Homotopy Mackey functors of equivariant algebraic K-theory
aa r X i v : . [ m a t h . A T ] F e b HOMOTOPY MACKEY FUNCTORS OF EQUIVARIANT ALGEBRAIC K -THEORY THOMAS BRAZELTON
Abstract.
Given a finite group G acting on a ring R , Merling constructed an equivariantalgebraic K -theory G -spectrum, and work of Malkiewich and Merling, as well as work ofBarwick, provides an interpretation of this construction as a spectral Mackey functor.This construction is powerful, but highly categorical; as a result the Mackey functorscomprising the homotopy are not obvious from the construction and have therefore notyet been calculated. In this work, we provide a computation of the homotopy Mackeyfunctors of equivariant algebraic K -theory in terms of a purely algebraic construction. Inparticular, we construct Mackey functors out of the n th algebraic K -groups of group ringswhose multiplication is twisted by the group action. Restrictions and transfers for thesefunctors admit a tractable algebraic description in that they arise from restriction andextension of scalars along module categories of twisted group rings. In the case wherethe group action is trivial, our construction recovers work of Dress and Kuku from the1980’s which constructs Mackey functors out of the algebraic K -theory of group rings.We develop many families of examples of Mackey functors, both new and old, including K -theory of endomorphism rings, the K -theory of fixed subrings of Galois extensions, and(topological) Hochschild homology of twisted group rings. Contents
1. Introduction 12. Preliminaries 43. Mackey functors on algebraic K -theory of twisted group rings 134. Comparison with equivariant algebraic K -theory 235. Families of Mackey functors 32Appendix A. Bimodule structures 35References 381. Introduction
Algebraic K -theory is a theory which encodes profound algebraic invariants of rings, re-vealing beautiful connections and patterns. The roots of such a theory run deep throughoutmathematics, and algebraic K -theory can now be found in almost every corner of algebra.For example, the class number formula generalizes to a statement about the torsion ofalgebraic K -groups of number fields. A complete understanding of the K -groups of the omotopy Mackey functors of equivariant algebraic K -theory Thomas Brazelton integers would resolve the Kummer–Vandiver conjecture, formulated over a century and ahalf ago at the time of writing, and which stands as one of the great unsolved problemsin number theory. However, the power and versatility of algebraic K -theory comes at thecost of its formulation relying on sophisticated categorical machinery.Following the development of algebraic K -theory, there was a surge of interest in definingan equivariant version. In the 1980’s, Fiedorowicz, Hauschild, and May gave a plus con-struction yielding the first topological space encoding equivariant algebraic K -theory forrings [FHM82], and Kuku, together with Dress, developed a Q -construction for equivariant K -theory for exact categories [Kuk84, DK81, DK82]. However both of these definitionsonly worked in the case where the rings in question were equipped with a trivial G -action.The slow progress in the development of a more robust equivariant algebraic K -theorycan be explained by the lack of equivariant homotopical tools available at the time. Therecent development of equivariant infinite loop space theory by Guillou, May, Merling, andOsorno [MMO17, GMMO19] was a major step in the direction towards laying the ground-work for equivariant algebraic K -theory. Using these new techniques, Merling was ableto build a genuine equivariant plus construction for algebraic K -theory for rings equippedwith a group action [Mer17].While classical algebraic K -theory produces a K -theory spectrum whose homotopygroups are the algebraic K -groups of a ring, equivariant algebraic K -theory produces agenuine G -spectrum, whose homotopy in each degree is a strictly richer algebraic objectcalled a Mackey functor . Ring AbSp K i K π i G - Ring
Mackeyfunctors G - Sp K i K G π i Mackey functors, as initially formulated by Dress [Dre71, Dre75] and Green [Gre71],are a collection of abelian groups indexed over subgroups of G , equipped with restrictionand transfer maps subject to various axioms (Definition 2.1.5). They may equivalentlybe defined as modules over an additive category called the Burnside category , denoted B G . Following the adage that spectra are the homotopy theorist’s abelian groups, one canenvision an analogue of Mackey functors which arise as modules over a spectrally enrichedversion of the Burnside category. Such a construction is called a spectral Mackey functor ,which have been explored in their greatest generality by Barwick [Bar17] together withGlasman and Shah [BGS20]. A celebrated result of Guillou and May demonstrates thatspectral Mackey functors are equivalent to genuine G -spectra [GM11, Theorem 0.1].Thus under the Guillou–May correspondence, one would anticipate that there is a con-struction of equivariant algebraic K -theory as a spectral Mackey functor. Indeed such aconstruction is given by Malkiewich and Merling, providing a Waldhausen spectral Mackeyfunctor associated to a Waldhausen G -category [MM20, Proposition 4.11], which is equiva-lently a spectral Mackey functor [MM20, Theorem 2.18]. This construction is closely related2 omotopy Mackey functors of equivariant algebraic K -theory Thomas Brazelton to that of Barwick [Bar17], together with Glasman and Shah [BGS20], which provides anequivariant algebraic K -theory in the more general setting of Waldhausen ∞ -categories.In this paper we will follow the setup from [MM20]. The construction of the equivariantalgebraic K -theory spectral Mackey functor is completely categorical, and in particular itshomotopy has not been investigated.The intent of this work is to provide an explicit algebraic description of the homotopyMackey functors of equivariant algebraic K -theory for a G -ring. If R is a G -ring with actionmap θ : G → Aut( R ), we can define a twisted/skew group ring R θ [ G ], which is the groupring R [ G ] equipped with a multiplication which is twisted by the group action. We provethat the algebraic K -groups of twisted group rings assemble to form a Mackey functor. Theorem 1.0.1. (As Theorem 3.0.1) Let R be a G -ring, where G is a finite group whoseorder is invertible over R . Then for any n ≥
0, there is a Mackey functor K Gn ( R ), whosevalue at G/H is the algebraic K -theory K n ( R θ [ H ]), and whose restriction and transfermaps are given by K n applied to extension and restriction of scalar functors over thetwisted group rings R θ [ H ].In particular in the case where the group action is trivial, we recover Mackey functorsof algebraic K -theory of group rings as built by Dress and Kuku (Proposition 3.6.2).As the construction of the Mackey functors K Gn ( R ) follow by defining transfer and re-strictions, applying the K -theory spectrum, then taking homotopy groups, we may seethat each K Gn ( R ) is actually the n th homotopy of a broader construction, namely a spec-tral Mackey functor K G ( R ). Theorem 1.0.2. (As Theorem 4.0.1) Let G be a finite group, and R a G -ring where | G | − ∈ R . Then the Mackey functors K Gn ( R ) comprise the homotopy of a spectral Mackeyfunctor K G ( R ), which is equivalent to equivariant algebraic K -theory of R , as defined in[MM19].This theorem allows us to conclude that K Gn ( R ) is the n th homotopy Mackey functor π n of equivariant algebraic K -theory. Remark 1.0.3.
In their recent book, Balmer and Dell’Ambrogio developed a detailedtheory of
Mackey 2-functors , which are an axiomatization of the idea of Mackey functors“valued” in additive categories rather than abelian groups [BD20]. The work laid out inSection 3 can be thought of as a new example of a Mackey 2-functor in their framework,namely categories of finitely generated projective modules over twisted group rings. Post-composition with any suitable functor from additive categories to abelian groups allows oneto recover a classical Mackey functor out of a Mackey 2-functor; in particular this holds for K n . In Section 5, we use this general idea to produce broad families of Mackey functors,however our specific functors are chosen to factor through connective spectra.In Section 2, we establish some categorical and algebraic background which will beneeded for the paper. We describe the orbit category and the Burnside category for agroup G , and discuss the categorical and axiomatic definitions of a Mackey functor. We3 omotopy Mackey functors of equivariant algebraic K -theory Thomas Brazelton define group actions on categories and their homotopy fixed points, culminating in thecomputation of the homotopy fixed points of a module category, arising from a groupaction on a ring, as being the module category of the twisted group ring. Finally, we defineequivariant algebraic K -theory as a Waldhausen spectral Mackey functor.In Section 3 we build various ring homomorphisms between twisted group rings, and usethese to construct restriction and extension of scalar functors, which will induce restrictionand transfer for our family of Mackey functors K Gn ( R ). We verify that all the axiomsfor a Mackey functor hold, modulo some work defining and comparing right actions oncertain modules, which is deferred until Appendix A. This section proves Theorem 3.0.1,that algebraic K -groups of twisted group rings form a Mackey functor. We compare theseMackey functors to those constructed by Dress and Kuku in Subsection 3.6.In Section 4, we compare our restriction, transfer, and conjugation data to that ofMalkiewich and Merling, and prove that they agree up to natural isomorphism. In par-ticular this allows us to prove Theorem 4.0.1; that the transfer data we defined turns the K -theory of twisted group rings into a spectral Mackey functor, which is isomorphic toequivariant algebraic K -theory. In Subsection 4.6 we explain how to recover the homotopyMackey functors from a spectral Mackey functor, following work of Bohmann and Osorno.This allows us to conclude that K Gn ( R ) is the n th homotopy Mackey functor of equivariantalgebraic K -theory.Finally, in Section 5, we demonstrate that our work generates various families of Mackeyfunctors. We approach the requirements used to prove Theorem 3.0.1 from a more ax-iomatic lens, and generate a Mackey functor on twisted group rings for any suitable in-variant in Subsection 5.1. As examples, we observe that Hochschild homology, topolog-ical Hochschild homology, topological cyclic homology, topological restriction homology,and topological periodic homology all yield Mackey functors in each degree when eval-uated on twisted group rings. In Subsection 5.2, we recover the classical example that G/H K n ( L H ) is a Mackey functor, where G is the Galois group of a Galois extension offields L/k , and we extend this result to hold for Galois ring extensions and for suitable in-variants such as THH. In Subsection 5.3, we use a classical theorem of Auslander to provethat the algebraic K -groups of endomorphism rings over fixed subrings K n (End R H ( R ))form a Mackey functor under certain technical conditions.1.1. Acknowledgements.
We would like to thank Mona Merling for guidance throughoutthis paper, and for inspiring a love of all things equivariant. We would also like to thankKirsten Wickelgren for being a constant source of mathematical support. Thank you alsoto Cary Malkiewich for helpful correspondence about homotopy invariants. The author issupported by an NSF Graduate Research Fellowship (DGE-1845298).2.
Preliminaries
The orbit and Burnside categories.
Denote by F G the category whose objectsare finite sets equipped with a group action by G , and whose morphisms are equivariantfunctions. It is a general fact that finite G -sets decompose into a disjoint union over their4 omotopy Mackey functors of equivariant algebraic K -theory Thomas Brazelton orbits, on which G acts transitively. Moreover, any transitive G -set can be seen to be of theform G/H for some subgroup H ⊆ G . To that end, we define the orbit category , denoted O G , to be the full subcategory of F G on the G -sets of the form G/H . One may easily seethat any morphism
G/H → G/K in the orbit category is determined by where it sends theidentity coset, and in particular such a morphism exists if and only if H is subconjugateto K , meaning that there is some g ∈ G so that gHg − ⊆ K . As we will be working withconjugation frequently, we incorporate some standard notation. Notation 2.1.1.
Given any subgroup H ⊆ G , and g ∈ G , one denotes by g H := gHg − the conjugation of the subgroup by g , and H g = g − Hg conjugation by its inverse.There are two natural classes of morphisms in the orbit category one considers. Thefirst is a projection morphism : for subgroups H ⊆ K , there is a map G/H → G/K sending xH to xK . The second is a conjugation morphism : for any g ∈ G there is a map G/H → G/ g H given by sending xH to ( xg − ) g H . One may check that these morphismsgenerate all morphisms in the orbit category, in the sense that any morphism in O G canbe written as a conjugation morphism followed by a projection morphism, or vice versa.Given the category F G of finite G -sets, we can consider its span category Span( F G ),where a morphism from S to T is an isomorphism class of a span of the form S ← U → T ,and composition of spans is given by pullback. We remark that hom sets in Span( F G )are equipped with a binary operation; given S ← U → T and S ← U → T , we canform the span S ← U ∐ U → T given by taking the disjoint union of U and U . Thisoperation endows each hom-set with an abelian monoid structure, which we may thengroup complete. Definition 2.1.2.
The
Burnside category B G is the category whose objects are finite G -sets, and whose morphisms are given byHom B G ( S, T ) := K (Span( S, T ) , ∐ ) , for S, T ∈ F G , where K denotes group completion.In particular, suppose that G/H → G/K is a morphism in O G . Then we can view it asa span in two natural ways, namely as G/H id ← G/H → G/K , or as
G/K ← G/H id −→ G/H .This defines two inclusion functors O G ֒ −→ B G and O op G ֒ −→ B G .The Burnside category is pre-additive , meaning that it is enriched in the category ofabelian groups. Recall that a group can be thought of as a category with a single object,whose group structure is encoded by composition of morphisms. Analogously, we maythink of a ring as a pre-additive category with a single object, as its hom-set will comeequipped with composition and addition, with the caveat that composition distributes overaddition. Just as groupoids generalize groups, pre-additive categories generalize rings. Inthe literature, pre-additive categories are sometimes referred to as ringoids or as rings withseveral objects .A module M over a ring R is determined by the data of a ring homomorphism R → End Ab ( M ), which is the same as an additive functor from R , viewed as a category with oneobject, to Ab . We can generalize this definition as follows.5 omotopy Mackey functors of equivariant algebraic K -theory Thomas Brazelton Definition 2.1.3.
Let A be a pre-additive category. Then we define a module over A tobe an additive functor A → Ab .With this in mind, we can provide a categorical definition of a Mackey functor. Definition 2.1.4. A Mackey functor is a module over the Burnside category, in the senseof Definition 2.1.3.As a Mackey functor is an additive functor, it will preserve disjoint unions. Thus we seethat a Mackey functor is completely determined on objects by where it sends transitive G -sets. Moreover, we may verify that a Mackey functor is completely determined by thecomposite functors O G ֒ −→ B G → Ab and O op G ֒ −→ B G → Ab . In particular, in order to definethe data of a Mackey functor, we must determine where it sends objects, projection mor-phisms in O G , and conjugation morphisms in O G under both composites, and then verifythat certain conditions hold. The images of projection morphisms under these inclusionswill be called restriction and transfer , while the image of conjugation morphisms will stillbe referred to as conjugation . With this data in mind, we may state an axiomatic definitionof Mackey functors, equivalent to the one above. This definition may be found in [Web00, § Definition 2.1.5. A Mackey functor is a function M : ob O G → Ab together with morphisms for all H ⊆ K ⊆ G subgroups and g ∈ G , called transfer , restriction , and conjugation , respectively: I HK : M ( G/K ) → M ( G/H ) R HK : M ( G/H ) → M ( G/K ) c g : M ( G/H ) → M ( G/ g H )satisfying the following axioms for all subgroups J, K, H ⊆ G and h ∈ H MF0 I HH , R HH , and c h are the identity on M ( G/H ). MF1 R KJ R HK = R HJ for all J ⊆ K ⊆ H MF2 I HK I KJ = I HJ for all J ⊆ K ⊆ H . MF3 c g c h = c gh . MF4 R g H g K c g = c g R HK for all K ⊆ H MF5 I g H g K c g = c g I HK for all K ⊆ H . MF6 (Mackey decomposition formula)
For
J, K ⊆ H subgroups of G , we have that R HJ I HK = X x ∈ [ J \ H/K ] I JJ ∩ x K c x R KJ x ∩ K . Mackey functors are rich algebraic objects, which are ubiquitous in algebra. For a morethorough account, we refer the reader to the survey paper of Webb [Web00]. We will revisitthis discussion when defining spectral Mackey functors, but we first take a detour to definehomotopy fixed points for G -categories. 6 omotopy Mackey functors of equivariant algebraic K -theory Thomas Brazelton G -categories and homotopy fixed points. We define a G -category to be any func-tor G → Cat , that is, it is a 1-category C equipped with a collection of endofunctors g : C → C for each group element, which we think of as inducing an action on the objectsand morphisms of C . We refer to a G -functor as a natural transformation between functors G → Cat , equivalently it is a functor of 1-categories which is equivariant in the sense that itpreserves the action on objects and morphisms. We say two G -categories are G -equivalent if there is an equivariant functor between them which is also an equivalence of categories. Example 2.2.1.
Given any finite G -set X , we can consider it as a G -category in twonatural ways. The first is by viewing it as a discrete category, which we also denote by X ,whose objects are points of X and which only has identity morphisms. Another categorywe denote by X , and refer to as the translation category . This has as objects elements x ∈ X , and a morphism x → y for any g ∈ G such that g · x = y . Any equivariantfunction between G -sets induces a G -functor between the associated discrete categories orthe associated translation categories in a natural way. Definition 2.2.2.
Let E G be the category whose objects are the elements in G , and whichhas exactly one morphism in each hom-set. This comes equipped with an action of G ,acting as g · c on objects, and on morphisms by g · ( c → c ′ ) := gc → gc ′ . We observe that E G and G are G -isomorphic, thus we will refer to E G as the translation category of thegroup G without ambiguity. Remark 2.2.3.
For H ⊆ G a subgroup, there is a natural inclusion of translation categories E H → E G . This is an equivalence of categories, however constructing an inverse can notbe done canonically in general. Writing H \ G = { Ha , . . . , Ha n } , we can define a functorin the other direction E G → E H as follows: any g ∈ G lies in a unique right coset Ha i , so g = ha i for some h , and we simply define E G → E H to send g h .We can also endow the functor category between two G -categories with an action. Definition 2.2.4.
For two G -functors C and D , we define an action of G on the objectsof Fun( C , D ) by ( gF )( c ) := g · (cid:0) F ( g − c ) (cid:1) . Thus the G -fixed points Fun( C , D ) G are precisely the G -functors. We define an action onthe morphisms (natural transformations) η : F ⇒ F ′ by( gη ) c := g · (cid:0) η g − c (cid:1) , which is a morphism gF ( g − c ) → gF ′ ( g − c ). A G -equivariant natural transformation isone fixed under this action of G . Definition 2.2.5.
Given a G -category C , and a subgroup H ⊆ G , we define the homotopyfixed point category as C hH := Fun( E H, C ) H . omotopy Mackey functors of equivariant algebraic K -theory Thomas Brazelton Remark 2.2.6.
There are two definitions of homotopy fixed points for a G -categorythat one might find in the literature, namely C hH could mean either Fun( E H, C ) H orFun( E G × G/H, C ) H , where G/H is the discrete category on the G -set G/H in the senseof Example 2.2.1. These two categories are easily seen to be G -equivalent, however we willbe a bit pedantic about this point, particularly since this equivalence does not admit acanonical inverse. Proposition 2.2.7.
Let C be a G -category, and H ⊆ G a subgroup. Then there is anequivalence of categories Fun( G/H × E G, C ) G → Fun( E H, C ) H , given by sending a functor F to F ( eH, − ), and by sending a natural transformation η : F ⇒ η ′ to the restricted components: η ( eH, − ) : F ( eH, − ) ⇒ F ′ ( eH ) . While this functor is suitably canonical, if we decided to construct a categorical inverse,we would be forced to confront the same ambiguities to defining an inverse to E H → E G as in Remark 2.2.3. Namely, we must pick explicit coset representatives. Notation 2.2.8.
Let F : E H → C be an H -equivariant functor between G -categories, andlet { g i } be a choice of right coset representatives of the subgroup H in G . Then we denoteby e F the lift of F to a G -equivariant functor e F : E G → C obtained by precomposing withan equivalence E G ∼ −→ E H . If η : F ⇒ F ′ is an H -natural transformation, we denote by e η : e F ⇒ f F ′ its lift, defined by e η hg i := η h . Proposition 2.2.9.
In the setting of Proposition 2.2.7, let H \ G = { Hg i } i be a choice ofright coset representatives. Then there is a functor exhibiting an equivalence of categoriesFun( E H, C ) H → Fun(
G/H × E G, C ) G F h(cid:0) gH, g ′ (cid:1) (cid:16) g · e F (cid:17) ( g ′ ) i = h(cid:0) gH, g ′ (cid:1) g · e F ( g − g ′ ) i , which sends η : F ⇒ F ′ to the natural transformation whose component at ( gH, g ′ ) isgiven by ( g · e η ) g ′ = g · (cid:0)e η g − g ′ (cid:1) .2.3. Homotopy fixed points of module categories and twisted group rings.
As aparticular example of the discussion above, we will look at a group action on the categoryof modules over a ring arising from a group action on a ring. In this setting, the homotopyfixed points admit a nice description as module categories over a twisted group ring , definedas follows.
Definition 2.3.1.
Given a group action θ : G → Aut
Ring ( R ), one may define the twistedgroup ring R θ [ G ], whose elements are finite formal sums of the form P i r i g i with r i ∈ R and g i ∈ G , and whose multiplication is defined by X i r i g i ! · X j r ′ j g ′ j = X i,j r i θ g i ( r ′ j ) g i g ′ j . omotopy Mackey functors of equivariant algebraic K -theory Thomas Brazelton Our goal in this section will be to see that, for a group action of G on R , we have anequivalence of categories Mod ( R ) hH ≃ Mod ( R θ [ H ]) for any subgroup H ⊆ G . First we mustsee how an action of G on R induces an action of G on the module category Mod ( R ). Definition 2.3.2.
Suppose that M is an R -module, and θ : G → Aut
Ring ( R ) is a groupaction. Then we define gM to be equal to M as an abelian group, but equipped with atwisted action map R × M θ g × id −−−→ R × M × −→ M. Given f : M → N an R -module homomorphism, we denote by gf : gM → gN the mor-phism ( gf )( m ) := f ( m ), that is, it is the same underlying abelian group homomorphism.We may verify that this data specifies a G -action on Mod ( R ), and we refer to this as theaction of G on Mod ( R ) induced by θ .We now look closer at the homotopy fixed point category Mod ( R ) hG . Recall that anelement of the homotopy fixed point category is a G -equivariant functor f : E G → Mod ( R ).As f ( g ) = g · f ( e ), we see that it suffices to determine f on objects by specifying the module M mapped to by the identity object e ∈ E G . As E G is a groupoid, f determines R -moduleisomorphisms f ( g ) : gM ∼ −→ M , where gM is as in Definition 2.3.2.As a brief point, we have that f ( g ) : gM ∼ −→ M is an isomorphism of R -modules,however gM is the same abelian group as M equipped with a different module structure.In particular f ( g ) can be thought of as an abelian group automorphism of M for any g ∈ G .This automorphism is not an R -module automorphism, since we have that f ( g )( rm ) = θ g ( r ) m. We refer to f ( g ) instead as a semilinear R -module automorphism of M . Definition 2.3.3.
Let R be a ring, and θ ∈ Aut
Ring ( R ) a ring automorphism of R . Thenwe define a θ - semilinear R -module homomorphism f : M → N to be a function satisfying • f ( m + m ′ ) = f ( m ) + f ( m ′ ) for all m, m ′ ∈ M • f ( rm ) = θ ( r ) · f ( m ) for all r ∈ R , and m ∈ M .Thus we can rephrase our discussion of the homotopy fixed point category Mod ( R ) hG to say that an element of the homotopy fixed point category is an R -module M equippedwith an action of G , acting via semilinear R -module automorphisms. This is also called a semilinear group action of G on M .We now return to our comparison of Mod ( R ) hG and Mod ( R θ [ G ]). It is a classical factthat modules over a twisted group ring are equivalent to modules equipped with a semi-linear G -action. We may see this, for example, as an explicit natural bijection as [Bra21,Corollary 3.6], which follows from an adjunction R/ Ring ⇄ Grp / Aut
Ring ( R ). This naturalbijection will provide us essential surjectivity for the following equivalence of categories.9 omotopy Mackey functors of equivariant algebraic K -theory Thomas Brazelton Proposition 2.3.4. (c.f. [Mer17, 4.3, 4.8]) Let R be a G -ring. Then there is an equivalenceof categories Fun( E G, Mod ( R )) G ∼ −→ Mod ( R θ [ G ]) , given by sending F to the abelian group M := F ( e ) equipped with the R θ [ G ]-modulestructure R θ [ G ] → End Ab ( M ) rg (cid:18) M F ( e,g ) −−−−→ M r ·− −−→ M (cid:19) , and sending a natural transformation η : F ⇒ F ′ to the R θ [ G ]-module homomorphism η e : M → N. Proof.
We must first verify that η e is indeed an R θ [ G ]-module homomorphism as claimed.By assumption it is an R -module homomorphism, so in particular it is additive, thus itsuffices to check that it preserves the R θ [ G ]-action. Let M := F ( e ), and N := F ′ ( e ). Thenwe must verify that η e (( rg ) · m ) = ( rg ) · η e ( m ) for all m ∈ M . That is, that the composites M F ( e,g ) −−−−→ M r ·− −−→ M η e −→ NM η e −→ N F ′ ( e,g ) −−−−→ N r ·− −−→ N agree. In the second composite, we remark that we can rewrite η e F ′ ( e, g ) as F ( e, g ) η g by thenaturality of η . Moreover, as η is G -equivariant, we see that η g = g · η e , and as gM = M asabelian groups, we see that η g = η e is an equality of abelian group homomorphisms. Thusit suffices to see that rη e ( m ) = η e ( rm ), which follows immediately by the component η e being an R -module homomorphism.Checking functoriality is immediate, since for F η ⇒ F ′ ε ⇒ F ′′ , we clearly have that( ε ◦ η ) e = ε e ◦ η e , and this can be seen to be associative. Moreover, (id F ) e = id F ( e ) , thusthe assignment Fun( E G, Mod ( R )) G → Mod R θ [ G ] is indeed a functor.If η, ε : F ⇒ F ′ are two G -equivariant natural transformations so that η e = ε e as R θ [ G ]-module homomorphisms, one can see that η e = ε e agree in particular as R -modulehomomorphisms. As they are equivariant, this implies that η g = gη e = gε e = ε g is anequality of R -module homomorphisms, implying that η = ε . Now if M f −→ N is any R θ [ G ]-module homomorphism, then we define functors F, F ′ : E G → Mod ( R ) given by F ( g ) := gM and F ′ ( g ) := gN . We may then define a natural transformation η : F ⇒ F ′ given by η g := g · f , which we may check is G -equivariant. This implies the functorFun( E G, Mod ( R )) G → Mod R θ [ G ] is fully faithful.For essential surjectivity, as we have remarked, every module over R θ [ G ] arises as an R -module equipped with a semilinear G -action [Bra21]. (cid:3) omotopy Mackey functors of equivariant algebraic K -theory Thomas Brazelton Corollary 2.3.5. [Mer17, 4.9, 4.10, 4.11] Under the conditions of Proposition 2.3.4, forany subgroup H ⊆ G , there is an equivalence of categoriesFun( E H, Mod ( R )) H ∼ −→ Mod ( R θ [ H ]) . Moreover, if | H | − ∈ R , then this equivalence descends to an equivalence on the categoryof finitely generated projective modulesFun( E H, P ( R )) H ∼ −→ P ( R θ [ H ]) . Waldhausen G -categories. Suppose we are handed a Waldhausen category C equippedwith an action of G , whose K -theory we want to study. If we wish to study the K -theory ina way that sees the symmetry coming from the G -action, then a natural expectation wouldbe that the group action is compatible with the Waldhausen structure in the followingsense. Definition 2.4.1. A Waldhausen G -category is a Waldhausen category C with an actionof G by exact functors.We could now attempt to study the K -theory of a Waldhausen G -category C by takingthe K -theory of the associated fixed point categories C H , obtaining a collection of spectraindexed over subgroups of G . This naive approach forces us to come to grips with thefact that ordinary fixed points are too coarse – in general, it will not be true that C H is aWaldhausen category [MM19, 2.1]. This failure is rectified by passing to homotopy fixedpoints, and is our main motivation for studying homotopy fixed points in greater detail. Theorem 2.4.2. [MM19, Theorem 2.15] Let C be a G -Waldhausen category, and H ⊆ G a subgroup. Then the homotopy fixed points C hH is a Waldhausen category.This theorem allows us to assign to every orbit G/H a spectrum K (cid:0) C hH (cid:1) , and moreoverwe will see that we are able to travel between these spectra along morphisms which aredirectly analogous to those found in Mackey functors. This data forms a homotopicalanalogue of a Mackey functor, which is referred to as a spectral Mackey functor . Thus forany group action on a ring, we will obtain a spectral Mackey functor, which encodes the K -theory of the associated homotopy fixed point module categories.2.5. Spectral Mackey functors.
One of the primary philosophies of stable homotopytheory is the analogy between abelian groups and spectra. From this perspective, wecan try to replicate the definition of a Mackey functor as a module over the pre-additiveBurnside category.
Definition 2.5.1.
The spectral Burnside category , denoted B G , is defined to be the cat-egory whose objects are finite G -sets, and whose hom sets are defined by taking the K -theory of the permutative category of finite equivariant spans between two finite G -sets(see [GM11] for details). Definition 2.5.2. A module over a spectrally enriched category A is a spectrally enrichedfunctor A → Sp . This is the perspective of Schwede and Shipley, see for example [SS01, § omotopy Mackey functors of equivariant algebraic K -theory Thomas Brazelton Definition 2.5.3. A spectral Mackey functor or B G -module is a spectrally enriched functor B G → Sp .The Guillou–May theorem establishes a powerful connection between spectral Mackeyfunctors and genuine G -spectra, namely that their homotopy theories are equivalent. Theorem 2.5.4. [GM11, Theorem 1.13] When G is finite, there is a string of Quillenequivalences between B G -modules and genuine G -spectra Mod ( B G ) ≃ G - Sp . In establishing the theory of equivariant A -theory, Malkiewich and Merling construct analternative version of the spectral Burnside category, which is more understandable for thecontext of Waldhausen G -categories. Definition 2.5.5.
We define B Wald G to be the spectrally enriched category whose objectsare transitive G -sets, and whose homs are given by first denoting by S H,K the category offinite G -sets containing G/H × G/K as a retract, and then definingHom B Wald G ( G/H, G/K ) := K ( S H,K ) , where K denotes the Waldhausen K -theory spectrum [MM19, 4.4].We refer to a module over B Wald G as a Waldhausen spectral Mackey functor . It is naturalto ask whether it is substantially different to be a B Wald G -module than to be a B G -module.The answer is that they are equivalent. Theorem 2.5.6. [MM20, Theorem 2.18] There is an equivalence of spectrally enrichedcategories between B Wald G and B G . Corollary 2.5.7. [SS03, 6.1] The module categories
Mod ( B G ) and Mod ( B Wald G ) are Quillenequivalent.One of the major results of [MM20] is that, given a G -Waldhausen category C , onemay endow the assignment G/H K (cid:0) C hH (cid:1) with the structure of a Waldhausen spectralMackey functor [MM20, Proposition 4.11]. In particular, the authors define certain trans-fer, conjugation, and restriction maps at the level of G -categories, and prove that thesedetermine the data of a B Wald G -module. By Corollary 2.5.7, this is then a spectral Mackeyfunctor. Definition 2.5.8.
Let R be a G -ring. When C = P ( R ) is the category of finitely generatedprojective R -modules, we refer to the spectral Mackey functor G/H K (cid:0) P ( R ) hH (cid:1) as equivariant algebraic K -theory .Our goal will be first to construct Mackey functors associated to the algebraic K -theoryof twisted group rings. Following this, we will compare the restriction, transfer, and conju-gation functors that we define at the level of G -categories to those found in [MM20], whichdetermine the structure of a B Wald G -module. In particular by seeing that they agree up tonatural isomorphism, the resulting spectral Mackey functors will be equivalent.12 omotopy Mackey functors of equivariant algebraic K -theory Thomas Brazelton Mackey functors on algebraic K -theory of twisted group rings The entirety of this section will be devoted to proving the following theorem.
Theorem 3.0.1.
For any G -ring R , where G is a finite group whose order is invertibleover R , and for any n ≥
0, there is a Mackey functor K Gn ( R ), whose value at G/H isthe algebraic K -group K n ( R θ [ H ]), and whose restriction and transfer maps are given by K n applied to extension and restriction of scalar functors of finitely generated projectivemodules over the twisted group rings R θ [ H ].In order to prove this theorem, we will construct ring homomorphisms between thetwisted group rings R θ [ H ], and use extension and restriction along these ring homomor-phisms to build the data of our restriction, transfer, and conjugation morphisms for theMackey functors K Gn ( R ). In proving Theorem 3.0.1, we avoid making reference to the al-gebraic K -theory functors until the very last step. The advantage of this approach is thatthe Mackey functor axioms can be proved up to natural isomorphism at the level of modulecategories. Then by applying K n for any n , the Mackey functor axioms will hold on thenose, due to the following remark, which can be seen as a consequence of additivity. Remark 3.0.2. If F, G : C → D are naturally isomorphic functors of exact categories,then K n ( F ) = K n ( G ) as abelian group homomorphisms.Before discussing twisted group rings in greater detail, we include the following remarkwhich will allow us to make use of more concise notation in the proofs for this section,without overburdening the reader with extraneous sums and indices. Remark 3.0.3.
Recall that the elements of a twisted group ring R θ [ G ] are finite sumsof the form P i r i g i , with r i ∈ R and g i ∈ G . At many points in this paper, it willsuffice to prove facts about the group rings R θ [ G ] on the “pure elements,” which are sumsover a singleton set, i.e. elements of the form rg . For example, in order to define a ringhomomorphism f : R θ [ G ] → S out of a twisted group ring, it suffices to define f on pureelements, extend this definition additively (i.e. define f ( P i r i g i ) := P i f ( r i g i )), and thenverify that f (( r g ) · ( r g )) = f ( r g ) · f ( r g ). Similarly when we know a function outof a twisted group ring to be additive, it suffices to verify that it is multiplicative on pureelements. Notation 3.0.4.
Given a ring homomorphism f : R → S , we will use f ∗ : Mod ( S ) → Mod ( R ) to denote restriction of scalars, and we will use f ! : Mod ( R ) → Mod ( S ) to denoteextension by scalars. This is intended to align with the notation found in [MM19, § Assignments of ring homomorphisms.
As discussed above, we would like to as-sign ring homomorphisms between group rings to certain classes of maps in the orbitcategory O G . Our lives will be easier if this can be done functorially, however ambiguitiesarising from choosing coset representative in the orbit category will hinder our ability to13 omotopy Mackey functors of equivariant algebraic K -theory Thomas Brazelton construct a functor O G → Ring . We can resolve these ambiguities by defining a “fattened”version of the orbit category, denoted e O G . Remark 3.1.1.
Suppose
G/H, G/K are two transitive G -sets. Then any morphism f : G/H → G/K in O G is determined uniquely by where it sends the identity coset; if f ( eH ) = xK , then because f is G -equivariant, we have that f ( gH ) = g · f ( eH ) = gxK . Thus wecan encode the data of f as a morphism in O G concisely by the triple ( H, x, K ). Wenote however that (
H, x, K ) = (
H, xk, K ) for any k ∈ K , so morphisms in O G can berepresented by many different triples. We define a version of the orbit category in whichmorphisms are uniquely represented by a choice of coset representative. Definition 3.1.2.
Define the category e O G to be the category whose objects are transitive G -sets G/H , and whose morphisms are triples (
H, x, K ) for each element x ∈ G satisfying x − Hx ⊆ K . Composition is given by( J, y, K ) ◦ ( H, x, J ) = (
H, xy, K ) . We contrast this with O G , where we had that ( H, x, K ) = (
H, hx, K ) for any h ∈ H inthis notation. These become distinct morphisms in e O G . Proposition 3.1.3.
There is a functor T : e O G → Ring
G/H R θ [ H ]( H, x, K ) (cid:2) τ x : rh φ x − ( r ) x − hx (cid:3) . Proof.
We must check that τ x := T ( H, x, K ) is a ring homomorphism. It is additive andpreserves multiplicative and additive identities, so it suffices to check it is multiplicative.We see that τ x ( r h ) · τ x ( r h ) = (cid:0) φ x − ( r ) x − h x (cid:1) · (cid:0) φ x − ( r ) x − h x (cid:1) = φ x − ( r ) φ x − h x ( φ x − ( r )) x − h xx − h x = φ x − ( r φ h ( r )) x − h h x = τ x ( r φ h ( r ) h h ) = τ x (( r h ) · ( r h )) . We observe that τ y ◦ τ x = τ xy , and we conclude that T is a functor. (cid:3) Proposition 3.1.4.
Let H ⊆ G a subgroup, and let x ∈ N G ( H ) be an element of itsnormalizer. Then T ( H, x, H ) is a ring automorphism of R θ [ H ], with inverse T ( H, x − , H ). Notation 3.1.5.
There are two classes of ring homomorphisms in the image of T that wewill interact with frequently, associated to our projection and conjugation morphisms inthe orbit category, so we introduce concise notation to distinguish these. For a subgroup H ⊆ K , we denote by ρ KH = T ( H, e, K ): ρ KH : R θ [ H ] → R θ [ K ] rh rk. omotopy Mackey functors of equivariant algebraic K -theory Thomas Brazelton Given H ⊆ G a subgroup and g ∈ G arbitrary, we denote by γ g the map T ( H, g − , g H ): γ g : R θ [ H ] → R θ [ g H ] rh θ g ( r ) ghg − . In our Mackey functors, extension of scalars along ρ KH will play the role of transfers,restriction of scalars along ρ KH will be our restrictions, and finally extension along γ g willplay the role of our conjugation maps.3.2. The module categories
Mod R θ [ H ] . The morphism ρ KH : R θ [ H ] → R θ [ K ] exhibits R θ [ K ] as an R θ [ H ]-module. Really this is an injective inclusion of a subring, which leadsus to wonder whether R θ [ K ] is a free R θ [ H ]-module. This turns out to be true. Thereare other isomorphic copies of R θ [ H ] embedded in R θ [ K ], and we can translate betweenthem by right multiplication by elements of K . Remark 3.2.1.
Given any twisted group ring R θ [ K ], and y ∈ K , we have a shift map sh y : R θ [ K ] → R θ [ K ] rk rky. This is not a ring automorphism of R θ [ K ], but it is an R θ [ H ]-module automorphism forany H ⊆ K . Proof.
Let R θ [ K ] ∈ Mod R θ [ H ] under the structure map ρ KH for any subgroup H ⊆ K , let rh ∈ R θ [ H ] and r ′ k be a pure element of the ring R θ [ K ]. Then we see thatsh y (cid:0) ( rh ) · (cid:0) r ′ k (cid:1)(cid:1) = sh y (cid:0) rφ h ( r ′ ) hk (cid:1) = rφ h ( r ′ ) hky = ( rh ) · ( r ′ ky ) = ( rh ) · sh y ( r ′ k ) . It is clear that sh y is bijective on pure elements, and therefore as an endomorphism of R θ [ K ] as an R θ [ H ]-module. (cid:3) Provided these shift maps, we can describe the structure of R θ [ K ] as an R θ [ H ]-modulevia the following result. Proposition 3.2.2.
Suppose G is finite, and let H ⊆ K be subgroups of G . Then R θ [ K ]is a free R θ [ H ]-module under the morphism ρ KH . Proof.
We remark that ρ KH is injective as an R θ [ H ]-module homomorphism, and that theshift maps were injective as in Remark 3.2.1, therefore sh y ◦ ρ KH : R θ [ H ] → R θ [ K ] isinjective. Picking right coset representatives y , . . . , y n for H \ K , we claim that the ringhomomorphism (sh y ◦ ρ KH , . . . , sh y n ◦ ρ KH ) : M y i ∈ H \ K R θ [ H ] → R θ [ K ]is an R θ [ H ]-module isomorphism. As this is a direct sum of injective module homomor-phisms, and their images can be checked to intersect trivially in R θ [ K ], thus the map is15 omotopy Mackey functors of equivariant algebraic K -theory Thomas Brazelton clearly injective. Moreover, it is easy to see it is surjective, as every element rk ∈ R θ [ K ]can be written as rhy i for some h ∈ H and i , and thus is hit by the homomorphismsh y i ρ KH . (cid:3) Corollary 3.2.3.
Let H ⊆ K subgroups of G , and let y , . . . , y n be right coset represen-tatives for H \ K . Then the set { R y , . . . , R y n } is an R θ [ H ]-basis for R θ [ K ], viewed as a left R θ [ H ]-module.We will also want to understand how bases of free modules are affected under extensionalong the morphisms γ x : R θ [ H ] → R θ [ x H ]. We remark that γ x is the restriction of aring automorphism R θ [ G ] → R θ [ G ] of the form rg φ x ( r ) xgx − . By keeping this fact inmind, we are able to better characterize extension of scalars along γ x for certain types offree R θ [ H ]-modules, as is shown by the following remark. Remark 3.2.4.
Let
R, S ⊆ T be subrings of a general ambient ring, and let f : T → T be a ring isomorphism which restricts to an isomorphism of subrings g := f | R : R ∼ −→ S .Suppose that M is a free R -module of the form ⊕ i Rt i , where t i ∈ T . Then the S -module g ! M obtained by extension of scalars along g is the free S -module of the form ⊕ i Sf ( t i ). Example 3.2.5.
As a particular case of this remark, suppose that we have a free R θ [ H ]-module M of the form M = ⊕ R θ [ H ] 1 R y i for some y i ∈ G . Then we have that γ x ! M = ⊕ R θ [ x H ] γ x (1 R y i ) = ⊕ R θ [ x H ] xy i x − . In Subsection 3.4 we will need to prove complicated natural isomorphisms related toextension and restriction of scalars along these homomorphisms of twisted group rings,and Example 3.2.5 will come into play.3.3.
Restriction and transfer.
In this section we will define extension and restriction ofscalars along homomorphisms of twisted group rings and verify their basic properties.
Proposition 3.3.1.
For subgroups H ⊆ K of G , the morphism ρ KH : R θ [ H ] → R θ [ K ]induces extension and restriction of scalar functors, which restrict to the subcategory offinitely generated projective modules, and induce functors which we denote byTr HK := (cid:0) ρ KH (cid:1) ! : P ( R θ [ H ]) → P ( R θ [ K ])Res HK := (cid:0) ρ KH (cid:1) ∗ : P ( R θ [ K ]) → P ( R θ [ H ]) . Proof.
We recall that extension of scalars always preserves projective modules. Restrictionof scalars does not in general preserve projective modules, however as R θ [ K ] is finitelygenerated and projective over R θ [ H ] by Proposition 3.2.2, restriction of scalars along ρ KH descends to the subcategories of finitely generated projective modules. (cid:3) We denote by γ g ! : P ( R θ [ H ]) → P ( R θ [ g H ]) the extension of scalars functor along thering homomorphism γ g : R θ [ H ] → R θ [ g H ]. 16 omotopy Mackey functors of equivariant algebraic K -theory Thomas Brazelton Proposition 3.3.2.
Let G be finite. Then Tr HK , Res HK , and γ g ! are exact. Proof.
By Proposition 3.2.2, we have that R θ [ K ] is free over R θ [ H ], and in particular itis flat. Therefore extension of scalars (which is tensoring R θ [ H ]-modules with R θ [ K ]) isexact. Extension along γ g is exact as γ g is a ring isomorphism. Finally, restriction ofscalars is always exact since any exact sequence of R θ [ K ]-modules will remain exact (asthe underlying sets and functions are unchanging) when viewed as an R θ [ H ]-module. (cid:3) Lemma 3.3.3.
Let J ⊆ K ⊆ H ⊆ G be subgroups, and let g ∈ G . Then the following hold(where the reader is encouraged to compare the indexing here with that of Definition 2.1.5): (0) Tr HH and Res HH are equal to the identity functor on P ( R θ [ H ]), and γ h ! is naturallyisomorphic to the identity for all h ∈ H . (1) We have a natural isomorphismRes KJ Res HK ∼ = Res HJ . (2) We have a natural isomorphismTr HK Tr KJ ∼ = Tr HJ . (3) We have a natural isomorphism γ g ! γ h ! ∼ = γ gh ! (5) We have a natural isomorphismTr g H g K γ g ! ∼ = γ g ! Tr HK . Proof.
For (0), it is clear that Tr HH and Res HH are the identity. To see that γ h ! is naturallyisomorphic to the identity on P ( R θ [ H ]), it suffices to see that γ h is a ring automorphismof R θ [ H ]. This is clear, since γ h ◦ γ h − = γ e = id.The statements (1), (2), (3), and (5) are basic consequences of the functoriality ofrestriction and extension of scalars. (cid:3) Proposition 3.3.4.
Let H ⊆ G be a subgroup and g ∈ G . Then we have the following. (4) There is a natural isomorphism of functorsRes g H g K γ g ! ∼ = γ g ! Res HK . Proof.
Consider the commutative square R θ [ H ] R θ [ K ] R θ [ g H ] R θ [ g K ] . ρ KH γ g γ g ρ gKgH omotopy Mackey functors of equivariant algebraic K -theory Thomas Brazelton Since γ g is an isomorphism of rings, extension along it is an equivalence of categories, andtherefore the unit id → ( γ g ) ∗ γ g ! and counit γ g ! ( γ g ) ∗ → id are natural isomorphisms. Thuswe see that (cid:0) ρ g K g H (cid:1) ∗ γ g ! ∼ = γ g ! ( γ g ) ∗ (cid:0) ρ g K g H (cid:1) ∗ γ g ! ∼ = γ g ! (cid:0) ρ KH (cid:1) ∗ ( γ g ) ∗ γ g ! ∼ = γ g ! (cid:0) ρ KH (cid:1) ∗ . (cid:3) Thus we have established natural isomorphisms of restriction, transfer, and conjugationfunctors of exact categories. By the additivity theorem, after applying K n , we will obtainequalities of abelian group homomorphisms, satisfying the axioms of a Mackey functor.Before we are able to do this, it remains to prove the most difficult of all the axioms for aMackey functor, which is the Mackey decomposition formula. (6) For
J, K ⊆ H subgroups of G , we have a natural isomorphism of functorsRes HJ Tr HK ∼ = X x ∈ [ J \ H/K ] Tr JJ ∩ x K γ x ! Res KJ x ∩ K . Verifying this last axiom will take up the entirety of Subsection 3.4, with some of thework deferred to Appendix A.3.4.
The Mackey decomposition formula for twisted group rings.
In order to provethat Res HJ Tr HK and P x ∈ [ J \ H/K ] Tr JJ ∩ x K γ x ! Res KJ x ∩ K are naturally isomorphic as functorsfrom P ( R θ [ K ]) to P ( R θ [ J ]), we will first verify that their images agree on R θ [ K ]. Notonly does one obtain isomorphic left R θ [ J ]-modules when plugging R θ [ K ] into each of thesefunctors, there is actually an induced right R θ [ K ]-module structure for which their imageson R θ [ K ] agree as bimodules. By a theorem of Eilenberg and Watts, this will be sufficientto demonstrate that the functors are naturally isomorphic. Proposition 3.4.1.
Let
J, K ⊆ H be subgroups, and let x , . . . , x n be a set of doublecoset representatives for J \ H/K . Then | H : J | = n X i =1 | K : J x i ∩ K | . Proof.
From the pullback square of finite H -sets ` x ∈ [ J \ H/K ] H/ ( J x ∩ K ) H/JH/K H/H, y we obtain an isomorphism of finite H -sets: H/J × H/K ∼ = ∐ x ∈ [ J \ H/K ] H/ ( J x ∩ K ) . omotopy Mackey functors of equivariant algebraic K -theory Thomas Brazelton As the underlying sets are bijective, we see that | H/J | · |
H/K | = X x ∈ [ J \ H/K ] | H/ ( J x ∩ K ) | = X x ∈ [ J \ H/K ] | H/K | · | K/ ( J x ∩ K ) | . Canceling | H/K | from each side gives the desired equality. (cid:3) Proposition 3.4.2.
For each x i ∈ [ J \ H/K ], let β i, , . . . , β i,r i denote a set of right cosetrepresentatives for ( J x i ∩ K ) \ K . Then the right cosets for J in H are given by H = n [ i =1 r i [ ℓ =1 J x i β i,ℓ . Proof.
By Proposition 3.4.1, the set { x i β i,ℓ } i,ℓ gives us the expected number of coset rep-resentatives. It thus suffices to check that they are genuine representatives.Let h ∈ H be arbitrary. Then there is a unique x i so that h lies in the double coset J x i K , that is, h = jx i k for some unique i . We may take the element k and see which rightcoset of ( J x i ∩ K ) \ K it lies in, so there is some unique β i,ℓ so that k ∈ ( J x i ∩ K ) β i,ℓ . Thusthere is some unique j ′ ∈ J for which k = x − i j ′ x i β i,ℓ , from which we observe h = jx i k = ( jj ′ ) x i β i,ℓ . (cid:3) Corollary 3.4.3.
There is an isomorphism of free R θ [ J ]-modules R θ [ H ] ∼ = n M i =1 r i M ℓ =1 R θ [ J ] x i β i,ℓ . Proof.
As we have seen in Corollary 3.2.3, right coset representatives give bases for twistedgroup rings as free modules. Combining this with Proposition 3.4.2 gives the desired result. (cid:3)
Corollary 3.4.4.
There is an isomorphism of left R θ [ J ]-modulesRes HJ Tr HK ( R θ [ K ]) ∼ = M x ∈ [ J \ H/K ] Tr JJ ∩ x K γ x ! Res KJ x ∩ K ( R θ [ K ]) . Proof.
We first see that Tr HK ( R θ [ K ]) = R θ [ H ]. Invoking Corollary 3.4.3, we see thatrestriction gives us Res HJ R θ [ K ] ∼ = n M i =1 r i M ℓ =1 R θ [ J ] x i β i,ℓ . omotopy Mackey functors of equivariant algebraic K -theory Thomas Brazelton On the other hand, for a fixed x i , we haveRes KJ xi ∩ K R θ [ K ] ∼ = r i M ℓ =1 R θ [ J x i ∩ K ] β i,ℓ . Via extension along γ x i , our basis elements are conjugated (see Example 3.2.5), so we seethat γ x i ! ( ⊕ j R θ [ J x i ∩ K ] β i,j ) ∼ = ⊕ j R θ [ J ∩ x i K ] x i β i,j x − i . Finally extending along the inclusion J ∩ x i K ⊆ K , we have thatTr JJ ∩ xi K (cid:0) ⊕ j R θ [ J ∩ x i K ] x i β i,j x − i (cid:1) ∼ = ⊕ j R θ [ J ] x i β i,j x − i . Taking a direct sum of all such paths, we see that n M i =1 Tr JJ ∩ xi K γ x i ! Res KJ xi ∩ K ( R θ [ K ]) ∼ = n M i =1 r i M ℓ =1 R θ [ J ] x i β i,j x − i . One sees that the shift map s x i : R θ [ J ] x i β i,j x − i → R θ [ J ] x i β i,j is an isomorphism of free R θ [ J ]-modules, and therefore a direct sum of shift maps exhibits an isomorphism n M i =1 ( ⊕ j s x i ) : n M i =1 Tr JJ ∩ xi K γ x i ! Res KJ xi ∩ K ( R θ [ K ]) ∼ −→ Tr HK Res HJ ( R θ [ K ]) . (cid:3) Proposition 3.4.5.
We have that the left R θ [ J ]-modules Res HJ Tr HK ( R θ [ K ]) and L x ∈ [ J \ H/K ] Tr JJ ∩ x K γ x ! Res KJ x ∩ K ( R θ [ K ]) inherit a compatible right R θ [ K ]-module structure,which allows us to view them as ( R θ [ J ] , R θ [ K ])-bimodules. Moreover, the shift mapexhibiting a left R θ [ J ]-module isomorphism in Corollary 3.4.4 is a right R θ [ K ]-modulehomomorphism as well. Therefore they are isomorphic as bimodules. Proof.
Deferred to Appendix A. (cid:3)
Lemma 3.4.6.
Let
P, Q : P ( R ) → P ( S ) be two exact functors from the category ofprojective left R -modules to the category of projective left S -modules. If P ( R ) ∼ = Q ( R ) as( S, R )-bimodules, then P and Q are naturally isomorphic. Proof.
This is a direct corollary of the Eilenberg–Watts theorem [Eil60, Wat60], whichstates that a right exact coproduct-preserving functor between module categories is natu-rally isomorphic to tensoring with a bimodule. We first remark that the proof of Eilenberg–Watts relies only upon our capacity to obtain free resolutions, so we may restrict our atten-tion to subcategories of projective modules without issue, provided that P ( R ) and Q ( R )are projective in Mod ( S ). For any such functors P and Q , we have that P ( R ) and Q ( R )obtain natural right R -module structures — this is discussed further in Appendix A.Applying the theorem of Eilenberg and Watts, we have that P ∼ = M ⊗ R − for some ( S, R )-bimodule M , while Q ∼ = N ⊗ R − , for some ( S, R )-bimodule N . The assumption that there20 omotopy Mackey functors of equivariant algebraic K -theory Thomas Brazelton is a bimodule isomorphism P ( R ) ∼ = Q ( R ) implies that M and N are isomorphic as ( S, R )-bimodules. In particular, there is a natural isomorphism of functors M ⊗ R − ∼ = N ⊗ R − .Combining all these facts, we have a string of natural isomorphisms P ∼ = M ⊗ R − ∼ = N ⊗ R − ∼ = Q. (cid:3) Corollary 3.4.7.
There is a natural isomorphism of functorsRes HJ Tr HK ∼ = M x ∈ [ J \ H/K ] Tr JJ ∩ x K γ x ! Res KJ x ∩ K between the categories of projective modules P ( R θ [ H ]) and P ( R θ [ J ]). Proof.
Apply Lemma 3.4.6 to the following situation: let R = R θ [ K ], let S = R θ [ J ], let P = L x ∈ [ J \ H/K ] Tr JJ ∩ x K γ x ! Res KJ x ∩ K , and let Q = Res HJ Tr HK . Then the assumption that P ( R ) ∼ = Q ( R ) as left S -modules is given by Corollary 3.4.4, while their isomorphism asbimodules follows from Proposition 3.4.5. The fact that we can apply the theorem in thissetting follows from the fact that all functors considered are exact (Proposition 3.3.2), anda direct sum of exact functors is exact as well. (cid:3) We are now equipped to prove the main theorem of this section.3.5.
Proof of Theorem 3.0.1.
As stated earlier, we have proven compatibility akin to theaxioms for a Mackey functor at the level of exact functors between categories of projectivemodules, and only up to natural isomorphism. Our goal is then to apply an algebraic K -group functor to obtain a genuine Mackey functor. We should first verify that this issomething we are allowed to do; i.e. will applying K n to extension and restriction of scalarfunctors induce algebraic K -theory homomorphisms? It turns out that the conditionswe used to ensure that projective modules were preserved suffice to induce abelian grouphomomorphisms on K -groups. Proposition 3.5.1. If f : R → S is a ring homomorphism exhibiting S as a finitelygenerated projective R -module, then extension and restriction of scalars induce grouphomomorphisms K n ( f ! ) : K n ( R ) → K n ( S ) K n ( f ∗ ) : K n ( S ) → K n ( R ) , for all n ≥
0. For a reference, see for example [Wei13, IV.6.3.2].Thus for any n ≥
0, we have abelian group homomorphisms K n (Tr HK ) : K n ( R θ [ H ]) → K n ( R θ [ K ]) K n (Res HK ) : K n ( R θ [ K ]) → K n ( R θ [ H ]) K n ( γ g ! ) : K n ( R θ [ H ]) → K n ( R θ [ g H ]) . omotopy Mackey functors of equivariant algebraic K -theory Thomas Brazelton Proof of Theorem 3.0.1.
Let n ≥ (0) — (6) proved in Lemma 3.3.3 and Corollary 3.4.7. By letting I HK : = K n (Tr HK ) R HK : = K n (Res HK ) c g : = K n ( γ g ! ) , one sees by the additivity theorem that these properties become equalities of abeliangroup homomorphisms, and in particular satisfy the Mackey functor axioms listed inDefinition 2.1.5. Thus there is a Mackey functor K Gn ( R ) whose value on G/H is the alge-braic K -group K n ( R θ [ H ]), and whose restriction and transfer maps arise from restrictionand extension of scalars along ring homomorphisms between twisted group rings. (cid:3) Comparison with Dress and Kuku’s Mackey functor for group rings.
In[DK81, DK82], Dress and Kuku proved the existence of a family of Mackey functors definedby taking functors from translation categories of G -sets into an exact category C , andthen taking their algebraic K -theory. In particular when the category C is a category ofprojective modules over a ring R , this has the interpretation of recovering the algebraic K -theory of the group ring R [ G ]. We will provide a short argument that, in the case where R is a ring with trivial action from G , our Mackey functor K Gn ( R ) constructed above agreeswith the Mackey functor K n ( R [ G ]) defined by Dress and Kuku. For further detail on theMackey functor K n ( R [ G ]) and the more general construction behind it, we refer the readerto the excellent exposition found in [Kuk07, Chapter 10].If C is an exact category and X is any G -set, then the functor category Fun( X, C )is exact, where we understand exactness of natural transformations to mean exactnesspointwise [Kuk07, Theorem 10.1.1]. Then by applying K n for any n , we obtain an abeliangroup K n Fun( X, C ). This is denoted by K Gn ( X, C ) in the relevant literature. Theorem 3.6.1. [Kuk07, Theorem 10.1.2] We have that K Gn ( − , C ) : G Set → Ab is a Mackey functor for all n ≥ C = P ( R ) is thecategory of projective left R -modules. In this context, we have that Fun( G/H , P ( R )) isequivalent to the category of finitely generated projective left R [ H ]-modules [DK82, Theo-rem 3.2]. In particular, the restriction and transfer maps can be understood as restrictionand extension of scalars for the associated group rings [Kuk07, Remarks 10.3.1(2)]. Fromthese facts, we conclude the following. Proposition 3.6.2.
Let G be a finite group, and let R be a ring which is viewed as havinga trivial G -action. Then there is an isomorphism of Mackey functors for all n ≥ K Gn ( R [ G ]) ∼ = K Gn ( R ) , omotopy Mackey functors of equivariant algebraic K -theory Thomas Brazelton where K Gn ( R [ G ]) is the Mackey functor constructed in [DK82, § K Gn ( R ) is the Mackeyfunctor from Theorem 3.0.1.4. Comparison with equivariant algebraic K -theory Given a G -Waldhausen category C , one may give the assignment G/H K (cid:0) C hH (cid:1) the structure of a Waldhausen spectral Mackey functor, that is, the structure of a B Wald G -module. We will refer to such an object as a homotopy fixed points Waldhausen spectralMackey functor . Malkiewich and Merling constructed restriction and transfer functors forsuch objects prior to taking K -theory, and then proved that their restriction and transfermaps satisfied the necessary axioms required to specify a spectral Mackey functor [MM19,4.11].Suppose that C = P ( R ) is the category of finitely generated projective modules overa ring R , where G acts on P ( R ) via an action of G on the ring R . Then we have shownthat one can define transfer and restriction functors on the categories of homotopy fixedpoints as extension and restriction of scalars along homomorphisms between twisted grouprings. Let K G ( R ) denote the data of the assignment G/H K ( R θ [ H ]), equipped withrestriction, transfer, and conjugation functors as defined in Subsection 3.3. If we can provethat our transfers and restrictions agree up to natural isomorphism with those constructedby Malkiewich and Merling, then we will have proven that K G ( R ) is a Waldhausen spectralMackey functor which is equivalent to the homotopy fixed points Waldhausen spectralMackey functor for P ( R ). Theorem 4.0.1.
Suppose that G is a finite group with an action on R , and | G | − ∈ R .Then we have that K G ( R ) is a B G -module, and is equivalent to the B G -module G/H K ( P ( R )) hH as defined by Malkiewich and Merling.As a corollary, we can compute the homotopy Mackey functors of equivariant algebraic K -theory in terms of the Mackey functors K Gn ( R ) constructed in Theorem 3.0.1, whosetransfers and restrictions admit a tractable algebraic description. Corollary 4.0.2.
We have that K Gn ( R ) is the n th homotopy Mackey functor of the equi-variant algebraic K -theory spectral Mackey functor.4.1. The homotopy fixed points Waldhausen spectral Mackey functor.
For thissection, let C be a G -category with arbitrary coproducts. Definition 4.1.1. [MM19, 4.7] Suppose f : S → T is a map of finite G -sets. Then wehave a restriction functor f ∗ : Fun( T × E G, Mod ( R )) G → Fun( S × E G, Mod ( R )) G F ( f ∗ F : ( s, g ) F ( f ( s ) , g )) f ∗ F (cid:0) s, g → g ′ (cid:1) = F ( f ( s ) , g → g ′ ) , and on α : F ⇒ F ′ by the formula( f ∗ F )( s, g ) = F ( f ( s ) , g ) α −→ F ′ ( f ( s ) , g ) = ( f ∗ F ′ )( s, g ) . omotopy Mackey functors of equivariant algebraic K -theory Thomas Brazelton Definition 4.1.2. [MM19, 4.7] Let f : S → T be a map of finite G -sets. Then we have a transfer functor f ! : Fun( S × E G, C ) G → Fun( T × E G, C ) G is given by sending F f ! F , where( f ! F )( t, g ) = g M i ∈ f − ( g − t ) F ( i, e ) . In particular when S → T is a map between transitive G -sets, these definitions providerestriction and transfer maps between homotopy fixed point categories. Proposition 4.1.3. [MM19, 4.11] The restriction and transfer functors defined above fortransitive G -sets turn K ( C hH ) into a spectral Mackey functor.In the setting where C = P ( R ), our goal will be to compare the restriction and transfermaps defined above with those given by restriction and extension of scalars along ringhomomorphism between twisted group rings. As we may see, it suffices to define restrictionand transfers on the orbit category, and therefore it suffices to consider Definition 4.1.1 andDefinition 4.1.2 in the cases where we have a projection map G/H → G/K for a subgroup H ⊆ K , or a conjugation map G/H → G/ g H .Our goal will be to first compare restriction along a projection G/H → G/K in the senseof Definition 4.1.1 with restriction of scalars along ρ KH . We will also compare restrictionalong a conjugation function c x : G/H → G/ x H with restriction of scalars along γ x : R θ [ H ] → R θ [ x H ]. As every morphism in the orbit category is a composite of projectionand conjugation, this will imply that our definitions of restriction are compatible up tonatural isomorphism.To show that transfers are compatible, we will remark that the definitions of restrictionand transfer given in Definition 4.1.1 and Definition 4.1.2 are adjoint, as are restriction andextension of scalars. By some categorical trickery, transfers will then agree up to naturalisomorphism.4.2. Restriction on fixed point subcategories.
In the case where S and T are transi-tive G -sets, the restriction and transfer maps defined above are between categories of theform Fun( G/H × E G, C ) G . This is not our working definition of homotopy fixed points, aswe were using C hH = Fun( E H, C ) G , although we had an explicit equivalence between thesetwo in Proposition 2.2.7. However in order to relate these restriction and transfer maps tothose we build on categories of modules over twisted group rings, we will want to use theequivalence of the twisted group ring module category with homotopy fixed points as inProposition 2.3.4. Thus we will want to rewrite the restriction and conjugation functors ofMalkiewich and Merling using our working definition of homotopy fixed points categories.24 omotopy Mackey functors of equivariant algebraic K -theory Thomas Brazelton Definition 4.2.1.
For f : G/H → G/K associated to a subgroup inclusion H ⊆ K , wedefine the restriction functor f ∗ : Fun( E K, C ) K → Fun( E H, C ) H to be precomposition with the canonical functor E H ֒ −→ E K . Proof.
We must see these are functorial and well-defined. For restriction, we remark thatif F : E K → C is a K -equivariant functor, then it is necessarily H -equivariant as H ⊆ K .Thus to see that the composite E H → E K → C is H -equivariant, it suffices to observethat the canonical inclusion E H → E K is H -equivariant. Verifying that K -equivariantnatural transformations are send to H -equivariant natural transformations is immediateas well. (cid:3) Proposition 4.2.2.
The diagram of restriction functors commutes(3) Fun(
G/H × E G, Mod ( R )) G Fun( E H, Mod ( R )) H Fun(
G/K × E G, Mod ( R )) G Fun( E K, Mod ( R )) K Φ H ∼∼ Φ K f ∗ f ∗ Proof.
We begin with F ∈ Fun(
G/K ×E G, Mod ( R )) G , and see that Φ K ( F ) ∈ Fun( E K, Mod ( R )) K sends k F ( eK, k ), while f ∗ Φ K ( F ) ∈ Fun( E H, Mod ( R )) H sends h F ( eK, h ).For the other direction, we see Φ H ( f ∗ F ) sends h F ( f ( eH ) , h ) = F ( eK, h ). Thus thediagram commutes on objects.Given a G -natural transformation η : F ⇒ F ′ in Fun( G/K × E G, C ) G , we see that f ∗ η is of the form F ( f ( − ) , − ) ⇒ F ′ ( f ( − ) , − ) by definition. One sees then that Φ H f ∗ η is the restriction F ( f ( eH ) , − ) ⇒ F ′ ( f ( eH ) , − ), which is precisely equal to F ( eK, − ) ⇒ F ′ ( eK, − ). One can see that this is the same as f ∗ Φ K η . (cid:3) Thus defining restriction on homotopy fixed points for a projection morphism in such away that it agrees with Definition 4.1.1 was rather straightforward. Doing an analogousprocedure for conjugation morphisms is more difficult, and we will have to use this non-canonical inverse dependent on coset representatives in order to handle this situation. Thisnon-canonical characteristic of our conjugation morphisms will produce agreement only upto natural isomorphism, although this is good enough for our purposes.
Definition 4.2.4.
Let
G/H be a transitive G -set, let x ∈ G be arbitrary, and let g , . . . , g n be a choice of right coset representatives for G/H . Then we define a conjugation functor c ∗ x ( g , . . . , g n ), depending on the choice of coset representatives, by c ∗ x ( g , . . . , g n ) : Fun ( E x H, C ) x H → Fun( E H, C ) H F h h x − · (cid:16) e F ( xh ) (cid:17)i , omotopy Mackey functors of equivariant algebraic K -theory Thomas Brazelton and on morphisms by sending η : F ⇒ F ′ to the natural transformation whose componentat y is x − · ( e η xgy ). We will see that this is a functor via the proof contained in the followingproposition. Proposition 4.2.5.
We have that c ∗ x ( g , . . . , g n ) fits into a commutative diagram with therestriction map c ∗ x : Fun( G/ x H × E G, C ) G Fun(
G/H × E G, C ) G Fun( E x H, C ) x H Fun( E H, C ) H . c ∗ x ∼ c ∗ x ( g ,...,g n ) ∼ Here the right vertical map is the equivalence from Proposition 2.2.7, where the left verticalmap is the coset-dependence equivalence in Proposition 2.2.9.
Proof.
We verify that the conjugation functor in Definition 4.2.4 is indeed a functor andagrees with this diagram above by simply tracing objects and morphisms clockwise throughthe diagram and verifying that this agrees with what we called conjugation. On objects,we see that we have( g x H, e g ) g · (cid:16) e F ( g − e g ) (cid:17) ( gH, e g ) gx − · ( F ( xg e g )) F h x − · (cid:16) e F ( xh ) (cid:17) . On morphisms, we see that e η ( g x H, e g ) := g · (cid:0)e η g − e g (cid:1) ( c ∗ x η ) ( gH, e g ) = e η c x ( gH, e g ) = e η gx − x H, e g η : F ⇒ F ′ ( c ∗ x η ) ( eH, e g ) = e η x − x H, e g = x − · (cid:0)e η x e g (cid:1) . (cid:3) Comparison of restriction.
We verify that restriction along projection and conju-gation morphisms in the orbit category agree in the sense of [MM19] and in the sense ofrestriction of scalars along ρ KH and γ x . 26 omotopy Mackey functors of equivariant algebraic K -theory Thomas Brazelton Lemma 4.3.1.
For f : G/H → G/K a quotient map associated to an inclusion of sub-groups H ⊆ K , the diagram commutesFun( G/H × E G, Mod ( R )) G Fun( E H, Mod ( R )) H Mod R θ [ H ] Fun(
G/K × E G, Mod ( R )) G Fun( E K, Mod ( R )) K Mod R θ [ K ] , Φ H ∼ Ψ H ∼∼ Φ K f ∗ f ∗ Ψ K ∼ Res KH Proof.
By Proposition 4.2.2, the left square commutes, so it suffices to verify the rightsquare agrees.Suppose F ∈ Fun( E K, Mod ( R )) K . Then Ψ K ( F ) is the module M := F ( e ) ∈ Mod ( R ) withtwisted group action given by the morphisms F ( e → k ). Under Res KH , we simply restrictof the action to the image of morphisms of the form F ( e → h ) where h ∈ H .For the other direction, we see ( f ∗ F )( e ) = F ( f ( e )) = F ( e ), so we obtain the same R -module M = F ( e ). Moreover, for any e → h in E H , it maps to e → h in E K . Thatis, under Ψ H ( f ∗ F ) we obtain the same twisted module structure on M as in Res KH Ψ K ( F ).Thus the diagram commutes on objects.Given any natural transformation η : F ⇒ F ′ in Fun( E K, C ) K , we have that Ψ K ( η ) = η e : F ( e ) → F ′ ( e ), and that Res KH sends this morphism of R θ [ K ]-modules to the morphism η e : F ( e ) → F ′ ( e ), viewed as an R θ [ H ]-module homomorphism under restriction of scalars.For the other direction, we see that f ∗ η is given by whiskering η with f , and that Ψ H ( f ∗ η )is precisely ( η ◦ f ) e : ( f ( e )) → F ′ ( f ( e )), which is η e . Thus the diagram commutes onmorphisms. (cid:3) Lemma 4.3.2.
Let H ⊆ G be a subgroup, and x ∈ G arbitrary, so that we have aconjugation morphism G/H → G/ x H . When the coset representatives g i are chosen sothat x = g , the diagram commutes up to natural isomorphism(3) Fun( G/ x H × E G, C ) G Fun( E x H, Mod ( R )) x H Mod R θ [ x H ] Fun(
G/H × E G, C ) G Fun( E H, Mod ( R )) H Mod R θ [ H ] c ∗ x ∼ c ∗ x ( g ,...,g n ) ∼ c ∗ x ∼ ∼ where c ∗ x is restriction of scalars along the ring homomorphism R θ [ H ] → R θ [ x H ]. Proof.
By Proposition 4.2.5, the left square commutes up to natural isomorphism, so itsuffices to check the right square. Let F ∈ Fun( E x H, Mod ( R )) x H . Then its image in Mod x H [ x H ] is given by the module M := F ( e ) equipped with the following R θ [ x H ]-module27 omotopy Mackey functors of equivariant algebraic K -theory Thomas Brazelton structure R θ [ x H ] → End Ab ( M ) ry (cid:18) M F ( e,y ) −−−−→ M r ·− −−→ M (cid:19) . Under restriction of scalars, it is sent to the R θ [ H ]-module M , with action given by(4) R θ [ H ] → End Ab ( M ) rh (cid:18) M F ( e,xhx − ) −−−−−−−→ M r ·− −−→ M (cid:19) . Conversely, let Ψ ∈ Fun( E H, Mod ( R )) H be c ∗ x ( g , . . . , g n )( F ), that is,Ψ( h ) = x − e F ( xh ) , where F is extended to a functor e F on E G by selection of coset representatives. A priori,there is no good way to relate e F ( xh ) to e F ( x ), since one must write xh = yg i for some y ∈ x H and g i a coset representative, and then express e F ( xh ) := F ( y ). However, weremark that we can choose our coset representatives . To that end, since | G : x H | ≥
2, wemay select g = x , and let g , . . . , g n be an arbitrary selection of representatives for theremaining cosets. This has the following advantage: we can write xh = ( xhx − ) x, and then one sees that since xhx − ∈ x H , and x is a coset representative, we have that e F ( xh ) = F ( xhx − ) for all h ∈ H . In particular, one remarks that e F ( x ) = F ( e ). Wetherefore see that Ψ takes the following form:Ψ( h ) = x − F ( xhx − ) . We see that Ψ is sent to the abelian group Ψ( e ) = x − F ( xex − ) = x − F ( e ), and since x − F ( e ) = F ( e ) =: M by assumption, we have that Ψ M . Thus Equation 3 producesthe same abelian group.To see that the R θ [ H ]-module structures on M are the same, we see that the oneproduced by Ψ yields(5) R θ [ H ] → End Ab ( M ) rh (cid:18) M Ψ( e,h ) −−−−→ M r ·− −−→ (cid:19) . We must see that Equations Equation 4 and Equation 5 produce the same R θ [ H ]-modulestructure on M . It suffices to see that Ψ( e, h ) and F ( e, xhx − ) agree as abelian grouphomomorphisms for every h ∈ H . We see that Ψ( e, h ) is the image under Ψ of the uniquemap e → h in E H , and that, as a morphism in Mod ( R ), it is a map between x − e F ( xe ) =28 omotopy Mackey functors of equivariant algebraic K -theory Thomas Brazelton x − M , and x − e F ( xh ) = x − F ( xhx − ). That is, it is precisely the map x − (cid:0) F ( e, xhx − ) (cid:1) .One sees therefore that the diagram of abelian group homomorphisms commutes M MM. x − ·− Ψ( e,h ) F ( e,xhx − ) As x − is the identity as an abelian group homomorphism by definition, we see thatΨ( e, h ) = F ( e, xhx − ) in Hom Ab ( M, F ( xhx − )), and therefore that Equation 4 and Equation 5produce the same R θ [ H ]-action on M .We now verify that Equation 3 agrees on morphisms. Let η : F ⇒ F ′ be an arbitrary x H -equivariant natural transformation between functors F, F ′ ∈ Fun( E x H, Mod ( R )) x H . Thenwe have that c ∗ x ( g , . . . , g n )( η ) is given by, for h ∈ E H , the component x − · ( e η xh ) : x − · (cid:16) e F ( xh ) (cid:17) → x − · (cid:16)f F ′ ( xh ) (cid:17) . Under the equivalence Fun( E H, Mod ( R )) H ∼ −→ Mod R θ [ H ] , we send this to its component atthe identity e ∈ H , which is of the form (by recalling that e F ( x ) = F ( e ) and then that e η x = η e ): x − · ( η e ) : x − ( F ( e )) . Conversely, under the functor Fun( E x H, Mod ( R )) x H → Mod R θ [ x H ] , we see that η is sent to η e : F ( e ) → F ′ ( e ). Under the conjugation functor γ ∗ x : Mod R θ [ x H ] → Mod R θ [ H ] , this is sentto η e , viewed as an R θ [ H ]-module homomorphism. This isn’t equal on the nose to themorphism x − η e , however we remark that, by post-composition with the R θ [ H ]-moduleisomorphism x − · − , these morphisms agree. Thus the diagram commutes up to naturalisomorphism. (cid:3) Comparison of transfers.Lemma 4.4.1.
Let f : G/H → G/K be a morphism in the orbit category.(1) If f is a projection morphism for H ⊆ K , then f ! agrees with Tr KH up to naturalisomorphism. Fun( G/H × E G, P ( R )) G P ( R θ [ H ])Fun( G/K × E G, P ( R )) G P ( R θ [ K ]) . ∼ f ! Tr KH ∼ omotopy Mackey functors of equivariant algebraic K -theory Thomas Brazelton (2) If f is a conjugation morphism, that is, K = g H , then f ! agrees with γ g ! up tonatural isomorphism.Fun( G/H × E G, P ( R )) G P ( R θ [ H ])Fun( G/ g H × E G, P ( R )) G P ( R θ [ g H ]) . ∼ f ! γ g ! ∼ This follows as a particular case of the following, more general proposition.
Proposition 4.4.2.
Suppose we have a diagram of functors which commutes (up to naturalisomorphism)
A BC D ′ p ∗ r ∗ s ∗ q ∗ so that each functor admits a left adjoint, which we decorate with a lower shriek, andmoreover assume that p ∗ and q ∗ are equivalences of categories. Then the diagram A BC D p ∗ r ! q ∗ s ! commutes up to natural isomorphism. Proof.
We first see that the intermediate diagram
A BC D p ∗ r ∗ s ∗ q ! commutes up to natural isomorphism. Indeed, this is witnessed by the following composite(where ε is the counit associated to the adjunction q ! q ∗ → id, which is a natural isomorphismas q ∗ is an equivalence of categories): q ! ( s ∗ p ∗ ) ∼ = q ! ( q ∗ r ∗ ) ε −→ r ∗ . We then claim that r ! ∼ = p ! s ! q ∗ . This follows from uniqueness of adjoints, as r ∗ ∼ = q ! s ∗ p ∗ ,and q ! s ∗ p ∗ admits a left adjoint, given by p ! s ! q ∗ (since g is an equivalence, we have that q ∗ ⊣ q ! ⊣ q ∗ ). Combining this with the natural isomorphism p ∗ p ! ∼ −→ id given by p ∗ beingan equivalence of categories, we have that p ∗ r ! ∼ = p ∗ p ! s ! q ∗ ∼ −→ s ! q ∗ . (cid:3) omotopy Mackey functors of equivariant algebraic K -theory Thomas Brazelton Proof of Theorem 4.0.1.
We first state a concise corollary of the work in the pre-vious few sections.
Corollary 4.5.1.
Let G be finite, and | G | − ∈ R . Then restriction, transfer and con-jugation for module categories of twisted group rings agree with restriction, transfer andconjugation for homotopy fixed points up to natural isomorphism. Moreover, this compat-ibility holds on the subcategories of finitely generated projective modules.After applying K -theory, we have that the equivalence of categories P ( R ) hH ≃ P ( R θ [ H ])from Corollary 2.3.5 yields an isomorphism of spectra K ( P ( R ) hH ) ≃ K ( P ( R θ [ H ])). Inparticular, our definitions of restriction, transfer, and conjugation agree on these spectra,since they agreed up to natural isomorphism on the module categories. As restriction,transfer, and conjugation endowed G/H K ( P ( R ) hH ) with the data of a B Wald G -module,by the compatibility we have proven, this implies that G/H K ( P ( R θ [ H ])) is a B Wald G -module, and is moreover isomorphic to the B Wald G -module G/H K ( P ( R ) hH ).Finally, under the Quillen equivalence between module categories over B G and B Wald G (Corollary 2.5.7), after possibly modifying by an equivalence, we have that G/H K ( P ( R θ [ H ]))and G/H K ( P ( R ) hH ) are equivalent as B G -modules. This proves Theorem 4.0.1.4.6. Mackey functors from spectral Mackey functors, and Corollary 4.0.2.
Fromour previous work, we have obtained a spectral Mackey functor K G ( R ). By taking homo-topy groups, we have hinted at the ability to obtain Mackey functors at each level. We willnow make this procedure explicit.Given any spectrally enriched category A , and any lax monoidal functor F : Sp → Ab ,we can define a new category F • A , which is a pre-additive category on the same objects.The homs in F • A are obtained as follows: for any a, a ′ ∈ A ,we defineHom F • A ( a, a ′ ) := F • Hom A ( a, a ′ ) . The assumption that F is lax monoidal implies that the resulting category has a well-defined composition enriched over Ab . For a more general version of this statement, see[BO15, Proposition 2.11]. Example 4.6.1. As π is lax monoidal, we may apply ( π ) • to the spectral Burnsidecategory in order to recover the ordinary Burnside category( π ) • B G = B G . Given any spectrally enriched functor
A → Sp , by applying ( π ) • , we obtain an additivefunctor ( π ) • A →
Ho( Sp ), valued in the homotopy category of spectra. Post-composingwith any stable homotopy group functor π n produces a module over ( π ) • A .In particular, if Φ : B G → Sp is any spectral Mackey functor, and n ≥
0, then we obtainthe n th homotopy Mackey functor , denoted π n Φ, via the following composite B G ∼ = ( π ) • B G ( π ) • Φ −−−−→ Ho( Sp ) π n −→ Ab . omotopy Mackey functors of equivariant algebraic K -theory Thomas Brazelton This is due to [BO15, Proposition 7.6]. As an immediate example, we observe that K Gn ( R ) = π n K G ( R ), yielding Corollary 4.0.2.5. Families of Mackey functors
To wrap up, we provide a few particular examples and applications of our work con-structing the Mackey functors K Gn ( R ) for a G -ring R . In particular, we can generalizethese results to provide a family of Mackey functors E n ( R ) for any suitable invariant E ofrings. We explain this in detail in the following section.5.1. Mackey functors arising from homotopy invariants of rings.
Once we hadconstructed the functors Tr HK , Res HK and γ g ! between the exact categories of finitely gener-ated projective modules over twisted group rings, and we had proved they satisfied axiomsrelated to Mackey functors up to natural isomorphism, the construction of the Mackeyfunctor K Gn ( R ) was basically immediate. In particular we relied on two key aspects of thefunctor K n :(1) If f : R → S exhibits S as a finitely generated projective R -module, then restrictionof scalars along f descends to a functor f ∗ : P ( S ) → P ( R ), and extension of scalarsis a functor f ! : P ( R ) → P ( S ) (even without these conditions). These in turn induceabelian group homomorphisms f ∗ : K n ( S ) → K n ( R ) and f ! : K n ( R ) → K n ( S ).(2) If F, G : C → D are naturally isomorphic exact functors of exact categories, then K n ( F ) = K n ( G ).Denote by ExCat the 2-category whose objects are exact categories, and whose mor-phisms are given by exact functors. Let Sp ≥ denote the category of connective spectra(i.e. infinite loop spaces). We see that condition (1) can be thought of as a consequence ofthe fact that we may view K -theory as a 1-functor ExCat → Sp ≥ . Consequence (2) saysthat K -theory sends natural isomorphisms to homotopies. Terminology 5.1.1.
We refer to any functor E : ExCat → Sp ≥ sending natural iso-morphisms to homotopies as a homotopy invariant of exact categories. Denoting by E n : ExCat → Ab the composite functor π n ◦ E , we have that E n satisfies the conditions (1)and (2) listed above. Thus any such functor will provide a family of Mackey functors viaan analogous proof to that of Theorem 3.0.1. For any ring R , denote by E ( R ) := E ( P ( R ))the space given by evaluating E on the exact category of finitely generated projective R -modules, and denote by E n ( R ) := π n E ( R ) its n th homotopy group. Corollary 5.1.2.
Let R be any G -ring, where G is a finite group whose order is invertibleover R . Let E be any homotopy invariant on exact categories, and let n ≥ G/H E n ( R θ [ H ])is a Mackey functor, where restriction and transfer are induced by restriction and extensionof scalars along exact categories of finitely generated projective modules over twisted grouprings. 32 omotopy Mackey functors of equivariant algebraic K -theory Thomas Brazelton Example 5.1.3.
We may let E be any one of HH , THH , TC , TP , TR in Corollary 5.1.2,yielding many new Mackey functors such as THH n ( R θ [ H ]), the topological Hochschildhomology of twisted group rings. The structure of THH n ( R θ [ H ]) was explored in thethesis of Daniel Vera [Ver10], although this Mackey functor structure was not known.The structure of the Mackey functors associated to these homotopy invariants, as wellas their relation to trace maps, will be explored in a later paper.5.2. The Mackey functors of K -theory of fixed subrings of Galois extensions. Given a finite Galois extension of fields G = Gal( L/k ), we may consider L as a G -ringunder the natural Galois group action. It is a classical example that the assignment G/H K n ( L H ) is a Mackey functor (see, for example [Th´e95, 53.10]). We can prove a new prooffor any Galois extension of rings, relying on Galois descent, which follows as an immediatecorollary of Theorem 3.0.1.Let R → S be a Galois extension of rings (see [AG60, p. 396] for the original definition)with Galois group G = Gal( S/R ), and denote by θ the action of G on S . We remark that an S θ [ G ] module is an S -module equipped with a semilinear action of the Galois group. Thisis the same as an S -module equipped with Galois descent data . We recall that moduleswith descent data are equivalent to modules over the fixed subring, explicitly we have anequivalence of categories
Mod ( S θ [ H ]) ≃ Mod ( S H ) for any subgroup H ⊆ G , and it is easyto see that this equivalence descends to finitely generated projective modules. Proposition 5.2.1.
Let R → S be a Galois extension of rings, and assume that the orderof the Galois group G is invertible over R . Then for any n ≥ G/H K n ( S H )is a Mackey functor, where restriction and transfer come from restriction and extension ofscalars between fixed subrings. Corollary 5.2.2.
Under the conditions of Proposition 5.2.1, we have that
G/H THH n ( S H )is a Mackey functor for any n ≥ E ).5.3. The Mackey functor of K -theory of endomorphism rings. Suppose that θ : G → Aut
Ring ( R ) is a group action on a ring R . Then let R G ⊆ R denote the subring of G -fixed points under this action. We may define a ring homomorphism(1) R θ [ G ] → End R G ( R ) rg ( t rθ g ( t )) . An equivalence of categories can easily be seen to preserve projective objects — if a morphism admitsleft lifting with respect to epimorphisms in the target, then the functor exhibiting an equivalence preservesepimorphisms (since it is a left adjoint perhaps after promoting to an adjoint equivalence), so the imageof this morphism admits left lifting with respect to all epimorphisms in the essential image, which is theentire target category by equivalence. omotopy Mackey functors of equivariant algebraic K -theory Thomas Brazelton In general, we shouldn’t expect this map to be injective or surjective. However by a clas-sical theorem of Auslander, we can state a sufficient condition for this ring homomorphismto be an isomorphism. In the following theorem we retain our ongoing assumption that G is finite and that | G | − ∈ R Theorem 5.3.2. [Aus62, Proposition 3.4] If R is a normal domain, and R is unramified incodimension one over R G , then the ring homomorphism in Equation 1 is an isomorphism.For a detailed proof of this statement, we refer the reader to [LW12, Chapter 5.2].Via our result in Theorem 3.0.1, we can then prove that the algebraic K -groups of suchendomorphism rings admit the structure of a Mackey functor. In particular the restrictionsand transfers would arise from passing through the isomorphism with twisted group rings.We can actually say a bit more, as we can show that there is a natural choice of restrictionand transfer coming intrinsically from endomorphism rings.Suppose that G and R satisfy the conditions of Theorem 5.3.2. Let H ⊆ K be asubgroup. Then we have that R K ⊆ R H . In particular, any endomorphism of R which isfixed over R H is also fixed over R K . Thus there is a natural forgetful ring homomorphismEnd R H ( R ) → End R K ( R ) fitting into the diagram R θ [ H ] End R H ( R ) R θ [ K ] End R K ( R ) . Similarly if H ⊆ G is a subgroup and g ∈ G any element, there is a natural ring homomor-phism of endomorphism rings End R H ( R ) → End R gH ( R ), given by sending φ θ g ◦ φ ◦ θ g − .We can easily see this fits into a diagram R θ [ H ] End R H ( R ) R θ [ g H ] End R gH ( R ) . γ g Proposition 5.3.3.
Let G be a finite group acting on a normal domain R , so that | G | − ∈ R and R is unramified of codimension one over R G . Then for any n ≥
0, we have that
G/H K n (End R H ( R ))is a Mackey functor, where • transfer Tr KH for a subgroup H ⊆ K is induced by extension of scalars along theinclusions End R H ( R ) → End R K ( R ) • restriction Res KH is induced by restriction of sclars along End R H ( R ) → End R K ( R ) • conjugation is induced by extension of scalars along the ring homomorphism End R H ( R ) → End R gH ( R ) sending φ θ g ◦ φ ◦ θ g − .34 omotopy Mackey functors of equivariant algebraic K -theory Thomas Brazelton Example 5.3.4.
Under the conditions of Proposition 5.3.3, we have that
G/H THH n (End R H ( R ))is a Mackey functor (where we may also replace THH by any homotopy invariant). Appendix A. Bimodule structures
Let F : Mod ( R ) → Mod ( S ) denote a right exact coproduct-preserving functor betweenleft module categories. Then F ( R ) inherits a right R -module structure, as described in[Wat60]. Let m ∈ M , and consider the multiplication morphism ω m : R → Mr rm. Applying F , this gives a map F ω m : F R → F M for all m ∈ M . Letting m vary, we havea morphism F R × M → F M ( x, m ) F ω m ( x ) . Letting M = R , we have a map of the form F R × R → F R ( x, r ) F ω r ( x ) . We may verify this defines a right R -module structure on F R .This holds in general for right exact functors preserving coproducts. In our case, we willbe interested in building up an understanding of these structures for the case where F isextension or restriction of scalars along the ring maps ρ KH and γ x , with the ultimate goalof comparing the right R θ [ H ]-module structures on L ni =1 Tr JJ ∩ x K γ x ! Res KJ x ∩ K ( R θ [ H ]) andRes HJ Tr HK R θ [ H ]. Proposition A.0.1.
Let ω r : R → R be right multiplication by r viewed as a left R -modulehomomorphism, and let f : R → S . Then f ! ω r is of the form f ! ω r : S ⊗ R R → S ⊗ R R ( s ′ ⊗ r ′ ) ( s ′ ⊗ r ′ )(1 ⊗ r ) = s ⊗ r ′ r = sf ( r ′ r ) . So we have that f ! ω r = ω f ( r ) is right multiplication by f ( r ). Proposition A.0.2.
Let ω r : R → R be right multiplication by r viewed as a left R -modulehomomorphism, and let g : T → R . Then g ∗ ω r is of the form g ∗ ω r : g ∗ R → g ∗ Rr ′ r ′ · r, where − · r is viewed as a T -module homomorphism.35 omotopy Mackey functors of equivariant algebraic K -theory Thomas Brazelton This leads us to the very rough rule that f ! ω r = ω f ( r ) , and f ∗ ω r = ω r , where one needsto be careful about the ambient context. Proposition A.0.3.
Let ρ KH : R θ [ H ] → R θ [ K ], and let H \ K = Hy , . . . , Hy n be a choiceof right coset representatives. Let γ x : R θ [ H ] → R θ [ x H ].(1) Consider extension of scalars ( ρ KH ) ! : Mod ( R θ [ H ]) → Mod ( R θ [ K ]). We have that theright R θ [ H ]-module structure on (cid:0) ρ KH (cid:1) ! ( R θ [ H ]) is given by right multiplication on R θ [ K ].(2) Consider restriction of scalars (cid:0) ρ KH (cid:1) ∗ : Mod ( R θ [ K ]) → Mod ( R θ [ H ]). Then theright R θ [ K ]-module structure on (cid:0) ρ KH (cid:1) ∗ ( R θ [ H ]) is given by right multiplicationby R θ [ K ] viewed as an R θ [ H ]-module isomorphism. Explicitly, it is a map of theform ⊕ i R θ [ H ] y i × R θ [ K ] → ⊕ i R θ [ H ] y i X i r i h i y i , rk ! X i r i φ h i ( r ) h i k ! , where we express P i r i φ h i ( r ) h i k = P i r ′ i h ′ i y i as a new sum in the y i ’s.(3) Consider extension of scalars ( γ x ) ! : R θ [ H ] → R θ [ x H ]. Then the right modulestructure on γ x ! ( R θ [ H ]) is given by right multiplication by R θ [ H ] under the map γ x . Explicitly it is R θ [ x H ] × R θ [ H ] → R θ [ x H ]( ry, rh ) ry · γ x ( rh ) . Corollary A.0.4.
Let
J, K ⊆ H , and let J \ H = ∪ ri =1 J y i , and let K \ H = ∪ Kz j . Then theright R θ [ K ]-module structure on Res HJ Tr HK R θ [ K ] ∼ = ⊕ ri =1 R θ [ J ] 1 R y i is given by viewing rk ∈ R θ [ H ], and writing rk = rjy ℓ for some y ℓ and j ∈ J , and then considering the map ⊕ ri =1 R θ [ J ] 1 R y i × R θ [ K ] → ⊕ ri =1 R θ [ J ] 1 R y i (cid:16)X r ′ j ′ y i , rjy ℓ (cid:17) (cid:16)X r ′ φ j ′ ( r ) y i jy ℓ (cid:17) . Corollary A.0.5.
Let
J, K ⊆ H , and let x be a choice of double coset representative for J \ H/K . Let J x ∩ K \ K = ∪ i ( J x ∩ K ) β i . Then the right module structure of R θ [ K ] onTr JJ ∩ x K γ x ! Res KJ x ∩ K ( R θ [ K ])is given by first writing rk = rx − jxβ ℓ for some β ℓ , and right multiplying. Then after γ x ! , it is given by right multiplication by γ x ( rx − jxβ ℓ ), which is φ x ( r ) jxβ ℓ x − , then finallytransferring, i.e. multiplying through by this as an R θ [ J ]-module homomorphism on theright. Proof of Proposition 3.4.5.
Let P r i k i ∈ R θ [ K ] be arbitrary, and let ω P r i k i denote theright multiplication by this element, viewed as a left R θ [ K ]-module homomorphism. Then36 omotopy Mackey functors of equivariant algebraic K -theory Thomas Brazelton we see that, for a given x , we haveTr JJ ∩ x K γ x ! Res KJ x ∩ K ω P r i k i = ω P φ x ( r i ) xk i x − , by first viewing ω P r i k i as an R θ [ J x ∩ K ]-module homomorphism under restriction, thenextending to obtain ω P φ x ( r i ) xk i x − as an R θ [ J ∩ x K ]-module homomorphism, and finallyrestricting to view ω P φ x ( r i ) xk i x − as an R θ [ J ]-module homomorphism.We see that Res HJ Tr HK ω P r i k i = ω P r i k i , by extending along ρ HK and restricting along ρ HJ .For the sake of notation, let P := ⊕ Tr JJ ∩ x K γ x ! Res KJ x ∩ K Q := Res HJ Tr HK . Let ε : P ( R θ [ K ]) ∼ −→ Q ( R θ [ K ]) denote the isomorphism of R θ [ J ]-modules given in Corollary 3.4.4.Then it suffices to verify that F ω P r i k i ε = εGω P r i k i .Recall that P ( R θ [ K ]) = n M i =1 r i M ℓ =1 R θ [ J ] x i β i,j x − i Q ( R θ [ K ]) = n M i =1 r i M ℓ =1 R θ [ J ] x i β i,j . Let r ℓ j ℓ x ℓ β ℓ,a x − ℓ be an arbitrary summand in P ( R θ [ K ]). Then we consider the following(a priori noncommutative) diagram(6) P ( R θ [ K ]) P ( R θ [ K ]) Q ( R θ [ K ]) Q ( R θ [ K ]) . F ω P riki ε εGω P riki Tracing through where the element r ℓ j ℓ x ℓ β ℓ,a x − ℓ maps, we see that r ℓ j ℓ x ℓ β ℓ,a x − ℓ P i r ℓ φ j ℓ x ℓ β ℓ,a ( r i ) j ℓ x ℓ β ℓ,a k i x − ℓ r ℓ j ℓ x ℓ β ℓ,a P i r ℓ φ j ℓ x ℓ β ℓ,a ( r i ) x ℓ β ℓ,a k i . −· P i φ xℓ ( r i ) x ℓ k i x − ℓ −· x ℓ −· x ℓ −· P i r i k i omotopy Mackey functors of equivariant algebraic K -theory Thomas Brazelton Thus since r ℓ j ℓ x ℓ β ℓ,a x − ℓ was arbitrary, we have that Equation 6 commutes. This impliesthat ε is a right R θ [ J ]-module homomorphism. (cid:3) References [AG60] Maurice Auslander and Oscar Goldman,
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