Mackey Functors Research Articles

Spectral Mackey functors and equivariant algebraic K-theory, II

We study the "higher algebra" of spectral Mackey functors, which the first named author introduced in Part I of this paper. In particular, armed with our new theory of symmetric promonoidal $\infty$-categories and a suitable generalization of the second named author's Day convolution, we endow the $\infty$-category of Mackey functors with a well-behaved symmetric monoidal structure. This makes it possible to speak of spectral Green functors for any operad $O$. We also answer a question of A. Mathew, proving that the algebraic $K$-theory of group actions is lax symmetric monoidal. We also show that the algebraic $K$-theory of derived stacks provides an example. Finally, we give a very short, new proof of the equivariant Barratt-Priddy-Quillen theorem, which states that the algebraic $K$-theory of the category of finite $G$-sets is simply the $G$-equivariant sphere spectrum.

G-symmetric monoidal categories of modules over equivariant commutative ring spectra

We describe the multiplicative structures that arise on categories of equivariant modules over certain equivariant commutative ring spectra. Building on our previous work on N-infinity ring spectra, we construct categories of equivariant operadic modules over N-infinity rings that are structured by equivariant linear isometries operads. These categories of modules are endowed with equivariant symmetric monoidal structures, which amounts to the structure of an "incomplete Mackey functor in homotopical categories". In particular, we construct internal norms which satisfy the double coset formula. We regard the work of this paper as a first step towards equivariant derived algebraic geometry.

The center of a Green biset functor

For a Green biset functor $A$, we define the commutant and the center of $A$ and we study some of their properties and their relationship. This leads in particular to the main application of these constructions: the possibility of splitting the category of $A$-modules as a direct product of smaller abelian categories. We give explicit examples of such decompositions for some classical shifted representation functors. These constructions are inspired by similar ones for Mackey functors for a fixed finite group.

K-theory of Hermitian Mackey functors, real traces, and assembly

We define a genuine $\mathbb{Z}/2$-equivariant real algebraic $K$-theory spectrum $KR(A)$, for every genuine $\mathbb{Z}/2$-equivariant spectrum $A$ equipped with a compatible multiplicative structure. This construction extends the real $K$-theory of Hesselholt-Madsen for discrete rings and the Hermitian $K$-theory of Burghelea-Fiedorowicz for simplicial rings. We construct a natural trace map of $\mathbb{Z}/2$-spectra $tr\colon KR(A)\to THR(A)$ to the real topological Hochschild homology spectrum, which extends the $K$-theoretic trace of B\"okstedt-Hsiang-Madsen. The trace provides a splitting of the real $K$-theory of the spherical group-ring. We use this splitting on the geometric fixed points of $KR$, which we regard as an $L$-theory of genuinely equivariant ring spectra, to reformulate the Novikov conjecture on the homotopy invariance of the higher signatures purely in terms of the module structure of the rational $L$-theory of the "Burnside group-ring".

An equivariant tensor product on Mackey functors

For all subgroups H of a cyclic p-group G we define norm functors that build a G-Mackey functor from an H-Mackey functor. We give an explicit construction of these functors in terms of generators and relations based solely on the intrinsic, algebraic properties of Mackey functors and Tambara functors. We use these norm functors to define a monoidal structure on the category of Mackey functors where Tambara functors are the commutative ring objects.

Computations in $$C_{pq}$$ C pq -Bredon cohomology

In this paper, we compute the $$RO(C_{pq})$$ -graded cohomology of $$C_{pq}$$ -orbits. We deduce that in all the cases the Bredon cohomology groups are a function of the fixed point dimensions of the underlying virtual representations. Further, when thought of as a Mackey functor, the same independence result holds in almost all cases. This generalizes earlier computations of Stong and Lewis for the group $$C_p$$ . The computations of cohomology of orbits are used to prove a freeness theorem. The analogous result for the group $$C_p$$ was proved by Lewis. We demonstrate that certain complex projective spaces and complex Grassmannians satisfy the freeness theorem.

Equivariant $A$-Theory

We give a new construction of the equivariant K -theory of group actions [ C. Barwick , “Spectral Mackey functors and equivariant algebraic K -theory (I)”, Adv. Math. 304, 646–727 (2017; Zbl 1348.18020) and C. Barwick et al., “Spectral Mackey functors and equivariant algebraic K -theory (II)”, Preprint (2015); ], producing an infinite loop G -space for each Waldhausen category with G -action, for a finite group G . On the category R(X) of retractive spaces over a G -space X , this produces an equivariant lift of Waldhausen's functor A(X) , and we show that the H -fixed points are the bivariant A -theory of the fibration X_{hH}\to BH . We then use the framework of spectral Mackey functors to produce a second equivariant refinement A_G(X) whose fixed points have tom Dieck type splittings. We expect this second definition to be suitable for an equivariant generalization of the parametrized h -cobordism theorem.

The dominant dimension of cohomological Mackey functors

We show that a separable equivalence between symmetric algebras preserves the dominant dimensions of certain endomorphism algebras of modules. We apply this to show that the dominant dimension of the category [Formula: see text] of cohomological Mackey functors of a [Formula: see text]-block [Formula: see text] of a finite group with a nontrivial defect group is [Formula: see text].

A $C_2$-equivariant analog of Mahowald’s Thom spectrum theorem

We prove that the $C_2$-equivariant Eilenberg-MacLane spectrum associated to the constant Mackey functor $\underline{\mathbb{F}}_2$ is equivalent to a Thom spectrum over ${\Omega^\rho S^{\rho + 1}}$.

Generalised Kawada–Satake method for Mackey functors in class field theory

We propose and study a generalised Kawada–Satake method for Mackey functors in the class field theory of positive characteristic. The root of this method is in the use of explicit pairings, such as the Artin–Schreier–Witt pairing, for groups describing abelian extensions. We separate and simplify the algebraic component of the method and discuss a relation between the existence theorem in class field theory and topological reflexivity with respect to the explicit pairing. We apply this method to derive higher local class field theory of positive characteristic, using advanced properties of topological Milnor K-groups of such fields.

On monoidal group actions on tensor categories and their Green functors

In this paper we introduce the notion of a categorical Mackey functor. This categorical notion allows us to obtain new Mackey functors by passing to Quillen’s K-theory of the corresponding abelian categories. In the case of an action by monoidal autoequivalences on a monoidal category the Mackey functor obtained at the level of Grothendieck rings has in fact a Green functor structure.

On Morita and derived equivalences for cohomological Mackey algebras

By results of the second author, a source algebra equivalence between two p-blocks of finite groups induces an equivalence between the categories of cohomological Mackey functors associated with these blocks, and a splendid derived equivalence between two blocks induces a derived equivalence between the corresponding categories ofcohomological Mackey functors. The main result of this paper proves a partial converse: an equivalence (resp. Rickard equivalence) between the categories of cohomological Mackey functors of two blocks of finite groups induces a permeable Morita (resp. derived) equivalence between the two block algebras.

On the André–Quillen homology of Tambara functors

We lift to equivariant algebra three closely related classical algebraic concepts: abelian group objects in augmented commutative algebras, derivations, and Kähler differentials. We define Mackey functor objects in the category of Tambara functors augmented to a fixed Tambara functor R_, and we show that the usual square-zero extension gives an equivalence of categories between these Mackey functor objects and ordinary modules over R_. We then describe the natural generalization to Tambara functors of a derivation, building on the intuition that a Tambara functor has products twisted by arbitrary finite G-sets, and we connect this to square-zero extensions in the expected way. Finally, we show that there is an appropriate form of Kähler differentials which satisfy the classical relation that derivations out of R_ are the same as maps out of the Kähler differentials.

On the Projective Dimensions of Mackey Functors

We examine the projective dimensions of Mackey functors and cohomological Mackey functors. We show over a field of characteristic p that cohomological Mackey functors are Gorenstein if and only if Sylow p-subgroups are cyclic or dihedral, and they have finite global dimension if and only if the group order is invertible or Sylow subgroups are cyclic of order 2. By contrast, we show that the only Mackey functors of finite projective dimension over a field are projective. This allows us to give a new proof of a theorem of Greenlees on the projective dimension of Mackey functors over a Dedekind domain. We conclude by completing work of Arnold on the global dimension of cohomological Mackey functors over ℤ.

The additive completion of the biset category

Let R be a commutative unital ring. We construct a category CR of fractionsX/G, where G is a finite group and X is a finite G-set, and with morphisms given by R-linear combinations of spans of bisets. This category is an additive, symmetric monoidal and self-dual category, with a Krull–Schmidt decomposition for objects. We show that CR is equivalent to the additive completion of the biset category and that the category of biset functors over R is equivalent to the category of R-linear functors from CR to R-Mod. We also show that the restriction of one of these functors to a certain subcategory of CR is a fused Mackey functor.

A note on Green functors with inflation

This note is motivated by the problem to understand, given a commutative ring F, which G-sets X, Y give rise to isomorphic F[G]-representations F[X]≅F[Y]. A typical step in such investigations is an argument that uses induction theorems to give very general sufficient conditions for all such relations to come from proper subquotients of G. In the present paper we axiomatise the situation, and prove such a result in the generality of Mackey functors and Green functors with inflation. Our result includes, as special cases, a result of Deligne on monomial relations, a result of the first author and Tim Dokchitser on Brauer relations in characteristic 0, and a new result on Brauer relations in characteristic p>0. We will need the new result in a forthcoming paper on Brauer relations in positive characteristic.

Mislin’s theorem for fusion systems through Mackey functors

ABSTRACTWe state and prove a fusion system version of Mislin’s theorem [9] on cohomology and control of fusion using Mackey functors. The issue of an algebraic proof is also discussed.

The Dade group of Mackey functors for p-groups

We introduce endo-permutation Mackey functors and the Mackey–Dade group of a p-group and study their basic properties. We exhibit relations between the Mackey–Dade group of a finite p-group P and the Dade groups of the Weyl groups of all subgroups of P. As a result, we show that the Mackey–Dade group tensored with Q over Z is the kernel of the linearization map for Mackey functors, introduced by the author.

Spectral Mackey functors and equivariant algebraic K-theory (I)

Spectral Mackey functors are homotopy-coherent versions of ordinary Mackey functors as defined by Dress. We show that they can be described as excisive functors on a suitable ∞-category, and we use this to show that universal examples of these objects are given by algebraic K-theory.More importantly, we introduce the unfurling of certain families of Waldhausen ∞-categories bound together with suitable adjoint pairs of functors; this construction completely solves the homotopy coherence problem that arises when one wishes to study the algebraic K-theory of such objects as spectral Mackey functors.Finally, we employ this technology to introduce fully functorial versions of A-theory, upside-down A-theory, and the algebraic K-theory of derived stacks. We use this to give what we think is the first general construction of π1ét-equivariant algebraic K-theory for profinite étale fundamental groups. This is key to our approach to the “Mackey functor case” of a sequence of conjectures of Gunnar Carlsson.

Direct limits and inverse limits of Mackey functors

Let G be a finite group, S the subgroup category of G, and M a Mackey functor on G. For full subcategories G and H of S, we have the direct limit M⁎(G) and the inverse limit M⁎(H) of M. In this paper we study relation between the canonical homomorphisms ind:M⁎(G)→M(G) and res:M(G)→M⁎(H).