack is called an atlas if it is representable, smooth and admits local sections.

(4) A stack in smooth manifolds is called smooth if it admits an atlas [10, Def. 2.4].

(5) A morphism (not necessarily representable) between smooth stacks f : G → H is called smooth if for an atlas A → G the composition A → G → H is smooth [10, Def. 2.10]. A composition of smooth maps is smooth.

(6) A smooth morphism U → G from a manifold to a smooth stack is representable.

2.2.4 Let C be the two-category of smooth stacks in smooth manifolds. We have a Yoneda embedding z : S → C. Note that in general we will omit the Yoneda embedding in the notation and consider S as a subcategory of C. We let rC be the subcategory of representable smooth morphisms. 2.2.5 The conventions introduced in 2.2.4 place us in the situation of 2.1.1. Let G ∈ C be a smooth stack. Then by G we denote the site according to Definitions 2.1 and 2.2. Note that this site is derived from the big site S on smooth manifolds. We now have the categories of presheaves Pr G and sheaves ShG on the stack G. The present definition is essentially the same as in [10, Sec. 4] execpt that we use the big site S on the category of smooth manifolds instead of the small site Mf^{∞}. Note that we use the descend property for the big site in the proof of Lemma 2.36. The relation between the sheaf theories on the big and the small sites will be discussed in [6]. 2.2.6 Let f : G → H be a representable morphism of smooth stacks in smooth manifolds. Then it induces a morphism of sites f^{♯} : H → G by the rule f^{♯}(U → H) := U ×_{H} G → G (it is easy to check the axioms listed in [24, 1.2.2]). 2.2.7 If f : G → H is a smooth morphism of smooth stacks in smooth manifolds, then we can define another morphism of sites f_{♯} : G → H by f_{♯}(V → G) := (V → G →^{f} H). 2.2.8 We call a functor left exact if it preserves arbitrary limits. If it preserves arbitrary colimits, then we call it right exact. A functor is said to be exact if it is right and left exact. Recall that a functor which is a left adjoint is right exact. Similarly, a right adjoint is left exact. 2.2.9 A morphism of sites q : H → G induces an adjoint pair

q_{∗} : Pr H ⇔ Pr G : q^{∗} .

(see [24, 2.3]). In the following we compare these maps with the ad-hoc definitions 2.3 and 2.4 and discuss some special properties. 2.2.10

Lemma 2.7 If f : G → H is a smooth morphism between smooth stacks, then we have f^{∗} ≅ (f_{♯})^{∗}. In particular, then f^{∗} is exact and preserves sheaves.

Proof. Let (V → G) ∈ G. According to the definition [24, 2.3] we have

(f_{♯})^{∗}(F)(V → G) := F (f_{♯}(V → G)) = F (V → G → H) .

If (V → G) ∈ G, then the category V/H has an initial object

[figure: V; G; f; V; id; H] .

Therefore

(f^{∗}F)(V → G) ≅ F (V → G → H) .  (2.8)

This implies that f^{∗} ≅ (f_{♯})^{∗}. It is well-known [24, 3.6] that the contravariant functor (in our case (f_{♯})^{∗}) associated to a morphism of sites preserves sheaves. Therefore f^{∗} preserves sheaves. The limit of a diagram of presheaves is defined objectwise. By (2.8) the functor f^{∗} commutes with limits. As a left adjoint (by Lemma 2.5) it also commutes with colimits. □

2.2.11 Let f : G → H be a representable and smooth morphism of smooth stacks.

Lemma 2.9 We have an isomorphism of functors (f^{♯})_{∗} ≅ f^{∗}.

Proof. Let F ∈ Pr H. For (V → G) ∈ G we have the category V/f^{♯} of pairs ((U → H) ∈ H, (V → f^{♯}(U)) ∈ Mor(G)). It has a natural evaluation ev_{V} : V/f^{♯} → H which maps ((U → H), (V → f^{♯}(U))) to (U → H). By definition (see [24, Proof of 2.3.1])

(f^{♯})_{∗}(F)(V) = colim_{V/f^{♯}}F ∘ ev_{V} .

Now we observe that V/f^{♯} can be identified with the category of diagrams

\xymatrix{V \ar[d] \ar[r] & U \ar[d] \\ G \ar[r]^{f} \ar@{=>}[ur] & H}.

Since f is smooth we see that (V → G →^{f} H) ∈ H and

\xymatrix{V \ar[d] \ar[r]^{^{id^{V}}} & V \ar[d] \\ G \ar[r]^{f} \ar@{=>}[ur]^{id} & H}

is the initial element of V/f^{♯}. We conclude that

(f^{♯})_{∗}(F)(V → G) ≅ F (V → G → H) .  (2.10)

The equality f^{∗} ≅ (f^{♯})_{∗} now follows from (2.8). □

One can not expect that f^{∗} is left exact for a general map f : G → H. In fact this problem occurs in the corresponding definition in [14] of the pull-back for the lisse-etale site of an algebraic stack. For more details and a solution see [23]. 2.2.12

Lemma 2.11 If f : G → H is a representable morphism of smooth stacks, then f_{∗} = (f^{♯})^{∗} : Pr G → Pr H. The functor f_{∗} is exact.

Proof. Let (U → H) ∈ H. Then f^{♯}(U → H) = (U ×_{H} G → G) is the final object in G/U. Therefore

f_{∗}F (U) ≅ F (U ×_{H} G) ≅ (f^{♯})^{∗}F (U) .  (2.12)

Since (f^{♯})^{∗} is a right adjoint it commutes with limits. Since colimits of presheaves are defined objectwise it follows from the formula (2.12) that f_{∗} also commutes with colimits. □

2.2.13 Let now f : G → H be a map of smooth stacks.

Lemma 2.13 The functor f_{∗} preserves sheaves.

Proof. Let F ∈ ShG. Consider (U → H) ∈ H and let (U_{i} → U) be a covering of U. Consider a diagram

\xymatrix{V \ar[d] \ar[r] & G \ar[d]^{f} \\ U \ar@{=>}[ur] \ar[r] & H} .  (2.14)

From this we obtain a collection of diagrams

\xymatrix{V_{i} := U_{i} ×_{U} V \ar[d] \ar[r] & G \ar[d]^{f} \\ U_{i} \ar@{=>}[ur] \ar[r] & H}

functorially in V. Observe that (V_{i} → V) is a covering in G. We now consider the map of diagrams

\xymatrix{f_{∗}F (U) \ar[d] \ar[r] & ∏_{i} f_{∗}F (U_{i}) \ar[d] \ar@{=>}[r] & ∏_{i,j} f_{∗}F (U_{i} ×_{U} U_{j}) \ar[d] \\ F (V) \ar[r] & ∏_{i} F (V_{i}) \ar@{=>}[r] & ∏_{i,j} F (V_{i} ×_{V} V_{j})} .

The vertical maps are given by specialization. We must show that the upper horizontal line is an equalizer diagram. The lower horizontal line has this property since F is a sheaf. We now take the limit over the category of diagrams (2.14) und use the fact that a limit preserves equalizer diagrams. We get the commutative diagram of sets

[figure: f_{∗}F (U); ∏_{i} f_{∗}F (U_{i}); s; ∏_{i,j} f_{∗}F (U_{i} ×_{U} U_{j}); f_{∗}F (U); lim ∏_{i} F (V_{i}); lim ∏_{i,j} F (V_{i} ×_{V} V_{j})] .

Let us assume that s is injective. Then the fact that the lower horizontal line is an equalizer diagram implies by a simple diagram chase that the upper horizontal line is an equalizer diagram. We now show that s is injective. Note that a priori the product of spezialization maps

s = ∏_{i} s_{i} : ∏_{i} f_{∗}F (U_{i}) → ∏_{i} lim F (V_{i})

may not be injective since the functors L_{i} : G/U ∋ V ↦ V_{i} ∈ G/U_{i} are not necessarily essentially surjective. But in our situation the maps s_{i} are injective since each object in G/U_{i} maps into an object in the image of L_{i}. To see this consider a diagram

\xymatrix{W \ar[d] \ar[r]^{t} & G \ar[d]^{f} \\ U_{i} \ar@{=>}[ur] \ar[r] & H} ∈ G/U_{i} .

Using the composition W → U_{i} → U we can form the diagram

\xymatrix{W \ar[d] \ar[r] & G \ar[d]^{f} \\ U \ar@{=>}[ur] \ar[r] & X} ∈ G/U

and define the morphism in G/U_{i}

\xymatrix{U_{i} ×_{U} W \ar[dd]_{_{pr_{1}}} \ar[dr]^{t∘pr_{2}} \\ W \ar[u]^{j} \ar[d] \ar[r] & G \ar[d]^{f} \\ U_{i} \ar@{=>}[ur] \ar[r] & H} ,

where j : W → U_{i} ×_{U} W is induced by W → U_{i} and id_{W} : W → W.

□

2.2.14 Assume that we have a diagram in smooth stacks

\xymatrix{G \ar[r]^{u} \ar[d]^{f} & H \ar[d]^{g} \\ M \ar@{=>}[ur] \ar[r]^{v} & N} ,  (2.15)

where u and v are smooth. Lemma 2.16 We have a natural map of functors Pr H → Pr M

v^{∗} ∘ g_{∗} → f_{∗} ∘ u^{∗}

which is an isomorphism if (2.15) is cartesian. Proof. We use the description (2.8) of v^{∗} obtained in the proof of Lemma 2.7. Let (U → M) ∈ M. Then we have

v^{∗} ∘ g_{∗}F (U) ≅ lim F (A) ,

where the limit is taken over a category D of diagrams

\xymatrix{ &  &  & A \ar[dl] \ar[ddd] \\  & G \ar[r]^{u} \ar[d]^{f} & H \ar@{=>}[ddr]^{σ} \ar[d]^{g} \\  & M \ar@{=>}[drr] \ar@{=>}[ur] \ar[r]^{v} & N \\ U \ar[ur] \ar[rrr]^{id} &  &  & U \ar[ul]}  (2.17)

(where A varies). On the other hand f_{∗} ∘u^{∗}(F)(U) ≅ lim F (V), where the limit is taken over the category E of diagrams

\xymatrix{V \ar[dr] \ar[ddd] \ar[rrr] &  &  & V \ar[dl] \\  & G \ar@{=>}[ddl] \ar@{=>}[urr] \ar[r]^{u} \ar[d]^{f} & H \ar[d]^{g} \\  & M \ar@{=>}[ur] \ar[r]^{v} & N \\ U \ar[ur]}  (2.18)

(V varies). We define a functor X : E → D which sends the diagram (2.18) to the diagram

\xymatrix{ & & & V \ar[ddd] \ar[dl] \\ & G \ar[r]^{u} \ar[d]^{f} & H \ar@{=>}[ddr] \ar[d]^{g} \\ & M \ar@{=>}[ur] \ar@{=>}[drr] \ar[r]^{v} & N \\ U \ar[ur] \ar[rrr]^{id} & & & U \ar[ul]} .

We write F_{E} and F_{D} for the functor F precomposed with the evaluations E → H and D → H. The identity F (V) →^{∼} F (V) induces an isomorphism F_{D} ∘X →^{∼} F_{E}. Therefore we have a natural map of limits

v^{∗} ∘ g_{∗}F (U) → f_{∗} ∘ u^{∗}F (U) .  (2.19)

This gives the required transformation of functors. If (2.15) is cartesian, then we can define a functor Y : D → E which maps the diagram (2.17) functorially to

[figure: U ×_{N} A; h; G; u; f; H; g; M; v; N; U; id; U] ,

which employs the map A → H → N. The map h is induced by the universal property of the cartesian diagram. Since U → M and A → H are smooth, the map h is smooth, too. The map A → U together with the two-isomorphism σ gives a map A → U ×_{N} A in H. This map induces the natural transformation F_{E} ∘ Y → F_{D}. It gives a map of limits

f_{∗} ∘ u^{∗}(F)(U) → v^{∗} ∘ g_{∗}F (U) .  (2.20)

One can check that (2.20) is inverse to (2.19). □

2.2.15 Assume again that we have a diagram in smooth stacks

\xymatrix{G \ar[r]^{u} \ar[d]^{f} & H \ar[d]^{g} \\ M \ar@{=>}[ur] \ar[r]^{v} & N} .  (2.21)

We now assume that f and g are representable, and that u, v are smooth.

Lemma 2.22 We have a natural map of functors Pr H → Pr M

v^{∗} ∘ g_{∗} → f_{∗} ∘ u^{∗}

which is an isomorphism if (2.21) is cartesian.

This is a special case of Lemma 2.16. But under the additional representablility assumptions on f and g the proof simplifies considerably. Proof. Let F ∈ Pr H. For (U → M) ∈ M we calculate

v^{∗} ∘ g_{∗}(F)(U) ≅ colim_{V ∈U/N} lim_{W ∈H/V} F (W)
≅^{(2.8)} lim_{W ∈H/(U→M→N)} F (W)
≅^{(2.12)} F (H ×_{N} U → H) .

On the other hand

f_{∗} ∘ u^{∗}(F)(U) ≅ lim_{Z∈G/U} colim_{W ∈Z/H}F (W)
≅^{(2.12)} colim_{W ∈(U×_{M} G)/H}F (W)
≅^{(2.8)} F (U ×_{M} G → G → H) .

The transformation v^{∗}∘g_{∗}(F)(U) → f_{∗}∘u^{∗}(F)(U) is now induced from the map (G×_{M} U → H×_{N} U) ∈ H. If the diagram is cartesian, then we have G×_{M} U ≅ (H ×_{N} M)×_{M} U ≅ H ×_{N} U so that the transformation is an isomorphism. □

2.2.16 Let us consider a pair of composable maps of smooth stacks

G ⟶^{f} H ⟶^{g} L .

In Lemma 2.6 we have found natural transformations of functors between presheaf categories

(g ∘ f)_{∗} → g_{∗} ∘ f_{∗} , f^{∗} ∘ g^{∗} → (g ∘ f)^{∗} .

Lemma 2.23 If g is representable, or if f is smooth, then these transformations are isomorphisms.

Proof. We consider the transformation f^{∗} ∘ g^{∗} → (g ∘ f)^{∗} which appears as a transformation of colimits induced by a functor between indexing categories A → W/L, where we use the notation introduced in the proof of Lemma 2.6. Under the present additional assumptions on f or g we have a functor W/L → A which induces the inverse of the transformation. In the following we describe these functors.

If g is representable, then each diagram

\xymatrix{W \ar[d] \ar[r] & G \ar[d]^{g∘f} \\ U \ar[r] \ar@{=>}[ur] & L}  (2.24)

in W/L naturally completes to

\xymatrix{W \ar[r] \ar[d] & G \ar[d]^{f} \\ U ×_{L} H \ar[r] \ar[d] \ar@{=>}[ur] & H \ar[d]^{g} \\ U \ar[r] \ar@{=>}[ur] & L}

in A. If f is smooth, then the diagram (2.24) can be naturally completed to

[figure: W; G; f; W; id; H; g; U; L]

in A. It follows from adjointness that under the additional assumptions on f or g the transformation (g ∘f)_{∗} → g_{∗} ∘ f_{∗} is an isomorphism, too. □

2.2.17 Let f : G → H be a smooth map of smooth stacks.

Lemma 2.25 There exists a functor f_{!} : Pr G → Pr H so that we get an adjoint pair

f_{!} : Pr G ⇔ Pr H : f^{∗} .

Proof. Let (V → G) ∈ G. Then by (2.8) we have f^{∗}F (V) ≅ F (V → G → H). Let (V → G) ∈ G and h_{V →G} ∈ Pr G be the corresponding representable presheaf. Then we have a natural isomorphism

Hom(h_{V →G}, f^{∗}F) ≅ f^{∗}F (V → G)
≅ F (V → G → H)
≅ Hom(h_{V →G→H}, F)

which leads us to the definition

f_{!}h_{V →G} := h_{V →G→H} .

If L ∈ Pr G, then we can write L ≅ colim_{h_{V →G}→L}h_{V →G}. Since a left-adjoint must commute with colimits we are forced to set

f_{!}L := colim_{h_{V →G}→L}h_{V →G→H} .

Then we have indeed

Hom(L, f^{∗}F) ≅ Hom(colim_{h_{V →G}→L}h_{V →G}, f^{∗}F)
≅ lim_{h_{V →G}→L} Hom(h_{V →G}, f^{∗}F)
≅ lim_{h_{V →G}→L} Hom(h_{V →G→H}, F)
≅ Hom(colim_{h_{V →G}→L}h_{V →G→H}, F)
≅ Hom(f_{!}L, F)
□

2.3 Presheaves of abelian groups and derived functors

2.3.1 In the previous Subsections 2.1 and 2.2 we have developed a theory of set-valued presheaves and sheaves on stacks. We are in particular interested in the abelian categories of presheaves and sheaves of abelian groups and their derived categories. The functors (f^{∗}, f_{∗}) and (i^{♯}, i) preserve abelian group valued objects. In the present subsection we study how these functors descend to the derived categories. Furthermore, we check some functorial properties of these descended functors which will be employed in later calculations. The derived version (Lemma 2.43) of the fact that pull-back commutes with push-forward in certain cartesian diagrams (Lemma 2.16) would fit into the present subsection, but can only be shown after the development of a computational tool in Subsection 2.4. A similar remark applies to Lemma 2.50 saying that sheafification commutes with pull-back along smooth maps between smooth stacks. We will show this Lemma in Subsection 2.5. 2.3.2 For a site G let Pr_{Ab} G and Sh_{Ab}G denote the abelian categogies of presheaves and sheaves of abelian groups on G. These categories have enough injectives [24, 2.1.1 and 2.1.2]. Let D^{+}(Pr_{Ab} G) and D^{+}(Sh_{Ab}G) denote the lower bounded derived categories of Pr_{Ab} G and Sh_{Ab}G. 2.3.3 If f : G → H is a morphism of smooth stacks then f_{∗} : Pr G → Pr H is left exact since it is a right adjoint. We therefore have the right derived functor Rf_{∗} : D^{+}(Pr_{Ab} G) → D^{+}(Pr_{Ab} H). If g : H → L is a second morphism of smooth stacks, then we have a natural transformation

R(g ∘ f)_{∗} → Rg_{∗} ∘ Rf_{∗} .

In fact, let F ∈ D^{+}(Pr_{Ab} G) be a lower bounded complex of injective presheaves. Then we choose an injective resolution f_{∗}F → J. Note that g_{∗}(J) represents Rg_{∗} ∘ Rf_{∗}(F). Then using (2.6) the required morphism is defined as the composition

R(g ∘ f)_{∗}(F) ≅ (g ∘ f)_{∗}(F) → g_{∗} ∘ f_{∗}(F) → g_{∗}(J) ≅ Rg_{∗} ∘ Rf_{∗}(F) .

Lemma 2.26 If f is smooth or g is representable, then

R(g ∘ f)_{∗} ≅ Rg_{∗} ∘ Rf_{∗} .

Proof. If f is smooth, then f^{∗} is exact. In this case f_{∗} preserves injectives and we can take J := f_{∗}(F). We can now apply Lemma 2.23 in order to see that the natural transformation (g ∘ f)_{∗}(F) → g_{∗} ∘ f_{∗}(F) is an isomorphism. If g is representable, then g_{∗} is exact by Lemma 2.11. In this case we have again by Lemma 2.23 that R(g ∘ f)_{∗}(F) ≅ (g ∘ f)_{∗}(F) ≅ g_{∗} ∘ f_{∗}(F) ≅ Rg_{∗} ∘ Rf_{∗}(F). □

2.3.4 Let i : ShG → Pr G denote the inclusion. It has a left adjoint i^{♯} : Pr G → ShG, the sheafification functor (see [24, 3.1.1, 3.2.1]). Since the functor i is a right adjoint, it is left exact. We can form its right derived Ri : D^{+}(Sh_{Ab}G) → D^{+}(Pr_{Ab} G). Let f : G → H be a morphism of smooth stacks.

Lemma 2.27 The functor i preserves injectives and we have an isomorphism R(f_{∗} ∘ i) ≅ Rf_{∗} ∘ Ri.

Proof. Since i^{♯} is exact (see [24, Thm. 3.2.1 (ii)]) the functor i preserves injectives. This implies the assertion. □

2.3.5 Let τ := (U_{i} → U)_{i∈I} ∈ cov_{G}(U → G) be a covering family of (U → G) ∈ G. For a presheaf F ∈ Pr_{Ab} G we form the Čech complex Č^{∗}(τ, F). Its pth group is

Č^{p}(τ, F) := ∏_{(i_{0},...,i_{p})∈I^{p+1}} F (U_{i_{0}} ×_{U} ⋯ ×_{U} U_{i_{p}}) ,

and the differential is given by the usual formula.

Definition 2.28 (see 3.5.1, [24]) A sheaf F ∈ Sh_{Ab}G is called flabby if for all (U → G) ∈ G and all τ ∈ cov_{G}(U → G) we have H^{k}(Č(τ, F)) ≅ 0 for all k ≥ 1.

2.3.6 Let f : G → H be a smooth map between smooth stacks.

Lemma 2.29 The functor f^{∗} : Pr_{Ab} G → Pr_{Ab} H preserves flabby sheaves.

Proof. We have the functor f_{♯} : G → H given by f_{♯}(V → G) := (V → G → H) (see 2.2.7). By Lemma 2.7 we know that f^{∗} preserves sheaves. Let (U → G) ∈ G and τ := (U_{i} → U) ∈ cov_{G}(U). Observe that f_{♯}τ := (f_{♯}(U_{i}) → f_{♯}(U)) is a covering family of f_{♯}U in H. Let F ∈ Sh_{Ab}H. By Lemma 2.8 we have f^{∗}F (U) ≅ F (f_{♯}U). We therefore have an isomorphism of complexes

Č^{·}(τ, f^{∗}F) ≅ Č^{·}(f_{♯}τ, F) .

If F is in addition flabby, then the cohomology groups of the right-hand side in degree ≥ 1 vanish.

□

2.3.7 Let f : G → H be a representable map between smooth stacks.

Lemma 2.30 If F ∈ Sh_{Ab}G is flabby, then F is (f_{∗} ∘ i)-acyclic.

Proof. Let F ∈ Sh_{Ab}G be flabby. We must show that R^{k}(f_{∗} ∘ i)(F) ≅ 0 for k ≥ 1. By Lemma 2.11 the functor f_{∗} is exact so that R^{k}(f_{∗} ∘ i)(F) ≅ f_{∗} ∘ R^{k}i(F). Since F is injective it is flabby. Since flabby sheaves are i-acyclic by [24, Corollary 3.5.3] we get R^{k}i(F) ≅ 0. □

2.3.8 Let G be a smooth stack, F ∈ D^{+}(Sh_{Ab}G), and p : G → ∗ the canonical morphism. Then we have the object Rp_{∗} ∘ Ri ∈ D^{+}(Pr_{Ab}Site(∗)). Let ev : Pr_{Ab}Site(∗) → Ab be the evaluation at the object (∗ → ∗) ∈ Site(∗). This functor is exact.

Definition 2.31 We define the cohomology of F ∈ D^{+}(Sh_{Ab}G) as

h(G; F) := ev ∘ Rp_{∗} ∘ Ri(F) ∈ D^{+}(Ab)

Furthermore we set H^{∗}(G; F) := H^{∗}h(G; F).

In particular, for an abelian group Z we have the constant presheaf Z_{G} with value Z.

Definition 2.32 We define the cohomology of the smooth stack G with coefficients in Z by

H^{∗}(G; Z) := H^{∗}(G; i^{♯}Z_{G}) .

2.4 Simplicial models

2.4.1 For a morphism f : G → X between smooth stacks we defined a functor f_{∗} : Pr G → Pr X (see Definition 2.4). We are in particular interested in its derived version Rf_{∗}∘Ri : D^{+}(Sh_{Ab}G) → D^{+}(Pr_{Ab} X). The definitions of f_{∗} in terms of a limit, and of Rf_{∗} using injective resolutions are very useful for the study of the functorial properties of f_{∗}. For explicit calculations we would like to work with more concrete objects. In the present subsection we associate to a flabby sheaf F ∈ Sh_{Ab}G an explicit complex of presheaves C_{A}^{·}(F) ∈ C^{+}(Pr_{Ab} X) which represents Rf_{∗} ∘i(F) ∈ D^{+}(Pr_{Ab} X) (see Lemma 2.41). It looks like presheaf of Čech complexes and depends on the choice of a surjective smooth and representable map A → G such that A → G → X is also representable (e.g. an atlas of G). In the present paper we consider three applications of this construction. The first is the derived version of Lemma 2.16 which says that pull-back and push-forward in certain cartesian diagrams commute (see Lemma 2.43). In the second application we use the complex C_{A}^{·} in order to get a de Rham model of the derived push-forward of the constant sheaf with value ℝ on G (see equation (3.3). Finally we use this construction in Lemma 3.8 in order to calculate the cohomology of the gerbe [∗/S^{1}] explicitely. 2.4.2 Let G be a smooth stack and (A → G), (B → G) ∈ G.

Lemma 2.33 The fibre product in stacks

[figure: A ×_{G} B; A; u; B; id; G]

is the categorial product (A → G) ×_{G} (B → G).

Proof. The fibre product (H → G), (L → G) ↦ H ×_{G} L of stacks H, L ∈ C over G is the two-categorical fibre product in the two-category C/G of stacks over G. Let C_{0} ⊂ C be the full subcategory of stacks which are equivalent to smooth manifolds, i.e. the essential image of the Yoneda embedding Mf^{∞} → C. We define the one-category \overline{C_{0}/G} by identifying two-isomorphic morphisms and observe that the canonical functor C_{0}/G → \overline{C_{0}/G} is an equivalence. Under this equivalence the restriction of the fibre product to C_{0} becomes the one-categorical product. This implies the result since the natural functor

G → C_{0}/G → \overline{C_{0}/G}

is an equivalence of categeories. □

2.4.3 Let f : G → X be a map of smooth stacks. Let further A be a smooth stack and A → G be a representable, surjective and smooth map such that the composition A → G → X is also representable. An atlas of G would have these properties, but in applications we will need this more general situation where A is not necessarily equivalent to a manifold. Let (U → X) ∈ X and form the following diagram of cartesian squares:

\xymatrix{A_{U} \ar[d] \ar[r] & A \ar[d] \\ G_{U} \ar@{=>}[ur] \ar[r]^{j^{U}} \ar[d] & G \ar[d] \\ U \ar@{=>}[ur] \ar[r] & X}.  (2.34)

Since smoothness is preserved by pull-back the horizontal maps are smooth. Since surjectivity is also preserved by pull-back the two upper vertical maps are surjective and smooth. Since A → X is representable, the stack A_{U} ≅ U ×_{X} A is equivalent to a manifold.

2.4.4 Note that (A_{U} → G_{U}) ∈ G_{U}. In view of Lemma 2.33 we can take powers of A_{U} in G_{U}. Using these powers we form a simplical object A_{U}^{·} ∈ G_{U}. Its n-th object is given by

\underbrace{A_{U} ×_{G_{U}} ⋯ ×_{G_{U}} A_{U}}_{n+1 factors} → G_{U} .

We let j_{!}^{U}A_{U}^{·} ∈ G denote the simplicial object in G with nth object (A_{U}^{n} → G_{U} →^{j^{U}} G). If V → U is a morphism in X, then we obtain an induced morphism of simplicial objects j_{!}^{V} A_{V}^{·} → j_{!}^{U}A_{U}^{·} in G. 2.4.5 If F ∈ Pr G, then we consider the cosimplicial object U ↦ F (j_{!}^{U}A_{U}^{·}) in Pr X. For a morphism V → U in G the structure map F (j_{!}^{U}A_{U}^{·}) → F (j_{!}^{V} A_{V}^{·}) is induced by the morphism of simplicial objects j_{!}^{V} A_{V}^{·} → j_{!}^{U}A_{U}^{·} in G.

Definition 2.35 For a presheaf of abelian groups F ∈ Pr_{Ab}G let

C_{A}^{·}(F) ∈ C^{+}(Pr_{Ab}X)

denote the chain complex of presheaves associated to the cosimplicial presheaf of abelian groups U ↦ F (j_{!}^{U}A_{U}^{·}). Its differential will be denoted by δ.

2.4.6 Let F ∈ Pr_{Ab} G.

Lemma 2.36 We have a natural transformation ψ : f_{∗}F → H^{0}C_{A}^{·}(F) which is an isomorphism if F is a sheaf.

Proof. Let (U → X) ∈ X. We recall definition of the push-forward 2.1.5:

f_{∗}F (U) = lim_{(V →G)∈G/U} F (V) .

Observe that

\xymatrix{ & G_{U} \ar@{.>}[dr]^{j^{U}} \\ A_{U} \ar@{.>}[ur] \ar[d] \ar[rr] &  & G \ar[d] \\ U \ar[rr] \ar@{=>}[urr] &  & X}

belongs to G/U so that we have an evaluation

\xymatrix{f_{∗}F (U) \ar[rr]^{evaluation} \ar[dr]^{ψ} &  & C_{A}^{0}(F)(U) \\  & H^{0}C_{A}^{·}(F)(U) \ar@{-->}[ur]}

with a canonical factorization ψ by the definition of H^{0}C_{A}^{·}(F)(U) as a kernel. Assume now that F is a sheaf. Then we must show that ψ is an isomorphism. Let

\xymatrix{V \ar[d] \ar[r] & G \ar[d] \\ U \ar@{=>}[ur] \ar[r] & X}  (2.37)

be in G/U. Then we have a canonical factorization

\xymatrix{V \ar[dr] \ar[ddr] \ar[drr] \\ & G_{U} \ar[d] \ar[r] & G \ar[d] \\ & U \ar@{=>}[ur] \ar[r] & X}.

Using the induced map V → G_{U} we form the diagram

\xymatrix{V ×_{G_{U}} A_{U} \ar[r] \ar[d] & A_{U} \ar[d] \ar[dr] \\ V \ar@{=>}[ur] \ar[r] & G_{U} \ar[r]^{j^{U}} & G}

We consider the composition (V ×_{G_{U}} A_{U} → A_{U} → G) as an object in G. Since A_{U} → G_{U} is smooth and surjective the map (V ×_{G_{U}} A_{U} → V) is a covering of V in G (it is here where we use the big site). For a sheaf F we have

F (V) ≅ lim ( \xymatrix{F (A_{U} ×_{G_{U}} V) \ar@{=>}[r] & F ((A_{U} ×_{G_{U}} V) ×_{V} (A_{U} ×_{G_{U}} V))} ) .

We further have

(A_{U} ×_{G_{U}} V) ×_{V} (A_{U} ×_{G_{U}} V) ≅ A_{U} ×_{G_{U}} A_{U} ×_{G_{U}} V

and a diagram

\xymatrix{F (A_{U} ×_{G_{U}} V) \ar@{=>}[r] & F (A_{U} ×_{G_{U}} A_{U} ×_{G_{U}} V) \\ F (j_{!}^{U}A_{U}^{0}) \ar@{=>}[r] \ar[u] & F (j_{!}^{U}A_{U}^{1}) \ar[u]}

(recall that A_{U}^{0} = A_{U} and A_{U}^{1} = A_{U} ×_{G_{U}} A_{U}) induced by the projection along V. Since H^{0}C_{A}^{·}(F)(U) is the limit of the lower horizontal part the left vertical map induces a map H^{0}C_{A}^{·}(F)(U) → F (V). Since this construction is natural in the object (2.37) of G/U we obtain finally a map H^{0}C_{A}^{·}(F)(U) → f_{∗}F (U) which is the inverse to ψ. □

2.4.7

Lemma 2.38 If F ∈ Pr_{Ab} G is injective, then H^{i}C_{A}^{·}(F) = 0 for i ≥ 1.

Proof. We follow the ideas of the last part of the proof of [24, Thm. 2.2.3]. Let (U → X) ∈ X and A_{U}^{·} denote the simplicial presheaf of sets represented by j_{!}^{U}A_{U}^{·}. Furthermore, let ℤ_{A_{U}^{·}} be the (non-positively graded) complex of free abelian presheaves generated by A_{U}^{·}. Then for any presheaf F ∈ Pr_{Ab} G we have

C_{A}^{·}(F)(U) ≅ Hom_{Pr_{Ab} G}(ℤ_{A_{U}^{·}} , F) .

Since F is injective Hom_{Pr_{Ab} G}(…, F) is an exact functor. Hence it suffices to show that H^{i}(ℤ_{A_{U}^{·}}) = 0 for i ≤ −1. For (V → G) ∈ G the complex ℤ_{A_{U}^{·}} (V) is the complex associated to the linearization of the simplicial set Hom_{G}(V, j_{!}^{U}A_{U}^{·}). We now rewrite

A_{U} ×_{G_{U}} ⋯ ×_{G_{U}} A_{U} ≅ (A ×_{G} ⋯ ×_{G} A) ×_{X} U ≅ (A ×_{G} ⋯ ×_{G} A) ×_{G} G_{U} .  (2.39)

We consider V, (A×_{G}⋯×_{G}A) and G_{U} with their canonical maps to G as objects of the two-category C/G of stacks over G. The first object is a manifold and therefore does not have non-trivial two-automorphisms.

Since the maps (A×_{G}⋯×_{G}A) → G and G_{U} → G are representable these objects of C/G also do not have non-trivial two-automorphisms. By the same reasoning as in the proof of Lemma 2.33 we can interpret the fibre product (2.39) as a one-categorical product. We get

Hom_{G}(V, j_{!}^{U}A_{U}^{·}) = Hom_{C/G}(V, (A ×_{G} ⋯ ×_{G} A) ×_{G} G_{U})
≅ (Hom_{C/G}(V, A) × ⋯ × Hom_{C/G}(V, A)) × Hom_{C/G}(V, G_{U})
≅ Hom_{C/G}(V, A)^{·} × Hom_{C/G}(V, G_{U})

For any set S, if we take the simplicial set S^{·} of the powers of S, the complex associated to the linearization ℤ_{S^{·}} is exact in degrees ≤ −1. Therefore the complex ℤ_{Hom_{C/G}(V,A)^{·}} is exact in degree ≤ −1. Since the tensor product with the free abelian group ℤ_{Hom_{C/G}(V,G_{U})} is an exact functor the complex

ℤ_{Hom_{G}(V,j_{!}^{U} A_{U}^{·})} ≅ ℤ_{Hom_{C/G}(V,A)^{·}} ⊗ ℤ_{Hom_{C/G}(V,G_{U})}

is exact in degree ≤ −1, too. □

2.4.8 Since exactness of complexes of presheaves is defined objectwise the functors C_{A}^{p} : Pr_{Ab} G → Pr_{Ab} X are exact for all p ≥ 0. Composing with the total complex construction we extend the functor C_{A}^{·} to a functor between the categories of lower bounded complexes C_{A} : C^{+}(Pr_{Ab} G) → C^{+}(Pr_{Ab} X) (in order to distinguish this from the double complex we drop the ^{·} at the symbol C_{A}). Since this functor is level-wise exact it descends to a functor C_{A} : D^{+}(Pr_{Ab} G) → D^{+}(Pr_{Ab} X) between the lower-bounded derived categories. 2.4.9 Assume that F is a presheaf of associative algebras on G. Then C_{A}^{·}(F) is a presheaf of DGalgebras in the following natural way. Pick

α ∈ F (j_{!}^{U}A_{U}^{p}) ≅ C_{A}^{p}(F)(U) , β ∈ F (j_{!}^{U}A_{U}^{q}) ≅ C_{A}^{q}(F)(U) .

We have natural maps u : j_{!}^{U}A_{U}^{p+q} → j_{!}^{U}A_{U}^{p} and v : j_{!}^{U}A_{U}^{p+q} → j_{!}^{U}A_{U}^{q} in G projecting onto the first p + 1 or last q + 1 factors, respectively. Then we define α · β ∈ F (j_{!}^{U}A_{U}^{p+q}) ≅ C_{A}^{p+q}(F)(U) by u^{∗}α · v^{∗}β. One easily checks that δ(α · β) = δα + (−1)^{p}α · δβ. 2.4.10 If F^{·} is a presheaf of commutative DG-algebras, then C_{A}(F^{·}) is a presheaf of associative DGalgebras central over the presheaf of commutative DG-algebras (ker(δ) : C_{A}^{0}(F^{·}) → C_{A}^{1}(F^{·})). 2.4.11 By Lemma 2.36 we have a map

ψ : f_{∗}F → H^{0}C_{A}^{·}(F)

which is an isomorphism if F is a sheaf.

Lemma 2.40 For all F ∈ Pr_{Ab} G we have a natural isomorphism RH^{0}C_{A}^{·}(F) ≅ C_{A}^{·}(F) in D^{+}(Pr_{Ab} X).

Proof. Let F → I^{·} be an injective resolution. Then we have RH^{0}C_{A}^{·}(F) ≅ H^{0}C_{A}^{·}(I^{·}). By Lemma 2.38 the inclusion H^{0}C_{A}^{·}(I^{·}) → C_{A}(I^{·}) is a quasi-isomorphism. Since C_{A}^{·} is exact the quasi-isomorphism F → I^{·} induces a quasi-isomorphism C_{A}^{·}(F) ≅ C_{A}(I^{·}). □

2.4.12 Recall that i : Pr_{Ab} G → Sh_{Ab}G is left exact and admits a right derived functor Ri : D^{+}(Sh_{Ab}G) → D^{+}(Pr_{Ab} G), and that C_{A} descends to a functor between the lower bounded derived categories (see 2.4.8).

Lemma 2.41 We have a natural isomorphism of functors

C_{A} ∘ Ri ≅ Rf_{∗} ∘ Ri : D^{+}(Sh_{Ab}G) → D^{+}(Pr_{Ab}X) .

Proof. By Lemma 2.36 we have an isomorphism of functors f_{∗} ∘ i ≅ H^{0}C_{A} ∘ i. Hence we have an isomorphism

Rf_{∗} ∘ Ri ≅^{!} R(f_{∗} ∘ i) ≅ R(H^{0}C_{A}^{·} ∘ i) ≅^{!} RH^{0}C_{A}^{·} ∘ Ri ≅ C_{A} ∘ Ri ,

where at the marked isomorphisms we use that i preserves injectives (compare Lemma 2.27). □

2.4.13 Assume that we have a diagram in smooth stacks

\xymatrix{G \ar[r]^{u} \ar[d]^{f} & H \ar[d]^{g} \\ X \ar@{=>}[ur] \ar[r]^{v} & Y} ,  (2.42)

where u and v are smooth. Note that u^{∗} and v^{∗} are exact (Lemma 2.7).

Lemma 2.43 (1) We have a natural transformation of functors D^{+}(Pr_{Ab} H) → D^{+}(Pr_{Ab} X)

v^{∗} ∘ Rg_{∗} → Rf_{∗} ∘ u^{∗} .

(2) The induced transformation D^{+}(Sh_{Ab}H) → D^{+}(Pr_{Ab} X)

v^{∗} ∘ Rg_{∗} ∘ Ri → Rf_{∗} ∘ u^{∗} ∘ Ri

is an isomorphism if (2.42) is cartesian.

Proof. The transformation (1) is induced by

v^{∗} ∘ Rg_{∗} ≅ R(v^{∗} ∘ g_{∗}) ≅^{2.16} R(f_{∗} ∘ u^{∗}) → Rf_{∗} ∘ u^{∗} .

In order to show the second part (2) we must show that

R(f_{∗} ∘ u^{∗}) ∘ Ri → Rf_{∗} ∘ u^{∗} ∘ Ri

is an isomorphism. We calculate Ri using injective resolutions. Note that i preserves injectives. Hence in order to show that this map is an isomorphism it suffices to show that u^{∗} maps injective sheaves to f_{∗}-acyclic presheaves. Note that u^{∗} preserves sheaves (Lemma 2.7). We let u_{s}^{∗} : Sh_{Ab}H → Sh_{Ab}G denote the restriction of u^{∗} to sheaves. Let F ∈ Sh_{Ab}H be injective. Since injective sheaves are flabby, flabby sheaves are i-acyclic, and u^{∗} preserves flabby sheaves (see Lemma 2.29) we have

Rf_{∗} ∘ i ∘ u_{s}^{∗}(F) ≅ Rf_{∗} ∘ Ri ∘ u_{s}^{∗}(F) ≅^{Lemma 2.41} C_{A} ∘ Ri ∘ u_{s}^{∗}(F) ≅ C_{A} ∘ u^{∗} ∘ i(F) .

We now show that the higher cohomology presheaves of C_{A}^{·} ∘ u^{∗} ∘ i(F) vanish. Let (U → X) ∈ X and choose an atlas B → H. Then we get the following extension of the diagram (2.34)

\xymatrix{A_{U} \ar[d] \ar[r] & A \ar[d] \ar[r] & B \ar[d] \\ G_{U} \ar@{=>}[ur] \ar[r]^{j^{U}} \ar[d] & G \ar@{=>}[ur] \ar[d] \ar[r]^{u} & H \ar[d]^{g} \\ U \ar@{=>}[ur] \ar[r] & X \ar@{=>}[ur] \ar[r]^{v} & Y}  (2.44)

such that all squares are cartesian. The three upper vertical maps are smooth and surjective. The composition A → G → X is representable. All horizontal maps are smooth. We have the simplicial

object (A_{U}^{·} → G_{U}) ∈ G_{U} and let u_{!}j_{!}^{U}A_{U}^{·} ∈ H be the induced simplicial object A_{U}^{·} → G_{U} →^{j^{U}} G →^{u} H in H. Then we have by the definition 2.4.5 of C_{A} and the formula (2.8) for u^{∗} that

C_{A}^{·} ∘ u^{∗} ∘ i(F)(U) = F (u_{!}j_{!}^{U}A_{U}^{·}) .

We now observe the isomorphisms

A_{U} ×_{G_{U}} ⋯ ×_{G_{U}} A_{U} ≅ (A ×_{G} ⋯ ×_{G} A) ×_{X} U
≅ (B ×_{H} ⋯ ×_{H} B) ×_{Y} U
≅ B_{v_{!}U} ×_{H_{v_{!}U}} ⋯ ×_{H_{v_{!}U}} B_{v_{!}U} ,

where the notation is explained by the cartesian diagram

\xymatrix{H_{v_{!}U} \ar[d] \ar[r]^{k} & H \ar[d] \\ U \ar[r] & Y},

and where v_{!}U := (U → X →^{v} Y) ∈ Y. We can thus identify the simplicial object u_{!}j_{!}A_{U}^{·} with the similar simplicial object k_{!}B_{v_{!}U} in H. In other words, we have an isomorphism of complexes

C_{A}^{·} ∘ u^{∗} ∘ i(F)(U) ≅ C_{B}^{·} ∘ i(F)(v_{!}U) .

Since i(F) is an injective presheaf the right-hand side is exact by Lemma 2.38. □

2.5 Comparison of big and small sites

2.5.1 Let X be a smooth stack and (U → X) ∈ X. A presheaf on X naturally induces a presheaf on the small site (U) of the manifold U consisting of the open subsets. This restriction functor will be used subsequently in order to reduce assertions in the sheaf theory over X to assertions in the ordinary sheaf theory on U. The goal of the present subsection is to study exactness properties of this restriction and its relation with the sheafification functors. 2.5.2 If U is a smooth manifold, then we let (U) denote the small site of U where covering families are coverings by families of open submanifolds. A presheaf with respect to the big site on U is in particular a presheaf with respect to (U). 2.5.3 Let G be a smooth stack and (U → G) ∈ G. Then we have a functor ν_{U} : Pr G → Pr(U) which associates to the presheaf F ∈ Pr G the presheaf ν_{U}(F) ∈ Pr(U) obtained by restriction of structure. Since limits and colimits in presheaves are defined objectwise the functor ν_{U} is exact. 2.5.4

Lemma 2.45 The functor ν_{U} preserves sheaves and indcues a functor ν_{U}^{s} : ShG → Sh(U).

Proof. An object V ∈ (U) gives rise to an object (V → U → G) ∈ G. Observe that covering families of objects of V ∈ (U) are also covering families of (V → G) ∈ G. For open subsets V_{1}, V_{2} ⊂ V the fibre products V_{1} ×_{V} V_{2} in (U) and in G coincide by the discussion in 2.1.3. Therefore the descent conditions on ν_{U}(F) to be a sheaf on (U) are part of the descent conditions for F to be a sheaf on G. Hence the functor ν_{U} restricts to ν_{U}^{s} : ShG → Sh(U).

□

2.5.5 Since limits of sheaves are defined objectwise the functor ν_{U}^{s} commutes with limits. The goal of the following discussion is to show that it also commutes with colimits.

Proposition 2.46 The functor ν_{U}^{s} : ShG → Sh(U) is exact.

Proof. If F is a diagram of sheaves, then we have

colim^{s}(F) ≅ i^{♯} ∘ colim ∘ i(F) ,

where colim^{s} is the colimit of sheaves. Note that ν_{U} ∘ i ≅ i ∘ ν_{U}^{s} and ν_{U} ∘ colim ≅ colim ∘ ν_{U}. In order to show that ν_{U}^{s} commutes with colim^{s} it remains to show the following Lemma.

2.5.6

Lemma 2.47 We have

i^{♯} ∘ ν_{U} ≅ ν_{U}^{s} ∘ i^{♯} : Pr G → Sh(U) .

Proof. For the moment it is useful to indicate by a subscript (e.g. i_{G} or i_{(U)}) the site for which the functors are considered. Following the discussion in [24, Section 3.1] we introduce an explicit construction of the sheafification functor. Consider the site G. We define the functor P_{G} : Pr G → Pr G as follows. Let (V → G) ∈ G. Then we have the category of covering families cov_{G}(V) whose morphisms are refinements. For τ := (V_{i} → V) ∈ cov_{G}(V) we define H^{0}(F)(τ) by the equalizer diagram

H^{0}(F)(τ) → ∏_{i} F (V_{i}) ⟹ ∏_{i,j} F (V_{i} ×_{V} V_{j}) .

We get a diagram τ → H^{0}(F)(τ) in Sets^{cov_{G}(V)} and define

P_{G}(F)(V) := colim_{τ∈cov_{G}(V)}H^{0}(F)(τ) .

Then we have

i_{G} ∘ i_{G}^{♯} := P_{G} ∘ P_{G} : Pr G → Pr G .

In a similar manner we define a functor P_{(U)} : Pr(U) → Pr(U) and get

i_{(U)} ∘ i_{(U)}^{♯} := P_{(U)} ∘ P_{(U)} : Pr(U) → Pr(U) .

In order to show the Lemma it suffices to show that

P_{(U)} ∘ ν_{U} ≅ ν_{U} ∘ P_{G} .

Let V ⊂ U be open and consider the induced (V → G) ∈ G. Then we have a functor

a : cov_{(U)}(V) → cov_{G}(V) .  (2.48)

If τ ∈ cov_{(U)}(V), then we have an isomorphism

H^{0}(F)(a(τ)) ≅ H^{0}(ν_{U}(F))(τ) .

We therefore have an induced map of colimits

P_{(U)} ∘ ν_{U}(F)(V) → ν_{U} ∘ P_{G}(F)(V) .

This map is in fact an isomorphism since we will show below that (2.48) defines a cofinal subfamily. Let σ := (U_{i} → V)_{i∈I} ∈ cov_{G}(V). Since the maps U_{i} → V are submersions they admit local sections. Hence there exists a covering τ : (V_{j} → V)_{i∈J} ∈ cov_{(U)}, a map r : J → I and a family of sections s_{j} : V_{j} → U_{r(j)} such that

\xymatrix{ & U_{r(j)} \ar[dr] \\ V_{j} \ar[ur]^{s_{j}} \ar[rr] &  & V}

commutes for all j ∈ J. This data defines a morphism σ → a(τ) in cov_{G}(V). □ This finishes the proof of Proposition 2.46. □

2.5.7 Recall the definition of a flabby sheaf 2.28.

Lemma 2.49 The functor ν_{U}^{s} : Sh_{Ab}G → Sh_{Ab}(U) preserves flabby sheaves.

Proof. Let V ⊂ U be an open subset and τ ∈ cov_{(U)}(V). Let a : cov_{(U)}(V) → cov_{G}(V) be as in (2.48). We have an natural isomorphism Č(τ, ν_{U}^{s}(F)) ≅ Č(a(τ), F). If F ∈ Sh_{Ab}G is flabby, then for k ≥ 1 we have H^{k}Č(τ, ν_{U}^{s}(F)) ≅ H^{k}(Č(a(τ), F)) ≅ 0. □

2.5.8 Since ν_{U} : Pr G → Pr(U) is exact it descends to a functor ν_{U} : D^{+}(Pr_{Ab} G) → D^{+}(Pr_{Ab}(U)) between the lower bounded derived categories. Since ν_{U}^{s} : ShG → Sh(U) is exact, it descends to a functor ν_{U}^{s} : D^{+}(Sh_{Ab}X) → D^{+}(Sh_{Ab}(U)). 2.5.9 Using the techniques above we show the following result which will be useful later. Let f : G → H be a smooth map between smooth stacks. Note that f^{∗} : Pr_{Ab} H → Pr_{Ab} G is exact.

Lemma 2.50 (1) We have an isomorphism of functors

f^{∗} ∘ i_{H} ∘ i_{H}^{♯} ≅ i_{G} ∘ i_{G}^{♯} ∘ f^{∗} : Pr_{Ab}H → Pr_{Ab}G .

(2) We have an isomorphism of functors

f^{∗} ∘ Ri_{H} ∘ i_{H}^{♯} ≅ Ri_{G} ∘ i_{G}^{♯} ∘ f^{∗} : D^{+}(Pr_{Ab}H) → D^{+}(Pr_{Ab}G) .

Proof. Let (U → G) ∈ G and f_{!}U := (U → G →^{f} H) ∈ H. We calculate for F ∈ Pr_{Ab} H that on the one hand

(f^{∗} ∘ i_{H} ∘ i_{H}^{♯}F)(U) ≅^{(2.7)} (i_{H} ∘ i_{H}^{♯}F)(f_{!}U)
≅  (ν_{f_{!}U} ∘ i_{H} ∘ i_{H}^{♯}F)(U)
≅^{2.45,2.47} (i_{(U)} ∘ i_{(U)}^{♯} ∘ ν_{f_{!}U}F)(U) .

On the other hand we have

(i_{G} ∘ i_{G}^{♯} ∘ f^{∗}F)(U)  ≅  (ν_{U} ∘ i_{G} ∘ i_{G}^{♯} ∘ f^{∗}F)(U)
≅^{2.45,2.47} (i_{(U)} ∘ i_{(U)}^{♯} ∘ ν_{U} ∘ f^{∗}F)(U) .

Finally we use the fact that ν_{U} ∘ f^{∗}F ≅ ν_{f_{!}U}F. Indeed, for V ⊂ U we have

ν_{U} ∘ f^{∗}F (V) ≅ F (f_{!}V) ≅ ν_{f_{!}U}F (V) .

The combination of these isomorphisms gives the first assertion. Since f^{∗} preserves sheaves we can consider the restriction f_{s}^{∗} : Sh_{Ab}H → Sh_{Ab}G of f^{∗}. Using the first part of the Lemma and the isomorphism i^{♯} ∘ i ≅ id we get

f_{s}^{∗} ∘ i_{H}^{♯} ≅ i_{G}^{♯} ∘ i_{G} ∘ f_{s}^{∗} ∘ i_{H}^{♯} ≅ i_{G}^{♯} ∘ f^{∗} ∘ i_{H} ∘ i_{H}^{♯} ≅^{(1)} i_{G}^{♯} ∘ i_{G} ∘ i_{G}^{♯} ∘ f^{∗} ≅ i_{G}^{♯} ∘ f^{∗} .  (2.51)

Note that f_{s}^{∗} is an exact functor. In order to see that it is right exact we use that f_{∗} preserves sheaves we consider its restriction f_{∗}^{s} to sheaves. For X ∈ ShH and Y ∈ ShG we get a natural isomorphism

Hom_{ShG}(f_{s}^{∗}X, Y) ≅ Hom_{ShG}(i_{G}^{♯} ∘ i_{G}f_{s}^{∗}X, Y)
≅ Hom_{Pr G}(i_{G}f_{s}^{∗}X, i_{G}Y)
≅ Hom_{Pr G}(f^{∗} ∘ i_{H}X, i_{G}Y)
≅ Hom_{Pr H}(i_{H}X, f_{∗} ∘ i_{G}Y)
≅ Hom_{Pr H}(i_{H}X, i_{H} ∘ f_{∗}^{s}Y)
≅ Hom_{ShH}(X, f_{∗}^{s}Y)

Therefore f_{s}^{∗} is a left adjoint and therefore right exact. We now write

f_{s}^{∗} ≅ i_{G}^{♯} ∘ i_{G} ∘ f_{s}^{∗} ≅ i_{G}^{♯} ∘ f^{∗} ∘ i_{H} .

Since i_{G}^{♯} is exact, and f^{∗} and i_{H} are left exact (since they are right adjoints, see Lemma 2.25 for a left adjoint of f^{∗} ) we conclude that f_{s}^{∗} is left exact, too Using the that f_{s}^{∗} preserves flabby sheaves (Lemma 2.29) and that it is an exact functor we get

f^{∗} ∘ Ri_{H} ≅ R(f^{∗} ∘ i_{H}) ≅ R(i_{G} ∘ f_{s}^{∗}) ≅ Ri_{G} ∘ f_{s}^{∗} .

Combining this with (2.51) we get the desired isomorphism

f^{∗} ∘ Ri_{H} ∘ i_{H}^{♯} ≅ Ri_{G} ∘ f_{s}^{∗} ∘ i_{H}^{♯} ≅ Ri_{G} ∘ i_{G}^{♯} ∘ f^{∗} .
□

3 The de Rham complex

3.1 The de Rham complex is a flabby resolution

3.1.1 We want to apply Lemma 2.41 to the sheafification i^{♯}ℝ_{G} of the constant presheaf with value ℝ on G. In particular, we must calculate Ri(i^{♯}ℝ_{G}). This can be done by applying i to a flabby resolution of i^{♯}ℝ_{G}. In the present subsection we introduce the de Rham complex G and show that it is a flabby resolution of i^{♯}ℝ_{G}. The de Rham complex of G is build from the de Rham complexes of the manifolds U for all (U → G) ∈ G. For each U equipped with the topology of the small site it is well known that the de Rham complex resolves the constant sheaf with value ℝ and is flabby. Our task here is to extend these properties to the stack G and the big site. 3.1.2 Let G be a smooth stack and fix an integer p ≥ 0. We define the presheaf Ω^{p}(G) by

Ω^{p}(G)(U) := Ω^{p}(U) .

If ϕ : U → V is a morphism in G, then Ω^{p}(G)(ϕ) := ϕ^{∗} : Ω^{p}(V) → Ω^{p}(U). Since ϕ^{∗} commutes with the de Rham differential we get a complex (Ω^{·}(G), d_{dR}) of presheaves.

Lemma 3.1 The presheaf Ω^{p}(G) is a sheaf and flabby.

Proof. Let (U → G) ∈ G. Observe that ν_{U}(Ω^{p}(G)) is the presheaf of smooth sections of the vector bundle Λ^{p}T^{∗}U. This is actually a sheaf. In order to show that Ω^{p}(G) is a sheaf it suffices to show that the unit Ω^{p}(G) → i_{G} ∘ i_{G}^{♯}(Ω^{p}(G)) of the adjoint pair (i_{G}^{♯}, i_{G}) is an isomorphism. This follows from the calculation

i_{G} ∘ i_{G}^{♯}(Ω^{p}(G))(U)  ≅  ν_{U} ∘ i_{G} ∘ i_{G}^{♯}(Ω^{p}(G))(U)
≅^{Lemma 2.45}  i_{(U)} ∘ ν_{U}^{s} ∘ i_{G}^{♯}(Ω^{p}(G))(U)
≅^{Lemma 2.47}  i_{(U)} ∘ i_{(U)}^{♯} ∘ ν_{U}(Ω^{p}(G))(U)
≅^{ν_{U} (Ω^{p}(G)) is a sheaf} ν_{U}(Ω^{p}(G))(U) .

A sheaf F ∈ Sh_{Ab}(U) on a paracompact space U is called soft if for all closed subsets Z ⊂ U the restriction Γ_{U}(F) → Γ_{Z}(F) is surjective. For a soft sheaf we have R^{i}Γ_{U}(F) ≅ 0 for all i ≥ 1 (see [13, Ex. II 5]). It now follows from [24, Corollary 3.5.3] that a soft sheaf is flabby.

A sheaf of smooth sections of a smooth vector bundle on a smooth manifold is soft. In particular, ν_{U}^{s}(Ω^{p}(G)) is soft and therefore flabby. In order to show that the sheaf Ω^{p}(G) is flabby it suffices by [24, Corollary 3.5.3] to show that R^{k}i(Ω^{p}(G)) ≅ 0 for k ≥ 1. We calculate

R^{k}i(Ω^{p}(G))(U)  ≅  Γ_{U} ∘ R^{k}i(Ω^{p}(G))
≅^{Γ_{U} exact}  H^{k}(Γ_{U} ∘ Ri(Ω^{p}(G)))
≅^{H^{k} objectwise}  H^{k}(Ri(Ω^{p}(G))(U))
≅^{definition of ν_{U}} H^{k}(ν_{U} ∘ Ri_{G}(Ω^{p}(G))(U))
≅^{ν_{U} exact}  R^{k}(ν_{U} ∘ i_{G})(Ω^{p}(G))(U)
≅^{Lemma 2.45}  R^{k}(i_{(U)} ∘ ν_{U}^{s})(Ω^{p}(G))(U)
≅^{Lemma 2.49}  R^{k}i_{(U)} ∘ ν_{U}^{s}(Ω^{p}(G))(U)
≅^{ν_{U}^{s} (Ω^{p}(G)) is flabby} 0 .
□

3.1.3 Let ℝ_{G} denote the constant presheaf on G with value ℝ and i^{♯}ℝ_{G} its sheafification. We have a canonical map ℝ_{G} → H^{0}(Ω^{·}(G)). Since H^{0}(Ω^{·}(G)) is a sheaf we get an induced map i^{♯}ℝ_{G} → H^{0}(Ω^{·}(G)).

Lemma 3.2 The map

i^{♯}ℝ_{G} → Ω^{·}(G)

is a quasi-isomorphism.

Proof. Note that the cohomology sheaves H_{s}^{k}(F^{·}) ∈ Sh_{Ab}G of a complex F^{·} of sheaves of abelian groups on G are defined by H_{s}^{k}(F^{·}) := i_{G}^{♯} ∘H^{k} ∘i_{G}(F^{·}), where H^{k} takes the cohomology of a complex of presheaves objectwise. We calculate

H_{s}^{k}(Ω^{·}(G))(U) ≅^{definition of H_{s}^{k}} (i_{G}^{♯} ∘ H^{k} ∘ i_{G})(Ω^{·}(G))(U)
≅^{Lemma 2.45} (ν_{U}^{s} ∘ i_{G}^{♯} ∘ H^{k} ∘ i_{G})(Ω^{·}(G))(U)
≅^{Lemma2.47} (i_{(U)}^{♯} ∘ ν_{U} ∘ H^{k} ∘ i_{G})(Ω^{·}(G))(U)
≅^{ν_{U} is exact}  (i_{(U)}^{♯} ∘ H^{k} ∘ ν_{U} ∘ i_{G})(Ω^{·}(G))(U)
≅^{Lemma 2.45} (i_{(U)}^{♯} ∘ H^{k} ∘ i_{(U)} ∘ ν_{U}^{s})(Ω^{·}(G))(U)
≅^{definition of H_{s}^{k}} (H_{s}^{k} ∘ ν_{U}^{s})(Ω^{·}(G))(U) .

Since ν_{U}^{s}(Ω^{·}(G)) is the de Rham complex of the manifold U its higher cohomology sheaves vanish by the Poincaré Lemma. This implies that H^{k}(Ω^{·}(G)) ≅ 0 for k ≥ 1. Furthermore, it is well-known that H^{0}(ν_{U}(Ω^{·}(G))) ≅ i_{(U)}^{♯}ℝ_{(U)}. It follows from the observation

ν_{U}^{s} ∘ i_{G}^{♯}ℝ_{G} ≅ i_{(U)}^{♯}ℝ_{(U)}

(proved by arguments similar as above) that i_{G}^{♯}ℝ_{G} ≅ H^{0}(Ω^{·}(G)). □

3.1.4 By Lemmas 3.1 and 3.2 the complex Ω^{·}(G) is a flabby resolution of i^{♯}ℝ_{G}. Therefore

Ri(i^{♯}ℝ_{G}) ≅ i(Ω^{·}(G))

in D^{+}(Pr_{Ab} G). By Lemma 2.41 we have the isomorphism

(Rf_{∗} ∘ Ri)(i^{♯}ℝ_{G}) ≅ C_{A}(Ω^{·}(G))  (3.3)

in D^{+}(Pr_{Ab} X).

3.2 Calculation for U(1)-gerbes

3.2.1 In this subsection we specialize the situation of 2.4.3 to the case where f : G → X is a gerbe with band U(1) according to the definition in 1.3.14 over a manifold X. We thus can assume that A → X is an atlas obtained from a covering of X by open subsets such that the lift s : A → G is an atlas of G. The U(1)-central extension of groupoids in manifolds (A ×_{G} A ⇒ A) → (A ×_{X} A ⇒ A) (we forget the structure maps to G for the moment) is the picture of a gerbe as presented in [11]. In order to compare the sheaf theoretic construction of the cohomology of G with the twisted de Rham complex we must choose some additional geometric structure on G, namely a connection in the sense of [11]. The comparison map will depend on this choice. 3.2.2 A connection on the gerbe f : G → X consists of a pair (α, β), where α ∈ Ω^{1}(A ×_{G} A) is a connection one-form on the U(1)-bundle A ×_{G} A → A ×_{X} A, and β ∈ Ω^{2}(A). Observe that Ω^{2}(A) and Ω^{1}(A ×_{G} A) are the first two spaces of the degree-two part of the graded commutative DG-algebra C_{A}(Ω^{·}(G))(X) ≅ Ω^{·}(G)(A^{·}) discussed in 2.4.9 and 2.4.5. To be a connection the pair (α, β) is required to satisfy the following two conditions:

(1) δβ = d_{dR}α (where δ is the Čech differential of the complex Ω^{·}(A^{·}), and d_{dR} is the de Rham differential) and

(2) δα = 0.

Note that δd_{dR}β = 0 so that there is a unique λ ∈ Ω^{3}(X) which restricts to d_{dR}β. We have d_{dR}λ = 0, and the class [λ] ∈ H^{3}(X; ℝ) represents the image under H^{3}(X; ℤ) → H^{3}(X; ℝ) of the Dixmier-Douady class of the gerbe G → X (see [11] for this fact and the existence of connections). 3.2.3 Let use choose a connection (α, β), and let λ ∈ Ω^{3}(X) be the associated closed three form. We consider (α, β) ∈ C_{A}(Ω^{·}(G))^{2}(X). We consider the sheaf of complexes Ω^{·}[[z]]_{λ} on X which associates to (U →^{i} X) ∈ X the complex

Ω^{·}(U)[[z]], d_{λ} := d_{dR} + λT ,

where T := \frac{d}{dz}, z has degree two, and λ acts by right multiplication by i^{∗}λ. In particular we have d_{λ}z = i^{∗}λ. Note that Ω^{·}[[z]]_{λ} is a sheaf of left Ω_{X}^{·}-DG-algebras. 3.2.4 Observe that z ∈ Ω^{·}[[z]]_{λ}(X) is central. Let L ∈ Pr_{Ab} X be a presheaf of graded unital central Ω_{X}^{·}-algebras. A map of presheaves of graded unital central Ω_{X}^{·}-algebras ϕ : iΩ^{·}[[z]]_{λ} → L determines a section ϕ(z) ∈ L(X). Vice versa, given a section l ∈ L(X) of degree two, there is a unique map of presheaves of graded unital central Ω_{X}^{·}-algebras ϕ : iΩ^{·}[[z]]_{λ} → L such that ϕ(z) = l. For (U →^{i} X) ∈ X the map ϕ_{U} : iΩ^{·}[[z]]_{λ}(U) → L(U) is given by

ϕ_{U}(∑_{k≥0} ω_{k}z^{k}) := ∑_{k≥0} ω_{k}i^{∗}(l)^{k} ,

where i^{∗} : L(X) → L(U) is determined by the presheaf structure of L. If (L, d^{L}) is a presheaf of DG-algebras over Ω_{X}^{·}, then ϕ is a homomorphism of DG-algebras over Ω_{X}^{·} if and only if d^{L}l = λ.

3.2.5

Proposition 3.4 We have an isomorphism

Ω^{·}[[z]]_{λ} ≅ i^{♯}C_{A}(Ω^{·}(G))

in D^{+}(Sh_{Ab}X).

Proof. C_{A}(Ω^{·}(G)) is a presheaf of DG-algebras by the 2.4.10. Given (U → X) ∈ X we have a natural projection π : A_{U}^{0} → U (see 2.4.5 for the notation). It induces a homomorphism of DG-algebras Ω_{X}^{·}(U) → (ker(δ) : Ω^{·}(G)(A_{U}^{0}) → Ω^{·}(G)(A_{U}^{1})) and therefore on Ω^{·}(G)(A_{U}^{·}) the structure of an Ω_{X}^{·}(U)-DG-module (see 2.4.10). In this way C_{A}(Ω^{·}(G)) becomes a sheaf of central Ω_{X}^{·}-DG-algebras. By the discussion in 3.2.4 we can define a map of presheaves of central Ω_{X}^{·}-algebras

ϕ̃: iΩ^{·}[[z]]_{λ} → C_{A}(Ω^{·}(G))

such that ϕ̃(z) = (α, β) ∈ C_{A}(Ω^{·}(G))^{2}(X). Because of dϕ̃(z) = d(α, β) = λ, the map ϕ̃ is a map of presheaves of DG-algebras over Ω_{X}^{·}, hence in particular a map of presheaves of complexes. We let

ϕ : Ω^{·}[[z]]_{λ} →^{∼} i^{♯} ∘ iΩ^{·}[[z]]_{λ} →^{i^{♯}ϕ̃} i^{♯}C_{A}(Ω^{·}(G))

be the induced map, where the first isomorphism exists since Ω^{·}[[z]]_{λ} is a complex of sheaves. 3.2.6 It remains to show that ϕ is a quasi-isomorphism of complexes of sheaves. This can be shown locally. We can therefore assume that X is contractible. We then have a pull-back diagram

\xymatrix{G \ar[r]^{q} \ar[d]^{f} & [∗/S^{1}] \ar[d]^{g} \\ X \ar[r]^{p} & *} .

Since p is smooth, so is q. By Lemma 2.43 we have a canonical isomorphism

p^{∗} ∘ Rg_{∗} ∘ Ri →^{∼} Rf_{∗} ∘ q^{∗} ∘ Ri .  (3.5)

Applying (3.5) to i^{♯}ℝ_{Site([∗/S^{1}])} we obtain

p^{∗} ∘ Rg_{∗} ∘ Ri(i^{♯}ℝ_{Site([∗/S^{1}])}) →^{∼} Rf_{∗} ∘ q^{∗} ∘ Ri(i^{♯}ℝ_{Site([∗/S^{1}])})  (3.6)

in D^{+}(Pr_{Ab} X). We now use (see 3.1.4) that

Ri(i^{♯}ℝ_{Site([∗/S^{1}])}) ≅ i(Ω^{·}([∗/S^{1}])) .

By the calculation of q^{∗} in Lemma 2.7 and the definition of the de Rham complex we have

q^{∗} ∘ i(Ω^{·}([∗/S^{1}])) ≅ i(Ω^{·}(G)) .

Therefore in D^{+}(Pr_{Ab} G) we have

q^{∗} ∘ Ri(i^{♯}ℝ_{Site([∗/S^{1}])}) ≅ i(Ω^{·}(G)) ≅^{3.1.4} Ri(i^{♯}ℝ_{G}) .

It follows by 3.1.4 that

Rf_{∗} ∘ q^{∗} ∘ Ri(i^{♯}ℝ_{Site([∗/S^{1}])}) ≅ Rf_{∗} ∘ Ri(i^{♯}ℝ_{G}) ≅ C_{A}(Ω^{·}(G)) .  (3.7)

3.2.7 We now must calculate the cohomology of the gerbe [∗/S^{1}] with real coefficients.

Lemma 3.8 We have an isomorphism

i^{♯} ∘ Rg_{∗} ∘ Ri(i^{♯}ℝ_{Site([∗/S^{1}])}) ≅ i^{♯}(ℝ[[z]]_{Site(∗)}) ,

where z has degree two.

Proof. We choose the atlas A := ∗ → [∗/S^{1}] and use the isomorphism

Rg_{∗} ∘ Ri(i^{♯}ℝ_{Site([∗/S^{1}])}) ≅ C_{A}(Ω^{·}([∗/S^{1}])) ∈ D^{+}(Pr_{Ab}Site(∗)) .

Note that Site(∗) is the category of smooth manifolds. Let U be a smooth manifold. We have

A_{U} ≅ U → U × [∗/S^{1}] ≅ [∗/S^{1}]_{U}

and

C_{A}(Ω^{·}([∗/S^{1}]))(U) ≅ Ω^{·}(A_{U}^{·}) ,

where

A_{U}^{p} = \underbrace{A_{U} ×_{[∗/S^{1}]_{U}} ⋯ ×_{[∗/S^{1}]_{U}} A_{U}}_{p+1 factors} ≅ U × (\underbrace{∗ ×_{[∗/S^{1}]} ⋯ ×_{[∗/S^{1}]} ∗}_{p+1 factors}) ≅ U × (S^{1})^{p} .

The simplicial manifold

A_{U}^{·} ≅ U × (S^{1})^{·}

is the simplicial model of the space U × BS^{1}, where BS^{1} is the classifying space of the group S^{1}. We can use the simplicial de Rham complex in order to calculate its cohomology. Note that H^{∗}(BS^{1}, ℝ) ≅ ℝ[[z]] with z in degree two. Let us fix a form ζ ∈ (Ω^{·}((S^{1})^{·})_{tot}^{2} which represents the generator z. Then we define a map

μ_{U} : Ω^{·}(U)[[z]] → Ω^{·}(U × (S^{1})^{·})

by

μ(ωz^{k}) := ω ∧ ζ^{k} .

This map induces a quasi-isomorphism of complexes of abelian groups. The family of maps μ_{U} for varying U defines a quasi-isomorphism of complexes of presheaves μ : iΩ(∗)[[z]] → C_{A}(Ω^{·}([∗/S^{1}])). It induces the quasi-isomorphism of complexes of sheaves

Ω(∗)[[z]] ≅ i^{♯} ∘ iΩ(∗)[[z]] ≅^{i^{♯}μ} i^{♯}C_{A}(Ω^{·}([∗/S^{1}])) .

Finally observe that the canonical map

i^{♯}ℝ[[z]]_{Site(∗)} → Ω(∗)[[z]]

is a quasi-isomorphism by Lemma 3.2 □

3.2.8 It follows from Lemma 3.8 by applying p^{∗} ∘ Ri that

p^{∗} ∘ Ri ∘ i^{♯} ∘ Rg_{∗} ∘ Ri(i^{♯}ℝ_{Site([∗/S^{1}])}) ≅ p^{∗} ∘ Ri ∘ i^{♯}ℝ[[z]]_{Site(∗)} .

We now use the second assertion of Lemma 2.50 in order to commute Ri ∘ i^{♯} with p^{∗}. We get

Ri ∘ i^{♯} ∘ p^{∗} ∘ Rg_{∗} ∘ Ri(i^{♯}ℝ_{Site([∗/S^{1}])}) ≅ Ri ∘ i^{♯} ∘ p^{∗}ℝ[[z]]_{Site(∗)} .

We now apply i^{♯} and use that i^{♯} ∘ Ri ≅ id in order to drop the functor Ri and get the quasi-isomorphism

i^{♯} ∘ p^{∗} ∘ Rg_{∗} ∘ Ri(i^{♯}ℝ_{Site([∗/S^{1}])}) ≅ i^{♯} ∘ p^{∗}ℝ[[z]]_{Site(∗)} .

By the explicite description of p^{∗} given in the proof of Lemma 2.9 we see that

p^{∗}(ℝ[[z]]_{Site(∗)}) ≅ ℝ[[z]]_{X} .

We thus have a quasi-isomorphism

i^{♯} ∘ p^{∗} ∘ Rg_{∗} ∘ Ri(i^{♯}ℝ_{Site([∗/S^{1}])}) ≅ i^{♯}(ℝ[[z]]_{X}) .  (3.9)

Combining the isomorphisms (3.6), (3.7) and (3.9) we obtain a quasi-isomorphism

i^{♯}C_{A}(Ω^{·}(G)) ≅ i^{♯}(ℝ[[z]]_{X}) .

In particular we see that z generates the cohomology. 3.2.9 Since X is contractibe we find γ ∈ Ω^{2}(X) such that d_{dR}γ = λ. We define a map of complexes of sheaves

ψ : i^{♯}ℝ[[z]]_{X} → Ω^{·}[[z]]_{0} →^{e^{−γT}} Ω^{·}[[z]]_{λ} .

The first map is given by the inclusion i^{♯}ℝ_{X} → Ω_{X}^{·} and is a quasi-isomorphism. The second map is an isomorphism of sheaves of complexes. Therefore ψ is a quasi-isomorphism. Note that ψ is multiplicative and ψ(z) = z − γ. We further define κ : i^{♯}ℝ[[z]]_{X} → i^{♯}C_{A}(Ω^{·}(G)) such that κ(z) = (α, β − γ) = ϕ(z − γ). Then we have a commutative diagram

\xymatrix{i^{♯}ℝ[[z]]_{X} \ar[rr]^{κ} \ar[dr]^{ψ} & & i^{♯}C_{A}(Ω^{·}(G)) \\ & Ω^{·}[[z]]_{λ} \ar@{-->}[ur]^{ϕ}} .

If we show that κ is a quasi-isomorphism, then since ψ is a quasi-isomorphism, ϕ must be a quasiisomorphism, too. It suffices to see that κ(z) := (α, β − γ) represents a non-trivial cohomology class. Assume that it is a boundary locally on (U → X) ∈ X. Then there exists x ∈ Ω^{0}(G)(A_{U}^{1}) and y ∈ Ω^{1}(G)(A_{U}^{0}) such that δx = 0, d_{dR}x + δy = α_{U} and d_{dR}y = (β − γ)_{U} (the subscript indicates that the forms are pulled back to A_{U}^{∗}). By exactness of the δ-complex we can in fact assume that x = 0. But then the equation δy = α_{U} is impossible since δy vanishes on vertical vectors on the bundles A_{U}^{1} → A_{U}^{0} given by the source and range projections while α as a connection form is non-trivial on those vectors. □

3.2.10 We now finish the proof of Theorem 1.1. We combine Proposition 3.4 with 3.1.4 and the fact that f_{∗} preserves sheaves (Lemma 2.13) in order to get

i^{♯} ∘ Rf_{∗} ∘ Ri(i^{♯}ℝ_{G}) ≅ i^{♯}C_{A}(Ω^{·}(G)) ≅ Ω^{·}[[z]]_{λ} .
□

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