aa r X i v : . [ m a t h . A T ] D ec EQUIVARIANT SHEAVES FOR PROFINITE GROUPS
DAVID BARNES AND DANNY SUGRUE
Abstract.
We develop the theory of equivariant sheaves over profinite spaces, where the groupis also taken to be profinite. We construct a good notion of equivariant presheaves, with asuitable sheafification functor. Using these results on equivariant presheaves, we give explicitconstructions of products of equivariant sheaves of R -modules. We introduce an equivariantanalogue of skyscraper sheaves, which allows us to show that the category of equivariant sheavesof R -modules over a profinite space has enough injectives.This paper also provides the basic theory for results by the authors on giving an algebraicmodel for rational G -spectra in terms of equivariant sheaves over profinite spaces. For thoseresults, we need a notion of Weyl- G -sheaves over the space of closed subgroups of G . We showthat Weyl- G -sheaves of R -modules form an abelian category, with enough injectives, that is a fullsubcategory of equivariant sheaves of R -modules. Moreover, we show that the inclusion functorhas a right adjoint. Contents
1. Introduction 22. Profinite groups, profinite spaces and discrete modules 33. Sheaves and presheaves 74. Equivariant sheaves 105. Sections of equivariant sheaves 136. Equivariant presheaves 147. G -sheaves of R -modules 168. Colimits and limits of equivariant sheaves and presheaves 199. Pull backs and push forwards 2210. Equivariant skyscraper sheaves 2311. Godemont resolutions of equivariant sheaves 2512. Weyl sheaves over spaces of subgroups 2713. Sheaves over a diagram of spaces 31References 36 The authors extend their thanks to Scott Balchin and Tobias Barthel for many useful conversations and a carefulreading of a draft. Introduction
We set forth the structure of G -equivariant sheaves (of sets or of abelian groups) with a focus onthe case where G is a profinite group and the base space is a profinite space. This theory underliesthe results of the authors in [BS20] and [BS21]. In the first paper, we construct an equivalencebetween the category of rational G -Mackey functors and Weyl- G -sheaves — a full subcategory ofrational G -equivariant sheaves over S G , the space of closed subgroups of G . In the second paper,we use that equivalence to give a classification of rational G -spectra in terms of rational Weyl- G -sheaves. As part of that classification, the injective dimensions of the categories of rational G -equivariant sheaves and rational G -Mackey functors are studied.The starting point of the new results of this paper is Proposition 5.1, which is similar to a resultof Schneider, see [Sch98, Statement (3)]. This proposition states that for a profinite group G , asection of a G -equivariant sheaf over a profinite base space X can be restricted to an N -equivariantsection, for some open subgroup N of G . We describe this as saying that “every section is locallysub-equivariant” and have the following useful rephrasing. Corollary A (Corollary 5.2) . Let F be a G -equivariant sheaf of sets over a profinite G -space X .For U a compact open subset of X , the set of sections F ( U ) is a discrete stab G ( U ) -set. This discrete action encodes the continuity of the stabiliser action on sections. This continuity isthe key to defining a notion of equivariant presheaf, as it ensures that the associated sheaf space hasa continuous action of a profinite group G . We summarise our results below, where G - PreSheaf ( B ) is the category of G -equivariant presheaves over B , a compact-open basis for the profinite G -space X . Theorem B (Theorems 6.3 and 6.5) . There is a notion of an equivariant presheaf and a construc-tion of equivariant sheafification that defines an adjunction with the forgetful functor ℓ : G - PreSheaf ( B ) / / G - Sheaf ( X ) : forget . o o The functor forget ◦ ℓ is idempotent, hence the forgetful functor is fully faithful. For R a ring, these results extend to G -sheaves of (left) R -modules and can be used to show thatwe have an abelian category and give explicit constructions of (small) colimits and limits, includinginfinite products. Theorem C (Theorem 7.9 and Proposition 8.7) . The category of G -equivariant sheaves of R -modules over a profinite G -space X is an abelian category with all small limits and colimits. Another useful construction that makes use of our results on equivariant presheaves is the con-struction of change of space functors (pull backs and push forwards) for equivariant sheaves, asgiven in Section 9.As well as the constant equivariant sheaf of Examples 4.10 and 7.5, we give an equivariant versionof a skyscraper sheaf in Example 10.5. Such a sheaf has non-zero stalks over exactly one orbit ofthe base space X (as equivariantly speaking, orbits take the place of points). This construction isadjoint to the functor which takes a stalk at a point of X . The class of equivariant skyscrapers areessentially the only example one needs, as the following makes clear. QUIVARIANT SHEAVES FOR PROFINITE GROUPS 3
Theorem D (Theorem 11.2) . The injective equivariant skyscraper sheaves provide enough injectivesfor the category of G -equivariant sheaves of R -modules over X . The next topic is a study of the category of Weyl- G -sheaves, which was central to [BS20]. Thisis a category of G -equivariant sheaves of R -modules over S G , such that the stalk over K is K -fixed.We show that the category is abelian and give explicit constructions of all small limits and colimits.We show that we can construct a Weyl- G -sheaf from a G -equivariant sheaf over S G by takingstalk-wise fixed points. Theorem E (Lemma 12.8 and Theorem 12.10) . There is an adjunction inc : Weyl - G - sheaf R ( S G ) / / G - Sheaf R ( S G ) : Weyl . o o The functor inc ◦ Weyl is idempotent, hence the inclusion functor is fully faithful.The category of Weyl- G -sheaves of R -modules is an abelian category with all small colimits andlimits. Limits are constructed by taking the limit of the underlying diagram in G - Sheaf R ( S G ) andapplying the functor Weyl . We end the paper by showing how to construct a G -equivariant sheaf over a profinite spaceby combining compatible equivariant sheaves over finite discrete spaces, see Theorem 13.6. Thisconstruction is extended to Weyl- G -sheaves in Corollary 13.8. Since equivariant sheaves over finitediscrete spaces are products of the stalks (with group actions), this gives an alternative method tosheafification of defining an equivariant sheaf from algebraic information.As well as those new results, the paper provides an introduction to sheaves and equivariantsheaves in Sections 2 and 3, reminding the reader of the definitions and surrounding terminologyof sheaves and profinite groups. We combine these notions to give two equivalent definitions of thecategory of G -equivariant sheaves (via equivariant local homeomorphisms and via pullback sheaves)in Section 4.Equivariant sheaves have been studied before, for example in the book of Bernstein and Lunts[BL94]. However, the results of this paper take a very different direction. In that book, the authors(very roughly speaking) want to make a derived category of sheaves over a free replacement of thebase space. We are quite happy for the base space to have G -fixed points. Indeed, S G — the spaceof closed subgroups of G , under conjugation by G — is the base space of greatest interest for ourapplications and will always have fixed points.2. Profinite groups, profinite spaces and discrete modules
We give a few reminders of useful facts on profinite groups, profinite spaces and discrete modules.More details can be found in Wilson [Wil98] or Ribes and Zalesskii [RZ00].A profinite group is a compact, Hausdorff, totally disconnected topological group. A profinitegroup G is homeomorphic to the inverse limit of its finite quotients: G ∼ = lim N P open G G/N ⊆ Y N P open G G/N.
The limit has the canonical topology which can either be described as the subspace topology onthe product or as the topology generated by the preimages of the open sets in
G/N under the
QUIVARIANT SHEAVES FOR PROFINITE GROUPS 4 projection map G → G/N , as N runs over the open normal subgroups of G . It follows that theopen normal subgroups form a neighbourhood basis of the identity. Moreover, the intersection ofall open normal subgroups is the trivial group. Lemma 2.1.
Let G be a profinite group and K a closed subgroup. Then K = \ N P open G N K.
Proof.
The intersection contains K , so assume g ∈ G is in the intersection. For any open normalsubgroup N , we may write g = nk for n ∈ N and k ∈ K . Hence n = gk − ∈ N ∩ gK . The sets N ∩ gK are non-empty and satisfy the finite intersection property as N is an open subgroup. Bycompactness of G , we have that ∅ 6 = \ N P open G N ∩ gK ⊆ \ N P open G N = { e } . Hence, for any N , e ∈ N ∩ gK ⊆ gK for any N . We have shown that g ∈ K . (cid:3) Closed subgroups and quotients by closed normal subgroups of profinite groups are also profinite.A subgroup of a profinite group is open if and only if it is of finite index and closed. The trivialsubgroup { e } is open if and only if the group is finite. Any open subgroup H contains an opennormal subgroup, the core of H in G , which is defined as the finite intersectionCore G ( H ) = \ g ∈ G gHg − . If a (not necessarily closed) subgroup H of G contains a subgroup K which is open in G , then H itself is open.We can also define a profinite topological space to be a Hausdorff, compact and totally discon-nected topological space. As with profinite groups, such a space is homeomorphic to the inverselimit of a cofiltered diagram of finite discrete spaces. Moreover, a profinite topological space has anopen-closed (hence compact-open) basis.For G a topological group, an action of G on X is a map G × X → X which is associative andunital. We often refer to such an X as a G -space . A map of G -spaces f : X −→ Y is a map oftopological spaces which commutes with the action of G , that is, f ( gx ) = gf ( x ) for all x ∈ X and g ∈ G . We often refer to such a map as a G -equivariant map.We now give a few key results about the action of a profinite group on topological spaces. Lemma 2.2.
Let G be a profinite group and X a G -space. If U is an open compact subset of X ,then there exists an open normal subgroup N of G such that N U = U .Proof. Let ψ be the action map ψ : G × X −→ X . For each x ∈ U we know that ( e, x ) ∈ ψ − ( U ) .As ψ − ( U ) is open, there are open neighbourhoods V x ∋ e and U ∋ x such that V x × U x ⊆ ψ − ( U ) .This implies that V x U x ⊆ U . As the open normal subgroups are a neighbourhood basis for theidentity, there are open normal subgroups N x ⊆ V x such that N x U x ⊆ U . QUIVARIANT SHEAVES FOR PROFINITE GROUPS 5 As { U x | x ∈ U } is an open cover of the compact set U , we may choose a finite subcover { U x i | ≤ i ≤ n } . We see that N = \ ≤ i ≤ n N x i is an open normal subgroup of G which satisfies N U = U . (cid:3) We may think of this as an application of the fact that profinite groups have many “small sub-groups”. By way of comparison, Lie groups have no non-trivial “small subgroups”. For example, theproper closed subgroups of S are the finite cyclic groups. Thus, a (sufficiently) small neighbourhoodof the identity will only contain the trivial subgroup. Lemma 2.3.
Let G be a profinite group and X a G -space. If U ⊆ X is compact and open, thenthe stabiliser subgroup of U stab G ( U ) = { g ∈ G | gU = U } is an open subgroup of G . Hence, stab G ( U ) has finite index in G .Proof. The preceding lemma guarantees an open normal subgroup is contained in stab G ( U ) , hencestab G ( U ) is open. (cid:3) We now look at the action of profinite groups on R -modules, where R is a unital ring. We assumethat R has trivial G -action. Definition 2.4.
Let M be a set with an (associative and unital) action of G . We say that M is discrete if, when M is equipped with the discrete topology, the action of G on M is continuous. Amap of such sets is a G -equivariant map of sets.Let M be an R -module with an (associative and unital) action of G that commutes with the actionof R , that is, an R [ G ] -module. We say that M is a discrete R [ G ] -module if, when M is equippedwith the discrete topology, the action of G on M is continuous. The category of R [ G ] -modules isdenoted R [ G ]- mod and the full subcategory of discrete R [ G ] -modules is denoted R d [ G ]- mod.We can describe discreteness in more algebraic terms and produce a discretisation functor usingthis description. We give the following results for the case of R [ G ] -modules, but similar statementscan be made for G -sets. Lemma 2.5.
Let G be a profinite group and M an abelian group with an action of G . Then M isdiscrete if and only if the canonical map colim N P open G M N ∼ = colim H open G M H ∼ = −→ M is an isomorphism. The isomorphism of colimits is a cofinality argument, based on the fact that every open subgroupcontains an open normal subgroup. The lemma can be rephrased as saying that M is discrete ifand only if for each element m ∈ M there is an open (normal) subgroup H of G such that m is H -fixed. Equally, we may say the the stabiliser of any point is open. QUIVARIANT SHEAVES FOR PROFINITE GROUPS 6
Examples 2.6.
The primary examples of discrete R [ G ] -modules are the modules R [ G/H ] (the R -module with basis the cosets of H in G ) for H an open subgroup of G . To see this is discrete,note that the basis element gH is fixed by the open subgroup gHg − . A finite sum of such elementsis fixed by the intersection of those subgroups, which is also open. Hence, every element is fixed bysome open subgroup. Conversely, the module R [ G ] is not discrete when G is infinite as the basiselement e is not fixed by any open subgroup.Given any R [ G ] -module M , the colimit of Lemma 2.5 defines a discrete R [ G ] -module, as thefollowing lemma states. This construction is part of an adjunction which also allows for a descriptionof limits in the category of discrete R [ G ] -modules. Lemma 2.7.
For G a profinite group, the inclusion R d [ G ]- mod −→ R [ G ]- mod has a right adjoint, disc , which is defined by disc ( M ) = colim H open G M H . We call this functor discretisation . The map disc ( M ) −→ M is a monomorphism, which is an isomorphism if M is already discrete. We sometimes call disc ( M ) the discrete part of M .The category of R [ G ] -modules has all small limits and colimits, defined in terms of underlying R -modules with induced G -actions. Colimits in R d [ G ]- mod are constructed similarly. A limit in R d [ G ]- mod is given by constructing the limit in R [ G ]- mod and then applying disc . That the limit is as described follows from the fact that the image of a discrete module under a G -equivariant map is contained in the discrete part of the codomain. Remark 2.8.
Note that nontrivial modules can have trivial discretisation, such as Q [ G ] when G isinfinite. Indeed, an element of Q [ G ] is a finite sum of (rational multiples of) basis elements. Sincethe sum is finite, it cannot be fixed by an open subgroup (which is necessarily infinite).Since finite limits commute with filtered colimits in module categories, a finite limit in R d [ G ]- modis given by the limit in R [ G ]- mod. The problem lies with the infinite product. Indeed, let G = Z ∧ p (for p a prime) and consider the following product of discrete Z ∧ p -modules. Y n > R [ Z /p n ] . Recall that the open subgroups are of the form p k Z ∧ p for k > . The element of the productconsisting of 1 in each factor is not fixed by any open subgroup. Of course, we already expected aproblem like this, as the infinite product of discrete spaces is not discrete, so an infinite product ofdiscrete R [ G ] -modules would have to be more complicated than the underlying product.We finish this section by noting that we have enough injective discrete R [ G ] -modules. QUIVARIANT SHEAVES FOR PROFINITE GROUPS 7
Lemma 2.9.
For G a profinite group and R a ring, the category of discrete R [ G ] -modules hasenough injectives.Proof. The inclusion functor from discrete R [ G ] -modules to R [ G ] -modules preserves monomor-phisms. Hence, its right adjoint discretisation preserves injective objects. Let A be a discrete R [ G ] -module, then as the category of R [ G ] -modules has enough injectives, there is an injective R [ G ] -module I and a monomorphism i : A −→ I . Since the image of A in I is discrete, i factorsover a monomorphism A −→ disc ( I ) . (cid:3) Sheaves and presheaves
We give the basic definitions and terminology, Swan [Swa64] and Tennison [Ten75] provide moredetails.
Definition 3.1.
Let X be a topological space. A presheaf of sets over X is a contravariant functorfrom the category of open sets of X and inclusions to sets F : O ( X ) op −→ Sets . A map of presheaves is a natural transformation.We call an element of F ( U ) a section of F supported on U . The stalk of F at x ∈ X is thefollowing colimit F x = colim U ∋ x F ( U ) and an element of the stalk is called a germ . For s ∈ F ( U ) , the image of s in F x is denoted s x . Definition 3.2. A sheaf of sets over X is a presheaf of sets over X that satisfies the sheaf condition :for each cover of an open set U , U = ∪ λ ∈ Λ U λ , we have an equaliser diagram F ( U ) ∼ = eq a λ ∈ Λ F ( U λ ) / / / / a ( λ,µ ) ∈ Λ × Λ F ( U λ ∩ U µ ) ! where the maps are induced by the restriction maps. A map of sheaves over X is a map of underlyingpresheaves. Remark 3.3.
If we write the equaliser as lim λ ∈ Λ F ( U λ ) , then we may describe the the sheafcondition as an isomorphism F ( U ) −→ lim λ ∈ Λ F ( U λ ) . That this map is surjective is also known as the patching condition , which says that a set of sections s λ ∈ F ( U λ ) , which agree under restrictions to intersections, can be patched into a section s ∈ F ( U ) which restricts to each s λ . That the above map is injective is also known as the separation condition ,which says that if two sections s, t ∈ F ( U ) agree on each F ( U λ ) then they are the same section. Definition 3.4. A local homeomorphism is a continuous map p : E → X such that for every e ∈ E ,there is an open set V ∋ e of E and an open set U ∋ p ( e ) of X such that p | V is a homeomorphismonto U . We call X the base space of the local homeomorphism. For X a topological space, a sheafspace over X is a local homeomorphism p : E → X . QUIVARIANT SHEAVES FOR PROFINITE GROUPS 8
Given a map of topological spaces p : E → X , a section of p over an open set U ⊆ X is acontinuous map s : U → E such that p ◦ s = Id U . By convention, we write s x for the value of s at x ∈ U .In particular, local homeomorphisms are open maps and the images of sections over the opensets of X form a basis for the topology of E , see Tennison [Ten75, Lemma 2.3.5]. Lemma 3.5. If p : E → X is a local homeomorphism, then the preimage of a point x ∈ X (as asubspace of E ) is a discrete topological space. Lemma 3.6.
A sheaf of sets over X defines a sheaf space over X . Conversely, a sheaf space over X defines a sheaf of sets over X . These two constructions are mutually inverse.Proof. Given a sheaf F , define a set E = a x ∈ X F x . We put a topology on E by defining a basis. An open set of the basis is given by B ( s, U ) = { s x | x ∈ U } for s a section of p over U . We define a map p : E → X by sending the elements of F x to x .Given a sheaf space p : E → X and U an open subset of X , define the set F ( U ) = Γ( U, E ) = { sections of p over U } . The functor F = Γ( − , E ) defines a sheaf over X . (cid:3) As the constructions are inverse to each other, we see that a section of a sheaf space is exactlya section of the corresponding sheaf. That is, given a sheaf F , we can make a sheaf space E ,and see that sections F ( U ) are precisely the sections of the sheaf space over U : Γ( U, E ) . ByTennison [Ten75, Proposition 2.3.6], the stalk of a sheaf is the preimage of the corresponding localhomeomorphism: colim U ∋ x Γ( U, E ) = E x = p − ( x ) = colim U ∋ x F ( U ) = F x . The construction of a sheaf space did not require that the input was a sheaf, only that we hada presheaf. thus we may take a presheaf F , construct a sheaf space from it E and then construct asheaf from E . We formalise this in the following. Definition 3.7.
The sheafification of a presheaf F is the sheaf corresponding to the sheaf space ofthe presheaf, which we write as ℓF . Lemma 3.8.
The sheafification construction ℓ is a functor that is left adjoint to the forgetful functorfrom sheaves to presheaves (this functor is often omitted from the notation).The unit of this adjunction gives a canonical map from a presheaf to a sheaf F → ℓF . Thecanonical map ℓF → ℓ F is an isomorphism. Furthermore, if E is a sheaf, a map F → E factorsover F → ℓF . We can alter the definition of a presheaf to land in abelian groups or modules over a ring R . Wethen define sheaves and sheaf spaces of R -modules in a similar fashion. QUIVARIANT SHEAVES FOR PROFINITE GROUPS 9
Definition 3.9.
Let X be a topological space. A presheaf of R -modules over X is a contravariantfunctor from the category of open sets of X and inclusions to sets F : O ( X ) op −→ R - mod . A sheaf of R -modules over X is a sheaf whose underlying presheaf is a presheaf of R -modules over X .A sheaf space of R -modules over X is a sheaf space whose underlying presheaf is a presheaf of R -modules over X .The given definition of a sheaf space of R -modules is not as useful as one would like, so a moreintrinsic characterisation is given below. Lemma 3.10.
A sheaf space p : E → X is a sheaf space of R -modules if and only if for each x ∈ X , p − ( x ) is an R -module and the map E × X E = { ( e, e ′ ) | p ( e ) = p ( e ′ ) } −→ E ( e, e ′ ) e − e ′ induced by the pointwise group operation is continuous and for r ∈ R , the map e re is acontinuous self map of E . The results on sheaves, sheaf spaces and sheafification all extend to R -module variants. Fromhere on we will no longer be as precise about the difference between a sheaf and a sheaf space.We will find it useful to restrict our definition of presheaves to a basis in Section 6 where we defineequivariant presheaves. The definition relies on the following result of Tennison [Ten75, Lemma4.2.6]. It says that a sheaf over a topological space X is uniquely determined by its behaviour ona basis for the topology of X . The lemma is stated for (pre)sheaves of sets but readily extends tosheaves of R -modules. Lemma 3.11.
Let X be a topological space with B a basis for the topology. Write O ( B ) to be thecategory of elements of B and inclusions.Assume there is a contravariant functor F : O ( B ) op −→ Sets such that whenever U = ∪ λ ∈ Λ U λ in O ( B ) , we have an equaliser diagram F ( U ) = eq a λ ∈ Λ F ( U λ ) / / / / a ( λ,µ ) ∈ Λ × Λ F ( U λ ∩ U µ ) ! . Then there is a sheaf F ′ on X , such that F ′ restricted to O ( B ) op is equal to F Moreover, this sheaf F ′ is unique up to isomorphism. The proof is based on the idea that one can define a sheaf space from F as for an ordinary(pre)sheaf. One also sees that F ′ ( V ) = lim O ( B ) ∋ U ⊆ V F ( U ) describes F ′ as a right Kan extension over the inclusion O ( B ) → O ( X ) . QUIVARIANT SHEAVES FOR PROFINITE GROUPS 10
We will need to consider change of base space constructions on sheaves, particularly in the caseof the maps g : X −→ X coming from the action of a group G on X . Again we state the theoremfor sets, but it can be written for R -modules just as well. Definition 3.12.
Let f : X → Y be a map of topological spaces. Let F be a presheaf over X andlet E be a sheaf space over Y .We define the presheaf f ∗ ( F ) over Y , the push forward of F over f , to be the functor O ( Y ) op f − −→ O ( X ) op F −→ Setswhere f − is the pre-image functor induced by f .We define a sheaf space f ∗ ( E ) over X , the pull back of E over f by the pullback f ∗ ( E ) / / (cid:15) (cid:15) E p (cid:15) (cid:15) X f / / Y. Proposition 3.13.
Let f : X → Y be a map of topological spaces. If F is a sheaf over X , then f ∗ ( F ) is a sheaf over Y . If E is a sheaf space over Y , then f ∗ ( E ) is a sheaf space over X . Given x ∈ X , there is an isomorphism of stalks f ∗ ( E ) x ∼ = E f ( x ) . The functors f ∗ and f ∗ form an adjunction f ∗ : Sheaf ( Y ) / / Sheaf ( X ) : f ∗ o o with the left adjoint f ∗ preserving finite limits. That f ∗ preserves finite limits follows from the analogous statement for categories of over-spaces,see also Construction 8.3. Definition 3.14.
When f : X → Y is an inclusion of a subspace, we call f ∗ the restriction to X functor .When working with sheaves of R -modules, if f : X → Y is an inclusion of a closed subset, then f ∗ is called extension by zero .Given a sheaf F over X and an inclusion of a closed subset f : X → Y , the extension by zero f ∗ F has zero stalks outside X . Moreover, the restriction of the extension by zero f ∗ f ∗ F , is isomorphicto F . We also note that if f is an open map and F a sheaf over Y , then the pull back sheaf f ∗ F can be defined by f ∗ F ( U ) = F ( f ( U )) .4. Equivariant sheaves
In this section we introduce the category of G -equivariant sheaves of sets. There are severalequivalent definitions, we start by giving a version from Bernstein and Lunts [BL94] in terms ofpullback sheaves. See also Scheiderer [Sch94, Chapter 8]. QUIVARIANT SHEAVES FOR PROFINITE GROUPS 11
Let G be a topological group and X a topological space with a left G -action. We name severalmaps relating to this action G × G × X e + + e / / e G × X d & & d X s o o e ( g, h, x ) = ( h, g − x ) d ( g, x ) = g − xe ( g, h, x ) = ( gh, x ) d ( g, x ) = xe ( g, h, x ) = ( g, x ) s ( x ) = ( e, x ) Definition 4.1. A G - equivariant sheaf of sets over X is a sheaf of sets F over X together with anisomorphism ρ : d ∗ ( F ) −→ d ∗ ( F ) such that the cocycle condition and the identity condition hold: e ∗ ρ = e ∗ ρ ◦ e ∗ ρ s ∗ ρ = Id F . A morphism f : F → F ′ of G -equivariant sheaves of sets over X is a map f of sheaves of sets suchthat the following square commutes. d ∗ ( F ) ρ (cid:15) (cid:15) d ∗ ( f ) / / d ∗ ( F ′ ) ρ ′ (cid:15) (cid:15) d ∗ ( F ) d ∗ ( f ) / / d ∗ ( F ′ ) We write G - Sheaf ( X ) for this category and often shorten the name G -equivariant sheaf to G - sheaf .For each pair ( g, x ) ∈ G × X , we can calculate the values on the stalks d ∗ ( F ) ( g,x ) = { ( g, x, f ) | f ∈ F x } ∼ = F x and d ∗ ( F ) ( g,x ) = (cid:8) ( g, x, f ) | f ∈ F g − x (cid:9) ∼ = F g − x . It follows that ρ induces isomorphisms on stalks ρ ( g,x ) : F x ∼ = d ∗ ( F ) ( g,x ) −→ d ∗ ( F ) ( g,x ) ∼ = F g − x . Taking the inverse gives maps g : F x −→ F gx for any g and x . The cocycle and identity conditionsimply that these maps are compatible under composition, so that F x g −→ F gx g ′ −→ F g ′ gx is the same as the map g ′ g and e acts as the identity. Consequently, F x has an action of stab G ( x ) .There is a second definition of G -equivariant sheaves, which is more focused on the local home-omorphism definition of a sheaf. Definition 4.2. A G -equivariant sheaf of sets over a topological space X is a G -equivariant localhomeomorphism p : E −→ X of spaces with continuous G -actions.We will write this as either the pair ( E, p ) or simply as E . A map f : ( E, p ) −→ ( E ′ , p ′ ) of G -equivariant sheaves of sets over X is a G -equivariant map f : E → E ′ such that p ′ f = p . QUIVARIANT SHEAVES FOR PROFINITE GROUPS 12
These two definitions of sheaves give equivalent categories. Just as one eventually ceases todistinguish between sheaves and sheaf spaces, one may freely use either definition of an equivariantsheaf.
Theorem 4.3.
The category of equivariant sheaves of Definition 4.1 is equivalent to the categoryof equivariant sheaves of Definition 4.2.Proof.
A proof in the case of sheaves of Q -modules can be found in work of the second author[Sug19, Theorem 4.1.5] and Kędziorek [Kęd14, Lemma 7.0.3]. The core of the argument holds inthe case of sheaves of sets. (cid:3) As non-equivariantly, we have a notion of sections of a sheaf.
Definition 4.4.
Let p : E → X be a a G -sheaf and U an open subset of X . Define Γ( U, E ) to bethe set of (non-equivariant) sections s : U −→ E .Given a (non-equivariant) section s : U −→ E and g ∈ G , we can define a section of gUg ∗ s = g ◦ s ◦ g − : gU −→ E which sends v = gu to gs ( u ) = gs ( g − v ) .If U is invariant under the action of a subgroup H (that is, hU = U for all h ∈ H ), theconstruction h ∗ − for h ∈ H defines a H -action on the space of sections Γ( U, E ) .For H a subgroup of G , the H -fixed points of this space are those sections which commute withthe action of H , which we call H -equivariant sections.Take a section s over U of a G -sheaf p : E → X , then for x ∈ X , g ∈ Gg ( s x ) = ( g ∗ s ) gx which is the key formula for relating the action of G on sections and stalks.That we are correct in taking the sections to be non-equivariant is given by the following, whichfollows from the proof of Theorem 4.3. Corollary 4.5.
Given a G -sheaf F as in Definition 4.1 we have a corresponding G -sheaf space E Definition 4.2. For U an open subspace of X , there is a canonical bijection F ( U ) ∼ = Γ( U, E ) and this bijection is natural in inclusions of subsets U → V and isomorphisms g : U → gU . Lemma 4.6.
Given a G -sheaf ( E, p ) over X and x ∈ X , the stalk E x is a discrete stab G ( x ) -set.Proof. Given a G -sheaf ( E, p ) over X and x ∈ X , the stalk E x is a discrete subspace of E , with acontinuous action of stab G ( x ) . (cid:3) Remark 4.7.
Let G δ be the group G , but with the discrete topology. Then a G -space X can beconsidered as a G δ -space via the identity map G δ → G . An interesting question is how G δ -sheavesover X and G -sheaves over X are related. Scheiderer provides an answer in [Sch94, Remark 8.3],which states that a G δ -sheaf over a G -space X is a G -sheaf if and only if gs x = s gx for all g in someneighbourhood of the identity of G . QUIVARIANT SHEAVES FOR PROFINITE GROUPS 13
Example 4.8.
If we let X be the trivial one point G -space, then a G -sheaf of sets over X is adiscrete G -set.We give a name to the special case where these actions on each stalk of a sheaf are trivial. Definition 4.9. A G -sheaf of sets E over a G -space X is said to be stalk-wise fixed if the actionof stab G ( x ) on E x is trivial for each x ∈ X .The evident example of a stalk-wise fixed sheaf is a constant sheaf. This example also demon-strates that the G -action on the sheaf space of a stalk-wise fixed sheaf is not trivial in general. Example 4.10.
Let X be a G -space and A a set (equipped with the discrete topology whereneeded). The equivariant constant sheaf at A over X is defined by the projection onto the X factor, which is a local homeomorphism p : A × X −→ X. Sections of equivariant sheaves
From this section onwards we will assume that both G and X are profinite. This assumptionallows us to show that any section of a G -equivariant is locally sub-equivariant in the sense of thefollowing result. A similar result can be found in Schneider [Sch98, Statement (3)]. Proposition 5.1.
Let F be a G -sheaf of sets over a profinite G -space X and let U be an opensubset of X .Any section of F ( U ) restricts to an N -equivariant section over an N -invariant domain, for someopen normal subgroup N of G .Proof. Write p : E −→ X be the sheaf space description of F . Given a section s : U −→ E , we canfind a compact-open subset of the form V contained in U since X has an open-closed basis. ByLemma 2.2, V is N -invariant for some open normal subgroup N of G .The set s ( V ) ⊂ E is compact and open, hence another application of Lemma 2.2 gives an opennormal subgroup N such that s ( V ) is N -invariant.Set N = N ∩ N , we claim that s is N -equivariant. As V and s ( V ) are invariant under N ,we have ny ∈ V and ns ( y ) ∈ s ( V ) for all y ∈ V and n ∈ N . Using this and the fact that p is a G -equivariant injection on V , we have p ◦ s ( ny ) = ny p ( ns ( y )) = np ( s ( y )) = ny, and so s ( ny ) = ns ( y ) for all y ∈ V and n ∈ N . (cid:3) The next result is key to Definition 6.2, which defines equivariant presheaves. That definitionencodes the continuity of the G action on a sheaf space in terms of its actions on sections, asTheorem 6.5 implies. Corollary 5.2.
Let F be a G -sheaf of sets over a profinite G -space X . For U a compact opensubset of X , the set of sections F ( U ) is a discrete stab G ( U ) -set. QUIVARIANT SHEAVES FOR PROFINITE GROUPS 14
Proof.
Let E be the sheaf space corresponding to F . Let s ∈ Γ (
U, E ) = F ( U ) , by Proposition 5.1there exists an open normal subgroup N such that U is N -invariant and s is N -equivariant. Thatis, s is fixed by the action of N and N P stab G ( U ) . As N was open, we see that F ( U ) = Γ ( U, E ) is discrete. (cid:3) Equivariant presheaves
As we saw, it was uncomplicated to define a G -action on a sheaf space: one simply requires thegroup to act on compatibility on the base and total space in a continuous manner. Definition 4.1defines an equivariant sheaf in terms of an isomorphism between certain pullback sheaves and thecocycle condition, which implicitly uses the sheaf space of sheaf. It is harder to directly define a G -equivariant presheaf, as the non-equivariant version has no topology on which the group can actcontinuously.For finite groups, there is a long-known description of a equivariant presheaves [Joh02, ExampleA2.1.11(c)]. Given a finite group G acting on the topological space X , define the category O G ( X ) to have objects the open sets of X with morphisms O G ( X )( U, V ) = { g ∈ G | gU ⊆ V } . A G -equivariant presheaf is a functor F : O G ( X ) op −→ Sets . We can think of this as a presheaf with additional maps F ( U ) → F ( gU ) for each g ∈ G . For a generaltopological group G and topological space this definition is insufficient, as it will not distinguishbetween G with its given topology and G with the discrete topology. We resolve this issue when G and X are profinite using Corollary 5.2. We construct a good definition of equivariant presheafand an equivariant sheafification functor with properties similar to the non-equivariant case.For the rest of this section, let G be a profinite group and X a profinite G -space. Definition 6.1.
For X a profinite G -space with open-closed basis B , the category O G ( B ) hasobjects given by the elements of B and morphisms given by the set O G ( B )( U, V ) = { g ∈ G | gU ⊆ V } . Any map in O G ( B ) is a composite of an inclusion map e : U → V and a translation map g : U → gU . Moreover, these maps commute in the sense that U e / / g (cid:15) (cid:15) V g (cid:15) (cid:15) gU e / / gV. The horizontal maps will induce the restriction maps that occur in non-equivariant sheaves.We now define G -equivariant presheaves in terms of an open-closed basis, the choice of which is(largely) unimportant by Remark 6.6. QUIVARIANT SHEAVES FOR PROFINITE GROUPS 15
Definition 6.2.
For B a open-closed basis of X , a G -equivariant presheaf of sets over B is a functor F : O G ( B ) op −→ Setssuch that the action of O G ( B ) op ( U, U ) = stab G ( U ) on F ( U ) is continuous, when F ( U ) is given thediscrete topology and stab G ( U ) G has the subspace topology.A map of G -presheaves of sets over B is a natural transformation. We write G - PreSheaf ( B ) todenote the category of G -presheaves of sets over B .Expanding the definition, F ( U ) is a a discrete stab G ( U ) -set under the action F ( g ) : F ( U ) −→ F ( U ) for g ∈ stab G ( U ) . For the sake of notational simplicity, we often write g := F ( g − ) : F ( U ) −→ F ( gU ) so that hg is a map from F ( U ) to F ( hgU ) .Let U ⊆ V be open sets and g ∈ G . Then for a G -presheaf F , the following diagram commutes F ( V ) F ( e ) / / g = F ( g − ) (cid:15) (cid:15) F ( U ) g = F ( g − ) (cid:15) (cid:15) F ( gV ) F ( e ) / / F ( gU ) . We can think of this diagram as saying that the restriction maps are equivariant.
Theorem 6.3.
Let B be an open-closed basis of the profinite G -space X and let F be a G -presheafof sets over B . Then the sheaf space constructed from F considered as a non-equivariant presheafhas a canonical G -action.Proof. We define a topological space E in the same way as the usual sheafification construction.The underlying set is E = a x ∈ X F x with projection map p : E → X sending f ∈ F x to x . The topology is generated by the sets B ( s, U ) = { s x | x ∈ U } for U ∈ B and s ∈ F ( U ) . The map p is a local homeomorphism.To define the G -action, take a germ f x ∈ F x and g ∈ G . Choose a section s ∈ F ( U ) representing f x , let gs ∈ F ( gU ) denote the image of s under g = F ( g − ) . We define g ( f x ) = ( gs ) gx This action is well-defined as restriction is equivariant and any two sections which represent a germagree on some neighbourhood. The action defines a map m : G × E −→ E such that the projectionmap is G -equivariant.It remains to show that the G -action on E is continuous. Let B ( s, U ) be a basis element, and ( g, e y ) ∈ G × E such that m ( g, e y ) = s x ∈ B ( s, U ) . QUIVARIANT SHEAVES FOR PROFINITE GROUPS 16
Then we see that gy = x , so y = g − x . Let t ∈ F ( W ) be a representative of e y , then s x = ge y = ( gt ) gy = ( gt ) x so there is some compact-open set V ⊆ U ∩ gW where s | V = ( gt ) | V = g ( t | g − V ) in F ( V ) . Let s ′ = s | V and t ′ = t | g − V , then s ′ = gt ′ .We know that F ( V ) is a discrete stab G ( V ) -set. Hence, there is an open normal subgroup N ofstab G ( V ) such that for any n ∈ N , ns ′ = s ′ = gt ′ = ngt ′ . By Lemma 2.3, the group stab G ( V ) isopen, hence N is an open subgroup of G and N g is an open neighbourhood of g in G .We claim that the open set N g × B ( t ′ , g − V ) ⊆ G × E is sent to B ( s, U ) by m . We see that m ( ng, t ′ g − v ) = ( ngt ′ ) nv = ( gt ′ ) nv = s ′ nv = s nv and nv ∈ N V = V ⊆ U , so the claim is true. (cid:3) Definition 6.4.
Let X be a profinite G -space with open-closed basis B and F a G -presheaf of setsover B . The G -sheaf constructed in Theorem 6.3 is called the equivariant sheafification of F anddenoted ℓF . Theorem 6.5.
Equivariant sheafification defines a functor ℓ : G - PreSheaf ( B ) −→ G - Sheaf ( X ) which is left adjoint to the forgetful functor.The unit of this adjunction gives a canonical map from a G -presheaf to a G -sheaf F → ℓF . Thecanonical map ℓF → ℓ F is an isomorphism. Furthermore, if E is a G -sheaf, a map F → E factorsover F → ℓF .Proof. Corollary 5.2 implies that the forgetful functor exists. The other statements follow from thenon-equivariant case. (cid:3)
Remark 6.6.
If we are interested in G -sheaves, then we claim that the choice of the basis B isunimportant. A presheaf over an open-closed basis B defines a presheaf on an open-closed basis B ′ ⊇ B by right Kan extension, as in Lemma 3.11. The (equivariant) sheafifications of those twopresheaves agree. Since any two open-closed bases B and B ′ can be included into a larger open-closedbasis, the claim is justified. 7. G -sheaves of R -modules From this section onwards, we will let R denote a fixed ring (with unit). We continue to assumethat G is a profinite group and X is a profinite G -space. We introduce the category of G -sheavesof R -modules over X . As earlier, we have two equivalent definitions, once these are introduced andrelated to G -equivariant presheaves of R -modules, we will verify that the category of G -sheaves of R -modules is an abelian category. Definition 7.1. A G - equivariant sheaf of R -modules over X is sheaf of R -modules F over X together with an isomorphism ρ : d ∗ ( F ) −→ d ∗ ( F ) QUIVARIANT SHEAVES FOR PROFINITE GROUPS 17 such that the cocycle condition and the identity condition hold: e ∗ ρ = e ∗ ρ ◦ e ∗ ρ s ∗ ρ = Id F . A morphism f : F → F ′ of G -sheaves of R -modules over X is a map f of sheaves of R -modulessuch that the following square commutes. d ∗ ( F ) ρ (cid:15) (cid:15) d ∗ ( f ) / / d ∗ ( F ′ ) ρ ′ (cid:15) (cid:15) d ∗ ( F ) d ∗ ( f ) / / d ∗ ( F ′ ) We write G - Sheaf R ( X ) for this category. Definition 7.2. A G -equivariant sheaf of R -modules over a topological space X is a map p : E → X such that:(1) p is a G -equivariant map p : E −→ X of spaces with continuous G -actions,(2) ( E, p ) is a sheaf space of R -modules,(3) each map g : p − ( x ) −→ p − ( gx ) is a map of R -modules for every x ∈ X, g ∈ G .We will write this as either the pair ( E, p ) or simply as E . A map f : ( E, p ) → ( E ′ , p ′ ) of G -sheavesof R -modules over X is a G -equivariant map f : E → E ′ such that p ′ f = p and f x : E x → E ′ x is amap of R -modules for each x ∈ X .Note that Points (1) and (2) give a map of sets for Point (3), but they do not imply that it is amap of R -modules. This last point is the compatibility between the equivariant structure and thesheaf structure.Similar to Theorem 4.3, these two definitions of equivariant sheaves are equivalent. See Sugrue[Sug19, Theorem 4.1.5] and Kędziorek [Kęd14, Lemma 7.0.3] for the proof. Theorem 7.3.
The definitions of G -sheaves of sets from Definition 7.1 and Definition 7.2 areequivalent. The earlier results on G -sheaves of sets hold in this new setting. Lemma 4.6 implies that thestalk of a G -sheaf F at x ∈ X is a discrete R [ stab G ( x )] -module. Similarly, the analogue of Corollary4.5 holds. This corollary implies the existence of a G -equivariant global zero section. That is, for F a G -sheaf of R -modules over a profinite space X , with sheaf space E , the element ∈ F ( X ) must correspond to a continuous G -equivariant section X → E . Proposition 5.1 and Corollary5.2 imply that any section is locally sub-equivariant and F ( U ) is a discrete R [ stab G ( U )] -modulewhenever U is a compact and open subset of X .The definition of a G -presheaf is easily adapted to the case of R -modules. Definition 7.4.
For X a profinite G -space with open-closed basis B , a G -equivariant presheaf of R -modules over B is a functor F : O G ( B ) op −→ R - modsuch that the action of O G ( B ) op ( U, U ) = stab G ( U ) on F ( U ) is continuous.A map of G -presheaf of R -modules over B is a natural transformation. QUIVARIANT SHEAVES FOR PROFINITE GROUPS 18
The analogues of Theorems 6.3 and 6.5 hold in the case of R -modules. Example 7.5.
As with Examples 4.8 and 4.10, if we let X denote the trivial one point G -space,then a G -sheaf of R -modules over X is a discrete R [ G ] -module.Let M be a discrete R [ G ] -module (equipped with the discrete topology where needed), then theconstant equivariant constant sheaf at M over X is defined by the projection onto the X factor,which is a local homeomorphism p : M × X −→ X. The G -action on M × X is the diagonal action g ( m, x ) = ( gm, gx ) which is continuous as M is discrete.As in the non-equivariant setting, the constant sheaf functor is part of an adjunction with theglobal sections functor. The simplest proof makes use of our definition of equivariant presheaves. Lemma 7.6.
Let X be a profinite G -space, then the constant sheaf functor Const is left adjoint tothe global sections functor
Const : R d [ G ]- mod / / G - Sheaf R ( X ) : Γ( X, − ) . o o Proof.
By Corollary 5.2, the global sections of an equivariant sheaf are a discrete R [ G ] -module, sothe right adjoint lands in the desired category.The adjunction argument is analogous to the non-equivariant case. The constant equivariantsheaf is the sheafification of the constant equivariant presheaf. The constant equivariant presheaffunctor is left adjoint to the global sections functor. (cid:3) Our next task is to show that the category of G -sheaves of R -modules is an abelian category.We start by defining an abelian group structure on hom( − , − ) , the set of maps of G -sheaves of R -modules. Definition 7.7.
Let ( E, p ) and ( F, p ′ ) be two G -sheaves over X . Then we have:(1) A zero morphism E −→ F , given by the map p : E → X followed by the global zerosection X → F .(2) If f, f ′ ∈ hom( E, F ) we define f − f ′ to be the morphism where: f + f ′ : E −→ Fs x f ( s x ) − f ′ ( s x ) ∈ F x . Proposition 7.8.
Let E and F be G -sheaves of R -modules. If f, f ′ ∈ hom( E, F ) then f − f ′ is anelement of hom( E, F ) and the zero map acts as a unit.Hence, the category of G -sheaves of R -modules is additive.Proof. Observe that the map f − f ′ is given by the composition m ◦ ( f πf ′ ) ◦ ι where: EπE = { ( e, e ′ ) ∈ E × E | p ( e ) = p ( e ′ ) } QUIVARIANT SHEAVES FOR PROFINITE GROUPS 19 ι : E −→ EπE, e ( e, e ) f πf ′ : EπE −→ EπE, ( e, e ′ ) ( f ( e ) , f ′ ( e ′ )) m : EπE −→ E, ( e, e ′ ) e − e ′ . The maps ι and m are continuous, see [Ten75, Section 2.5]. Since f πf ′ is continuous, it followsthat f − f ′ is a continuous map.We see that f − f ′ is a G -equivariant map of sheaf spaces as p ′ (( f − f ′ )( s x )) = p ′ ( f ( s x ) − f ′ ( s x )) = x = p ( s x ) g (( f − f ′ )( s x )) = g ( f ( s x ) − f ′ ( s x )) = f ( g ( s x )) − f ′ ( g ( s x )) = ( f − f ′ )( g ( s x )) for all g ∈ G . It follows that the zero map is a unit for this addition and hom is bilinear, see also[Ten75, Section 3.2]. This implies that the category is additive. (cid:3) Theorem 7.9.
The category of G -sheaves of R -modules over X is an abelian category.Proof. The category of G -sheaves of R -modules is an additive category via the addition on hom defined earlier. The constructions of (finite) limits and colimits is given in Constructions 8.2 and8.3. From that work it is clear that finite product and finite coproducts agree. Moreover, kernelsand cokernels exist. It remains to prove that the category is normal, this follows from Tennison[Ten75, Theorem 2.4.13]. (cid:3) Colimits and limits of equivariant sheaves and presheaves
We construct colimits and limits of G -equivariant presheaves of R -modules then apply this toconstruct colimits and limits of G -sheaves of R -modules. Point set models for the constructions canbe found in [Sug19, Sections 4.2 and 4.4]. As every limit is an equaliser of products, it suffices toconstruct finite limits and infinite products. We do this separately, as infinite products are muchmore difficult to construct than finite limits.To distinguish presheaf products from sheaf products, we use a superscript P and S . Lemma 8.1.
The category of G -presheaves of R -modules has all colimits, given by taking thetermwise colimit. Finite limits are constructed termwise.If { F i } i ∈ I is a set of G -presheaves of sets over B , with I an indexing set, then on U ∈ B (cid:16)Y i ∈ IP F i (cid:17) ( U ) = disc Y i ∈ I F i ( U ) is the infinite product in the category of G -presheaves of sets.Proof. The most important case is that of infinite products. Given maps f i : E → F i of G -presheavesof R -modules we have a map of R -modules f : E ( U ) −→ Y i ∈ I F i ( U ) . By Corollary 5.2, E ( U ) is a discrete stab G ( U ) -module, so the image of f must be in the discretepart of the product giving a map f ′ : E ( U ) −→ disc Y i ∈ I F i ( U ) . QUIVARIANT SHEAVES FOR PROFINITE GROUPS 20
The monomorphism disc Y i ∈ I F i ( U ) −→ Y i ∈ I F i ( U ) implies that composing f ′ with the projection maps recovers the f i . Hence the map f ′ is unique. (cid:3) The sheafification functor allow us to construct colimits of diagrams of G -sheaves of R -modulesin terms of the presheaf colimit. Construction 8.2.
Suppose we have a diagram F i of G -sheaves of R -modules (with maps omittedfrom the notation). We may construct the colimit F of F i in the category of G -presheaves of R -modules as earlier.The sheaf ℓF is the colimit of the diagram F i in the category of G -sheaves of R -modules. Aswith the non-equivariant case, the stalks of a colimit are the colimits of the stalks. Construction 8.3.
Finite limits of sheaves of R -modules over a base space X may be constructedby taking the limit in the category of sheaves of sets, then using the induced R -action on the stalks.Since the limit is finite, the stalks of the limit are exactly the limits of the stalks, so this R -actionis well-defined. If E i is the diagram of sheaves (with maps omitted) and E the limit, then thesheaf space of E is given by taking the limit of the diagram E i , in the category of spaces over X .To illustrate, consider the case of a product of two sheaves of R -modules over X , p : E → X and p ′ : E ′ → X . The sheaf space of the product is the pullback diagram given by p and p ′ , with limit E × X E ′ = { ( e, e ′ ) | p ( e ) = p ′ ( e ′ ) } whose map to X is given by ( e, e ′ ) p ( e ) = p ′ ( e ′ ) .If we started with G -sheaves of R -modules, then we can follow the same process, but now thesheaf space of the limit is constructed in the category of G -spaces over X .Equally, we can use Definition 7.1 as our starting point. Since the pullback functor of Definition3.12 preserves finite limits, a finite limit of G -sheaves of R -modules is given by taking the limitof sheaves of R -modules and equipping it with the canonical G -action coming from moving thefunctors d ∗ and d ∗ past the finite limit.The preceding construction of finite limits does not extend to the infinite case. Instead, we givean explicit construction of infinite product of G -sheaves of R -modules in Proposition 8.7. A point-set model can be found in [Sug19, Section 4.4]. That model can be compared with the point-setconstruction of finite limits in [Sug19, Subsection 4.2.2], which makes explicit use of the finitenessassumption. Formally, the existence of infinite products is already known, as the category of G -sheaves of R -modules is a Grothendieck topos by Giraud’s theorem. See Moerdijk [Moe88] and[Moe90] for this and related results.The problem is much easier in the case where G is finite (and hence has the discrete topology).With this assumption, the construction of infinite products of non-equivariant sheaves can be readilyextended to the G -equivariant case as the continuity of the G -action may be verified by looking atthe map induced by each g ∈ G . In particular, if G is a finite discrete group, then infinite productsof G -sheaves are products taken in the category of non-equivariant sheaves with the additionalstructure of a G -action. This does not hold when G is allowed to be infinite as the followingexample demonstrates. QUIVARIANT SHEAVES FOR PROFINITE GROUPS 21
Example 8.4.
Let G = Z ∧ p be the p -adic integers. The category of G -sheaves of Q -modules overthe one-point space is equivalent to the category of discrete Q [ Z ∧ p ] -modules. The infinite productin the category of discrete Q [ Z ∧ p ] -modules is the discretisation of the infinite product. Hence, thismust be the product in the category of G -sheaves of Q -modules over the one-point space.The product of non-equivariant sheaves of Q -modules over the one-point space is the product ofthe stalks, equipped with the box topology. In this case, the box product is the discrete topologyon the infinite product. This is not usually the discretisation of the infinite product. For example,the discretisation of Y n > Q [ Z /p n ] is a proper subset of the product. Lemma 8.5.
Let X be a profinite G -space with open-closed basis B . Let { F i } i ∈ I be a set of G -sheaves of R -modules. The presheaf product F = Q Pi ∈ I F i satisfies the sheaf condition as a functor F : O G ( B ) op −→ R - mod . Proof.
Take an open cover of U ∈ O G ( B ) by sets V i ∈ O G ( B ) , with i ∈ I an indexing set. As U iscompact, there is a finite subcover V i , . . . , V i n . Consider the patching and separation conditionsfor this subcover. Since the subcover is finite, the limit over j = 1 , . . . , n is finite and commuteswith the filtered colimit defining disc. Thus we have isomorphisms lim j F ( V i j ) = lim j disc Y i ∈ I F i ( V i j ) ∼ = disc lim j Y i ∈ I F i ( V i j ) ∼ = disc Y i ∈ I F i ( U ) = F ( U ) . One can check that this implies the patching and separation conditions for the original cover. (cid:3)
Remark 8.6.
We can apply the sheafification functor to Q Pi ∈ I F i from the preceding lemma. Asthe presheaf product is already a sheaf over B , all we are doing is taking a right Kan extension ofthis object, extending the domain from O G ( B ) to O G ( X ) . Hence, the value of ℓ Q Pi ∈ I F i on some U ∈ B is given by (cid:16) ℓ Y i ∈ IP F i (cid:17) ( U ) = disc Y i ∈ I ( F i ( U )) . Proposition 8.7.
The infinite product in G -sheaves of R -modules is given by the sheafificationof the underlying product of G -presheaves of R -modules and is independent (up to natural isomor-phism) of the choice of basis for the category of presheaves.That is, let { F i } i ∈ I be a set of G -sheaves of R -modules and use a superscript P for the presheafproduct and a superscript S for the sheaf product, then F = Y i ∈ IS F i := ℓ Y i ∈ IP F i is the product in the category of G -sheaves of R -modules.Proof. Choose an open-closed basis B to obtain a suitable category of G -presheaves of R -modules.Let f i : E −→ F i be a map of sheaves for each i ∈ I . Then we have a map f : E −→ Y i ∈ IP F i QUIVARIANT SHEAVES FOR PROFINITE GROUPS 22 in the category of G -presheaves of R -modules (over B ). Moreover, if we compose f with theprojection from p i : Y i ∈ IP F i → F i we recover the map f i as the presheaf product on U ∈ B is a subset of the product of the F i ( U ) .Applying sheafification gives a map f ′ : E ∼ = ℓE −→ ℓ Y i ∈ IP F i = F post-composing f ′ with ℓp i gives f i .We must now show that the map f ′ is uniquely determined. By Remark 8.6, the sheafificationfunctor does not change the value of the presheaf product on open-closed sets in some basis B .Hence, on these sets the map f ′ : E −→ F is uniquely determined. The value of the sheaves E and F on any other open set of X is then determined by the sheaf condition using open sets in the basis B and therefore is also unique up to isomorphism.By Remark 6.6 we see the construction is (up to natural isomorphism) independent of the choiceof basis B for the category of G -presheaves. (cid:3) Pull backs and push forwards
Recall the definition of pull backs and push forwards from Section 3. We prove that these functorspass to categories of equivariant sheaves of sets. These results can also be extended to shaves of R -modules. Lemma 9.1.
Let f : X → Y be an equivariant map of profinite G -spaces, for G be a profinitegroup. If E is a G -sheaf space over Y , then the pull back f ∗ ( E ) f ∗ ( E ) / / (cid:15) (cid:15) E (cid:15) (cid:15) X f / / Y is a G -sheaf space over X .Proof. The space f ∗ ( E ) has a G -action and the induced map to X is a G -equivariant local home-omorphism. (cid:3) The non-equivariant statement implies that for x ∈ X , there is an isomorphism of stalks f ∗ ( E ) x ∼ = E f ( x ) and that if f is open, f ∗ ( E )( U ) = E ( f ( U )) . Lemma 9.2.
Let f : X → Y be an equivariant map of profinite G -spaces, for G be a profinitegroup. If F is a G -sheaf of sets over X , then the push forward f ∗ ( F ) = F ◦ f − O ( Y ) op f − −→ O ( X ) op F −→ Sets is a G -sheaf of sets over X . QUIVARIANT SHEAVES FOR PROFINITE GROUPS 23
Proof.
We already know f ∗ ( F ) is a sheaf of sets over X . We show that on compact-open sets, thegroup action is discrete. The results of Section 6 will then imply that f ∗ ( F ) is a G -sheaf of setsover X .Take K ⊆ Y an open-closed set, which is therefore compact. The set f − K ⊆ X is also open-closed and hence compact. Thus, F ( f − K ) is a discrete stab G ( f − K ) -set by Corollary 5.2. As f is G -equivariant, stab G ( K ) ⊆ stab G ( f − K ) , so the stab G ( K ) -action on f ∗ ( F )( K ) = F ( f − K ) isalso discrete. (cid:3) Proposition 9.3.
Let f : X → Y be a G -equivariant map of profinite G -spaces, for G a profinitegroup. The functors f ∗ and f ∗ form an adjunction f ∗ : G - Sheaf ( Y ) / / G - Sheaf ( X ) : f ∗ . o o Proof.
Write the left action maps of G on X as g : X → X and g : Y → Y , using context todistinguish them. These maps are homeomorphisms, hence g ∗ = g ∗ .Let E be a G -sheaf of sets over X . Then the G -action on E defines, and is defined by, maps E → g ∗ E = g ∗ E . Similarly, for F a G -sheaf of sets over Y , the G -action on F defines, and isdefined by, maps F → g ∗ F = g ∗ F .Assume that we have a G -map of G -sheaves F −→ f ∗ E . Then we have a commutative squareas below-left, with its adjoint square below-right. F (cid:15) (cid:15) / / f ∗ E (cid:15) (cid:15) f ∗ F (cid:15) (cid:15) / / E (cid:15) (cid:15) g ∗ F / / g ∗ f ∗ E = f ∗ g ∗ E g ∗ f ∗ F = f ∗ g ∗ F / / g ∗ E Since the right hand square above commutes for each g ∈ G , the map f ∗ F −→ E is a G -map aswell. The converse is similar. (cid:3) Equivariant skyscraper sheaves
In this section we construct an equivariant analogue of a skyscraper sheaf – a sheaf with only onenon-zero stalk. When working with G -sheaves over a profinite G -space X , we see that if a stalk isnon-zero at x , it must be non-zero at all points of the form gx , for g ∈ G . This is an instance of howin equivariant settings, one should consider G -orbits in the place of points. We conclude that anequivariant skyscraper should be a sheaf which is non-zero on precisely one orbit. We work towardsthis definition, starting by studying sheaves on transitive G -spaces (that is, spaces homeomorphic toa homogeneous space, G/H ). As previously, the group G will be profinite throughout this section. Example 10.1.
Let H be a closed subgroup of G and M a discrete R [ H ] -module. Then the map G × H M −→ G/H, [ g, m ] gH defines a G -sheaf of R -modules over G/H . To see that it is a local homeomorphism, choose [ e, m ] ∈ G × H M . Then as M is a discrete H -module, there is some open normal subgroup of H which fixes m . We take that subgroup to be of the form N ∩ H for N an open normal subgroup of G . We wantto find an open set containing [ e, m ] which is homeomorphic to some open subset of G/H . QUIVARIANT SHEAVES FOR PROFINITE GROUPS 24
Let q : G × M → G × H M be the quotient map. Consider a basic open subset of the domain, K × { l } ( K an open subgroup of G , l ∈ M ). Since q − q ( K × { l } ) = K × ( K ∩ H ) { l } is also an open set of G × M , we see that q is an open map. It follows that q ( N × m ) is an openset of G × H M .We now claim that q ( N × m ) is an open set upon which p induces a homeomorphism to the openset N H/H ⊂ G/H . That this restriction of p is a bijection can be checked directly, indeed thedomain is homeomorphic to N/N ∩ H . As the sets are all compact and Hausdorff, this restrictionof p is a homeomorphism as claimed.As with equivariant vector bundles, requiring that the base space be transitive is a very strongrestriction, as the following lemma makes clear. Lemma 10.2.
For H a closed subgroup of G , there is an equivalence of categories between G -sheaves of R -modules over G/H and discrete R [ H ] -modules. The following adjunction realises thisequivalence G × H ( − ) : R d [ H ]- mod / / G - Sheaf R ( G/H ) : ( − ) eH o o where G × H − sends a discrete R [ H ] -module M to the sheaf G × H M −→ G/H of Example 10.1.Proof.
Let M be a discrete R [ H ] -module and E a G -sheaf of R -modules over G/H . Consider a mapof sheaves f : G × H M −→ E . As a map of G -spaces, f is uniquely determined by its restriction to M eH ∼ = M . Since f is also a map over G/H , it follows that the image of this restriction must be in E eH ⊆ E . Thus, f is uniquely determined by the map f eH : M −→ E eH and we have shown that the adjunction exists. The unit is the isomorphism M → ( G × H M ) eH . The counit is the map G × H E eH → E , which is an isomorphism on the stalk at eH . As the basespace G/H is transitive, the counit is an isomorphism on all stalks and [Ten75, Theorem 3.4.10]implies that the counit is an isomorphism of sheaves. (cid:3)
We can relate this example and lemma to general sheaves by fixing a specific point x in the basespace and considering the orbit O = Gx of x . Lemma 10.3.
Let O = Gx be the orbit of a point x in a G -space X and E be a G -sheaf of R -modules over X . There is a natural isomorphism of G -sheaves over Oψ : G × stab G ( x ) E x ∼ = E | O . Proof.
We begin by considering the continuous map: ψ ′ : G × E x −→ E | O ( g, e ) ge which factors over the balanced product to give the desired map ψ . The map ψ is an isomorphismof stalks over x and hence an isomorphism of sheaves as in the proof of Lemma 10.2. (cid:3) QUIVARIANT SHEAVES FOR PROFINITE GROUPS 25
We combine this result with the extension by zero and restriction functors of Proposition 3.13and Definition 3.14 to give a functor from discrete R [ H ] -modules to G -sheaves over a profinite G -space X . To use extension and restriction (rather than the more general push forward or pullback functors) we need to know that orbits are closed. Proposition 10.4. If G is a profinite group and X a profinite G -space, then the G -orbit of anypoint x ∈ X is closed.Proof. As G and X are compact and Hausdorff, the action map sends closed sets to closed sets.The subset G × { x } ⊆ G × X is closed, hence its image Gx (the orbit of x ) is also closed. (cid:3) Example 10.5.
Let O = Gx be an orbit of a profinite G -space X , with i : O −→ X the inclusion.For a discrete R [ stab G ( x )] -module M there is a sheaf over Xi ∗ G × stab G ( x ) M where i ∗ is the extension by zero functor. We call this the equivariant skyscraper of M over O . Thestalk of this sheaf at x is M , the stalks at other points in O are isomorphic to M as R -modules andthe stalks at z / ∈ O are zero.Thus we have a definition of an equivariant generalisation of a skyscraper sheaf. If we think ofthis construction as a functor, we find it has a right adjoint given by taking a stalk of the sheaf. Proposition 10.6.
Let O = Gx be an orbit of a profinite G -space X , with i : O −→ X the inclusion.There is an adjunction ( − ) x : G - Sheaf R ( X ) / / R d [ stab G ( x )]- mod : i ∗ G × stab G ( x ) ( − ) . o o Proof.
Since the restriction functor does not alter stalks, the right adjoint is the composite of theequivalence of Lemma 10.2 and the restriction functor of Definition 3.14. (cid:3)
Godemont resolutions of equivariant sheaves
A particularly useful property of skyscraper sheaves over a space X is that they contain a set ofenough injectives for the category of (non-equivariant) sheaves over X . Hence, they can be used togive a canonical injective resolution of a sheaf, often known as the Godemont resolution, see Bredon[Bre97, pp. 36, 37]. We replicate this for equivariant sheaves. Our aim is to show that equivariantskyscraper sheaves give enough injectives for G -sheaves of R -modules over X . We first see that theequivariant skyscraper functor preserves injective objects. We remind the reader that R is a unitalring, G is a profinite group and X is a profinite G -space. Proposition 11.1.
Let x in X . If A is an injective discrete R [ stab G ( x )] -module, then the G -sheaf i ∗ G × stab G ( x ) A is injective in the category of G -sheaves of R -modules over X .Proof. As the left adjoint of Proposition 10.6 preserves monomorphisms, the right adjoint preservesinjective objects. (cid:3)
QUIVARIANT SHEAVES FOR PROFINITE GROUPS 26
For any G -sheaf of R -modules we may construct a monomorphism into an injective sheaf. Non-equivariantly this is usually done by taking a product of skyscraper sheaves indexed over the pointsof the base space. Equivariantly, we use equivariant skyscraper sheaves and index over orbits of thebase space. Theorem 11.2.
The injective equivariant skyscraper sheaves provide enough injectives for thecategory of G -sheaves of R -modules over X .Proof. Let E be a G -sheaf of R -modules over X . For each orbit O of X , pick an element x O ∈ O .By Lemma 10.3 there is an isomorphism of sheaves over OG × stab G ( x O ) E x O ∼ = E | O . For each x O ∈ X , we use Lemma 2.9 to obtain a monomorphism into an injective discrete R [ stab G ( x )] -module E x O −→ I x O . By Tennison [Ten75, Theorem 3.3.5], this induces a monomorphism of G -sheaves over OE | O ∼ = G × stab G ( x O ) E x O −→ G × stab G ( x O ) I x O into an injective G -sheaf over O . By the extension–restriction adjunction, this gives a map of G -sheaves over X E −→ i ∗ G × stab G ( x O ) I x O = I O . This map is induces a monomorphism on stalks at every y ∈ O .We claim the product of these maps E −→ Y O P I O is a monomorphism into an injective sheaf. To see it is a monomorphism we compose with theprojection to I O and check stalkwise for points y ∈ O . By Proposition 11.1, the codomain isinjective, so we have finished our claim and the proof. (cid:3) Definition 11.3. If E is a G -sheaf of R -modules over a profinite G -space we define the equivariantGodement resolution as follows.Let δ : E −→ I ( E ) be the monomorphism into an injective sheaf constructed in the proof ofTheorem 11.2 applied to the sheaf E . / / E δ / / I ( E ) p (cid:15) (cid:15) / / I ( Coker δ ) = I ( E ) / / . . . Coker δ δ ♠♠♠♠♠♠♠♠♠♠♠♠♠ Where δ and I ( E ) come from Theorem 11.2 applied to the sheaf Coker δ . QUIVARIANT SHEAVES FOR PROFINITE GROUPS 27 As δ is a monomorphism, we see that the sequence is exact at I ( E ) . Inductively, we see thatthis is an injective resolution of E . This resolution is particularly sensitive to the stalks at isolatedpoints and was used in [Sug19, Chapter 7] to give calculations of the injective dimension of thecategory of rational G -sheaves over a profinite G -space X . The authors will revisit this point in[BS21]. 12. Weyl sheaves over spaces of subgroups
The second author used the theory of equivariant sheaves to construct an equivalence of categoriesbetween the category of rational G -equivariant Mackey functors with a full subcategory of thecategory of G -sheaves of Q -modules. This equivalence requires G to be profinite and can be foundin [Sug19, Chapter 5] and [BS20]. In this section we examine this subcategory, known as thecategory of Weyl- G -sheaves. We start by introducing the space of closed subgroups of a profinitegroup G , which is the base space used in the equivalence. This section overlaps slightly with [BS20,Section 2.3]. Definition 12.1.
For G a profinite group, let S G denote the set of closed subgroups of G . For N an open normal subgroup of G , we define surjective maps p N : S G −→ S G/N K
KN/N. .We use the maps p N to put a topology on S G . This gives open sets p − N { KN/N } = O ( N, N K ) = { L ∈ S G | N L = N K } for K a closed subgroup of G . These sets form a basis for the topology. Indeed, if N and M areopen normal subgroups and K and L closed subgroups, then O ( N ∩ M, ( N ∩ M ) C ) ⊆ O ( N, N K ) ∩ O ( M, M L ) for any C ∈ O ( N, N K ) ∩ O ( M, M L ) . Moreover, if N M , then O ( N, N K ) ⊆ O ( M, M K ) . Lemma 12.2.
Let G be a profinite group and K a closed subgroup. Then K = \ N P open G N K and { K } = \ N P open G O ( N, N K ) . Proof.
The intersection contains K , so assume g ∈ G is in the intersection. For any open normalsubgroup N , we may write g = nk for n ∈ N and k ∈ K . Hence n = gk − ∈ N ∩ gK . The sets N ∩ gK are non-empty and satisfy the finite intersection property for varying N as N is an opensubgroup. By compactness of G , we have that ∅ 6 = \ N P open G N ∩ gK ⊆ \ N P open G N = { e } . Hence, for any N , e ∈ N ∩ gK ⊆ gK for any N . We have shown that g ∈ K .For the second equality, consider some closed subgroup L such that N L = N K for all N . Then K = \ N P open G N K = \ N P open G N L = L. (cid:3) QUIVARIANT SHEAVES FOR PROFINITE GROUPS 28
The space S G has a continuous G -action given by G × S G −→ S G ( g, K ) gKg − . The stabiliser of a closed subgroup K of G in S G is therefore the normaliser of K in G , N G K . Wemay also calculate the stabiliser the basic open sets:stab G ( O ( N, N K )) = N G ( N K ) . Definition 12.3. A Weyl- G -sheaf of R -modules E is a G -sheaf of R -modules over S G such that E K is K -fixed and hence is a discrete R [ W G ( K )] -module. A map of Weyl- G -sheaves is a map of G -sheaves of R modules over S G . We write this category as Weyl - G - sheaf R ( S G ) , with the inclusioninto G -sheaves of R -modules over S G denoted by inc.Let F be a G -sheaf of R -modules over S G . A Weyl section of F is a section s : U → F such thatfor each K ∈ U , s K is K -fixed. Example 12.4.
The constant sheaf at an R -module M is a Weyl- G -sheaf. It should be rememberedthat here we are taking an R -module, not an R [ G ] -module.Let K be a closed subgroup of G , if A is a K -fixed discrete R [ N G K ] -module, then the equivariantskyscraper sheaf i ∗ G × N G K A is a Weyl- G -sheaf of R -modules that we may call a Weyl skyscraper sheaf .The definitions directly imply that every section of a Weyl- G -sheaf is a Weyl section. Weylsections occur quite often in general G -equivariant sheaves. We restate the local sub-equivarianceof sections of sheaves of S G and show that K -fixed germs over K ∈ S G are represented by Weylsections. This result and proof also appears as [BS20, Proposition 2.21]. Proposition 12.5. If E is a G -sheaf of R -modules over S G and s K ∈ E K is fixed by K , thenthere there is an open normal subgroup N of G such that s K is represented by an N K -equivariantsection: s : O ( N, N K ) −→ E. Proof.
Let r : O N ′ K ( N ′ , N ′ K ) = U −→ E be a section representing s K . Since the set V = r ( O N ′ K ( N ′ , N ′ K )) is open (by definition of the topology on a sheaf space) and compact, thereis an open normal subgroup N ′′ of G such that N ′′ V = V .Let N ∗ = M ′ ∩ M ′′ , then U and V are N ∗ -invariant. We want to show that r | U : U → V is N ∗ -equivariant. For n ∈ N ∗ and u ∈ U we may write pr ( nu ) = nu = np ( r ( u )) = p ( nr ( u )) As p is N ∗ -equivariant and injective when restricted to V , we see that r ( nu ) = n ( r ( u )) .Let t = r | U , we must restrict this section to a smaller domain to ensure it is also K -equivariant.Consider the set of sections k ∗ t for k ∈ K . We know that n ∗ t = t for all n ∈ N ∗ ∩ K and that N ∗ ∩ K is open in K and hence of finite index. Thus, there are only finitely different sections k ∗ t, k ∗ t, . . . , k n ∗ t. QUIVARIANT SHEAVES FOR PROFINITE GROUPS 29 where the k i are a set of coset representatives of N ∗ ∩ K in K (we may choose k to be the identityelement). As s K is fixed by all of K , ( k i ∗ t ) K = k i ( t K ) = t K for each i n . Hence, for each i n there exists N i N ∗ open and normal in G so that: t | O ( N i ,N i K ) = ( k i ∗ t ) | O ( N i ,N i K ) . Set N to be the intersection of the N i (with N = N ∗ ). Then t | O ( N,NK ) = ( k ∗ t ) | O ( N,NK ) foreach k ∈ K , so s = t | O ( N,NK ) is K -equivariant. As t was N ∗ -equivariant, s = t | O ( N,NK ) is also N equivariant. Hence, s is N K equivariant. (cid:3)
Corollary 12.6.
Let E be a G -sheaf of R -modules over S G . A K -fixed germ in E K can berepresented by a Weyl section.Proof. By Proposition 12.5, we have an
N K -equivariant representative s : O ( N, N K ) −→ E for a K -fixed germ s K ∈ E K . Let L ∈ O ( N, N K ) , then L N L = N K , so s is L -equivariant. As L ∈ S G is fixed by the action of L , s L is L -fixed. (cid:3) Using these results we can construct a Weyl- G -sheaf from a G -equivariant sheaf of R -modulesover S G . Construction 12.7.
Let E be a G -sheaf of R -modules over S G . We define a Weyl- G -sheafWeyl ( E ) . The stalk over K is given by the K -fixed points of E K Weyl ( E ) K = E KK . We then equip Weyl ( E ) with the subspace topology. Equally, we define the topology using onlythe Weyl sections of E . As every germ of Weyl ( E ) can be represented by a Weyl section, we havethe local homeomorphism condition.Since we are working with sheaves of R -modules, we note that the stalks of Weyl ( E ) will benon-empty, as the zero element is always fixed by the group actions.Since the image of a K -fixed element of a G -space must be K -fixed, we have the adjunction ofthe following lemma. The last statement of the lemma is equivalent to the statement that inc isfully faithful. Lemma 12.8.
For G a profinite group there is an adjunction inc : Weyl - G - sheaf R ( S G ) / / G - Sheaf R ( S G ) : Weyl . o o Moreover, the counit of this adjunction inc Weyl ( E ) −→ E is a monomorphism of G -sheaves of R -modules. and the unit map F −→ Weyl ( inc F ) is an isomorphism. QUIVARIANT SHEAVES FOR PROFINITE GROUPS 30
Remark 12.9.
As mentioned, earlier work of the authors [BS20] gives an equivalence betweenrational G -equivariant Mackey functors and Weyl- G -sheaves of Q -modules Mackey :
Weyl - G - sheaf Q ( S G ) / / Mackey Q ( G ) : Sheaf . o o The functor
Mackey can in fact take a G -sheaf of Q -modules over S G as input. By [BS20, Propo-sition 5.5] the construction Sheaf ◦ Mackey( E ) for E a G -sheaf (not necessarily Weyl) gives Weyl ( E ) .The construction of a rational Mackey functor Mackey( E ) defines Mackey( E )( H ) = E ( S H ) H . If K ∈ S H , then any H -fixed (or equally, H -equivariant) section s ∈ E ( S H ) is K -equivariant, so s K is K -fixed. Hence, the construction of a Mackey functor from a G -sheaf only makes use of theWeyl sections. As a consequence, Mackey(
Weyl E ) = Mackey( E ) giving a diagram of functors as below.Weyl - G - sheaf Q ( S G ) inc / / Mackey (cid:15) (cid:15) G - Sheaf Q ( S G ) Weyl o o Mackey r r Mackey Q ( G ) Sheaf O O Theorem 12.10.
The category of Weyl- G -sheaves of R -modules is an abelian category with allsmall colimits and limits. Limits are constructed by applying Weyl to the underlying diagram of G -sheaves of Q -modules over S G .Proof. Take a colimit diagram of Weyl- G -sheaves of R -modules. Applying inc and taking the colimitgives a G -sheaf whose stalk at K ∈ S G is the colimit of the stalks of the diagram. It follows thatthe stalk at K of the colimit is K -fixed. Hence, the colimit is a Weyl- G -sheaf. Finite limits maybe constructed similarly.The addition operation for maps of G -sheaves from Definition 7.7 applies equally well to Weyl- G -sheaves. The global zero section of a G -sheaf is G -equivariant, hence a Weyl- G -sheaf has a globalzero section. This gives a zero map in the set of maps from one Weyl- G -sheaf to another Weyl- G -sheaf. Arguments similar to those in the proof of Theorem 7.9 show that we have an abeliancategory.It remains to construct (small) infinite products. Let E i be a set of Weyl- G -sheaves of R -modulesand F another Weyl- G -sheaf. Using a superscript S for the sheaf product we have isomorphismsWeyl - G - sheaf R ( S G )( F, Weyl Y Si inc E i ) ∼ = G - Sheaf R ( S G )( inc F, Y Si inc E i ) ∼ = Y i G - Sheaf R ( S G )( inc F, inc E i ) ∼ = Y i Weyl - G - sheaf R ( S G )( F, E i ) QUIVARIANT SHEAVES FOR PROFINITE GROUPS 31
Hence, the product in the category of Weyl- G -sheaves is given by Weyl Q Si inc E i . (cid:3) When working rationally, Remark 12.9 and [Sug19, Subsection 5.3.3] give another constructionof limits via the category of rational G -Mackey functors. This description has the advantage thatthe product of Mackey functors is constructed objectwise.We can relate the ( inc , Weyl ) -adjunction to equivariant skyscraper sheaves by the following com-mutative diagram of adjunctions (which means the diagram of left adjoints commutes up to naturalisomorphism and the diagram of right adjoints commutes up to natural isomorphism). Lemma 12.11.
For any closed subgroup K of G , there is a commutative diagram of adjunctions Weyl - G - sheaf R ( S G ) inc (cid:15) (cid:15) ( − ) K / / R d [ W G K ]- mod inc (cid:15) (cid:15) i ∗ G × NGK ( − ) o o G - Sheaf R ( S G ) Weyl O O ( − ) K / / R d [ N G K ]- mod . ( − ) K O O i ∗ G × NGK ( − ) o o Since the above mentioned stalk functors preserves monomorphisms, the horizontal right adjointspreserve injective objects. This fact and the evident analogues of Theorem 11.2 and Definition 11.3(using discrete Q [ W G K ] -modules in place of discrete Q [ N G K ] -modules) give the following corollary. Corollary 12.12.
Let K be a closed subgroup of a profinite group G . If A is an injective discrete Q [ W G K ] -module, then the Weyl skyscraper sheaf i ∗ G × N G K A is an injective object of Weyl- G -sheaves of R -modules. Hence, the category of Weyl- G -sheaves of R -modules has enough injectives and the Godement resolution can be used for Weyl- G -sheaves. Sheaves over a diagram of spaces
A profinite space is an inverse limit of finite discrete spaces. It is logical to ask if this structurecan be applied to describe the category of sheaves over a profinite space X in terms of a diagramof sheaves over the finite spaces from the limit defining X . Working equivariantly for a profinitegroup G , we further want a finite group action on each of the finite spaces, with G the limit of thosefinite groups. In this section we provide an equivalence between a class of diagrams of equivariantsheaves and G -sheaves of R -modules over X in Theorem 13.6. Analogous statements and proofshold for equivariant sheaves of sets.The starting point is to extend the notion of push forward and pull back sheaves of Section 9 toinclude a change of group. A full treatment of change of group functors would take too long, so wefocus on the case of most interest to the aim of this section. Lemma 13.1.
Let N be an open normal subgroup of a profinite group G and p N : G → G/N theprojection. Let p : X → Y be a surjective map from a profinite G -space X to a G/N space Y suchthat p N ( g ) p ( x ) = p ( gx ) . QUIVARIANT SHEAVES FOR PROFINITE GROUPS 32
There is an adjunction (¯ p ) ∗ : G/N - Sheaf R ( Y ) / / G - Sheaf R ( X ) : (¯ p ) ∗ o o with the functors explained in the following construction. Construction 13.2.
Let D be a G/N -sheaf of R -modules over Y . Ignoring the group actions, p ∗ D is a sheaf of R -modules over X taking values U D ( pU ) for U an open subset of X (as the map p is open). As Y is a G/N -space, D ( pU ) has an action ofstab G/N ( pU ) . As the map p G sends stab G ( U ) G to stab G/N ( pU ) , we may let stab G ( U ) act on D ( pU ) via p G . This change of group functor (an instance of inflation) would normally be called ( p G ) ∗ , we omit it from out notation to avoid confusion with the change of base space functors.With this G -action, we have a G -sheaf over X which we call ¯ p ∗ D .We construct the adjoint to ¯ p ∗ . We will use the fact that for any open subset V of Y , N is asubgroup of the stabiliser of p − V . Let E be a G -sheaf of R -modules over X , then the functor V E ( p − V ) N for V an open subset of Y , defines a G/N sheaf over Y . We denote this sheaf by (¯ p ) ∗ E .To see that we have an adjunction, one checks the triangle identities. The unit of this adjunctionis the identity map, the counit at an open set UE ( p − ( pU )) N −→ E ( U ) is given by combining the inclusion of fixed points with the restriction map. That the triangleidentities hold comes from the triangle identity for the inflation-fixed point adjunction and the factthat p ( p − V ) = V for any open V in Y .We now turn to the context of interest to us. Fix a cofiltered category I and diagram of finitediscrete groups and surjective group homomorphisms G • = { G i , φ Gij : G i → G j } over I . Let G be the limit of this diagram, with projection maps p Gi , so that φ Gij ◦ p Gi = p Gj . We let N i be the kernel of p Gi . Similarly, fix a diagram of finite discrete spaces and surjective maps X • = { X i , φ ij : X i → X j } with projections maps p i such that X i is a G i -space and the following diagram commutes. G i × X i / / φ Gij × φ ij (cid:15) (cid:15) X iφ ij (cid:15) (cid:15) G j × X j / / X j It follows that G acts on X and the evident diagram of projection maps and action maps commutes. QUIVARIANT SHEAVES FOR PROFINITE GROUPS 33
Definition 13.3. A G • -sheaf of R -modules over X • is a G i -equivariant sheaf D i −→ X i of R -modules for each i , with maps of sheaves of R -modules over X i α ij : ( ¯ φ ij ) ∗ D j −→ D i for each i → j in I . A map f • : D • −→ D ′• of G • -sheaves of R -modules over X • consists of maps f i : D i → D ′ i of G i -sheaves of R -modules over X i that commute with the structure maps ( ¯ φ ij ) ∗ D j α ij / / φ ∗ ij f j (cid:15) (cid:15) D if i (cid:15) (cid:15) ( ¯ φ ij ) ∗ D ′ j α ′ ij / / D ′ i . We write G • - Sheaf R ( X • ) for this category.Note the contravariance introduced in the definition, the maps of sheaves reverse the maps inthe diagram I . It can be useful to write out how the functor ( ¯ φ ij ) ∗ acts. For U an open subset of X j , the sheaf D i is defined by ( ¯ φ ij ) ∗ D i ( U ) = D i ( φ − ij U ) N j /N i . Moreover, N j /N i is a subgroup of G/N i = G i which is in the stabiliser of φ − ij U .We can construct a G • sheaf of R -modules over X • from any G -sheaf over X from push forwardsheaves. Construction 13.4.
Let E be a G -sheaf of R -modules over X . For each i we have a push forwardsheaf E i = (¯ p i ) ∗ E over X i . For i → j in I , we have an equality E j = (¯ p j ) ∗ E = −→ ( ¯ φ ij ) ∗ (¯ p i ) ∗ E = ( ¯ φ ij ) ∗ E i of sheaves over X j , using the equality (( − ) N i ) N j /N i = ( − ) N j and composition of the non-equivariantpush forward functors. Taking the adjoint gives a map of sheaves over X i α ij : ( ¯ φ ij ) ∗ E j −→ E i . These sheaves and maps define a G • -sheaf of R -modules ¯ p • E over X • .In the reverse direction, we can construct a G -sheaf of R -modules over X from a G • -sheaf of R -modules over X • by taking a colimit of pullback sheaves. Construction 13.5.
Given a G • -sheaf of R -modules D • over X • , we have a sheaf (¯ p i ) ∗ D i for each i ∈ I . For i → j in I , we have maps (¯ p j ) ∗ D j = (¯ p i ) ∗ ( ¯ φ ij ) ∗ D j −→ (¯ p i ) ∗ D i of G j -sheaves of R -modules over X j . Define D = colim i (¯ p i ) ∗ D i .The functor ¯ p • is part of an adjunction and is fully faithful, that is, G -sheaves over X are a fullsubcategory of G • -sheaves over X • . We can restrict that adjunction to an equivalence of categories. QUIVARIANT SHEAVES FOR PROFINITE GROUPS 34
Theorem 13.6.
There is an adjunction colim i (¯ p i ) ∗ : G • - Sheaf R ( X • ) / / G - Sheaf R ( X ) : ¯ p • . o o whose counit is an isomorphism. This adjunction restricts to an equivalence of categories colim i (¯ p i ) ∗ : colim G • - Sheaf R ( X • ) / / G - Sheaf R ( X ) : ¯ p • o o where the left hand category is the full subcategory of G • - Sheaf R ( X • ) of those sheaves D • of R -modules over X • whose adjoint structure maps ¯ α ij : D j −→ ( ¯ φ ij ) ∗ D i are isomorphisms.Proof. That the two constructions fit into an adjunction is similar to the colimit–constant functoradjunction combined with the change of base space and group functors of Lemma 13.1. The counitstatement is mentioned (non-equivariantly) by Goodwillie and Lichtenbaum in the proof of [GL01,Lemma 3.4]. Following that reference, we check that the counit is an isomorphism on stalks.Take some x ∈ X and E a G -sheaf of R -modules over X . We show the counit is an isomorphismby considering the stalk of E at x , as follows. (colim i (¯ p i ) ∗ (¯ p i ) ∗ E ) x = colim i ((¯ p i ) ∗ (¯ p i ) ∗ E ) x = colim i ((¯ p i ) ∗ E ) p i x = colim i ((¯ p i ) ∗ E )( { p i x } ) = colim i E ( p − i { p i x } ) N i ∼ = −→ E x . The final map uses two facts. The first is that x has a neighbourhood basis given by the open sets p − i { p i x } . The second is that E x is a discrete R [ stab G ( x )] -module, so every element is fixed bysome N i .For the second statement, consider a G • -sheaf D • of R -modules over X • whose adjoint structuremaps ¯ α ij : D j −→ ( ¯ φ ij ) ∗ D i are isomorphisms. We want to show that D i −→ (¯ p i ) ∗ colim k (¯ p k ) ∗ D k is an isomorphism of sheaves over the discrete space X i . Choose x ∈ X i , then we have isomorphismsas follows. Since we are looking at stalks of a sheaf over a discrete space, we do not need to considerthe effect of any sheafification functors. The first step is the definition of the functors, the second isthe fact that fixed points commute with filtered colimits as the terms of all the colimits are discretemodules. We are also able to replace N i by N i /N k in the third step as D k is a G/N k -sheaf, that is QUIVARIANT SHEAVES FOR PROFINITE GROUPS 35 G acts on (¯ p k ) ∗ D k via the projection to G/N k . ((¯ p i ) ∗ colim k (¯ p k ) ∗ D k ) x = (colim k (¯ p k ) ∗ D k )( p − i { x } ) N i ∼ = colim k D k ( p k ( p − i { x } )) N i ∼ = colim k D k ( p k ( p − i { x } )) N i /N k = colim k D k ( φ − ki { x } ) N i /N k = colim k (( ¯ φ ki ) ∗ D k )( { x } ) ∼ = D i ( { x } ) = ( D i ) x The penultimate isomorphism is where we use our assumption that the adjoint structure maps of D • are isomorphisms. (cid:3) Fixing X = S G and R = Q , we may ask if the adjunctions of Lemma 13.1 and Theorem 13.6pass to categories of Weyl sheaves. Lemma 13.7.
Let N be an open normal subgroup of G , with p N : G → G/N the projection and p : S G → S G/N the induced map on spaces of closed subgroups.The functor (¯ p ) ∗ sends Weyl- G/N -sheaves to Weyl- G -sheaves. Using (¯ p ) ∗ for the functor onWeyl-sheaves, there is an adjunction (¯ p ) ∗ : Weyl - G/N - sheaf Q ( S G/N ) / / Weyl - G - sheaf Q ( S G ) : Weyl ◦ (¯ p ) ∗ ◦ inc o o . Proof.
Let E be a Weyl- G/N -sheaf, and choose K a closed subgroup of G . Then ((¯ p ) ∗ E ) K = ε ∗ E pK = ε ∗ E KN/N . Since E is a Weyl- G/N -sheaf, E KN/N is fixed by
KN/N = K/ ( K ∩ N ) . The group K acts on ε ∗ E KN/N by passing to the quotient and hence acts trivially.That we have an adjunction as stated follows from the preceding result and the fact that theinclusion of Weyl sheaves into equivariant sheaves is fully faithful, see Lemma 12.8. (cid:3)
Combining this lemma with Theorem 13.6 gives the following corollary.
Corollary 13.8.
Let S • G be the diagram coming from the finite spaces S G/N and the finite groups
G/N , indexed over the diagram of open normal subgroups N of G and inclusions.There is an equivalence of categories colim i (¯ p i ) ∗ : colim Weyl - G • - sheaf Q ( S G • ) / / Weyl - G - sheaf Q ( S G ) : ¯ p • o o where the left hand category is the full subcategory of G • - Sheaf Q ( S G • ) of those Weyl sheaves D • of Q -modules over S G • whose adjoint structure maps ¯ α ij : D j −→ ( ¯ φ ij ) ∗ D i are isomorphisms. QUIVARIANT SHEAVES FOR PROFINITE GROUPS 36
We can summarise the adjunctions of this paper into the following diagram, where arrows indicateinclusions of full subcategories. colim
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