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 $\mathbb Q$ over $\mathbb Z$ is the kernel of the linearization map for Mackey functors, introduced by the author.
In this paper, we describe the induction functor from the category of native Mackey functors to the category of biset functors for a finite group $G$ over an algebraically closed field $k$ of characteristic zero. We prove two applications of this description. As the first application, we exhibit that any projective biset functor over $k$ is induced from a (rational) virtual native Mackey functor. The second application is the explicit description of the projective indecomposable biset functors parameterized by simple groups.
By results of Rognerud, 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 of cohomological Mackey functors. We prove this by giving an intrinsic description of cohomological Mackey functors of a block in terms of the source algebras of the block, and then using this description to construct explicit two-sided tilting complexes realising the above me…
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 Dedeki…
We study cohomological finiteness conditions for groups associated to Mackey and cohomological Mackey functors, proving that the cohomological dimension associated to cohomological Mackey functors is always equal to the $\mathcal{F}$-cohomological dimension, and characterising the conditions Mackey-$\mathrm{FP}_n$ and cohomological Mackey-$\mathrm{FP}_n$. We show that all finiteness conditions for cohomological Mackey functors are unchanged when considering only the family of $p$-subgroups, and we characterise cohomological Mackey-$\mathrm{FP}_n$ conditions over the finite field $\mathbb{F}_p$…
Systematically using the language of groupoids, we survey the theory of global Mackey functors, global Green functors and global power functors. Given a global power functor, we study rings with similar operations. The example of n-class functions leads to the notion of an n-special lambda ring.
We consider analogs of Jacobson's $F$-Burnside construction and Boltje's $(-)_+$-construction for biset functors, using Mackey-functor theoretic interpretation of biset functors.
In this article, we consider a formulation of biset functors using the 2-category of finite sets with variable finite group actions. We introduce a 2-category $\mathbb{S}$, on which a biset functor can be regarded as a special kind of Mackey functors. This gives an analog of Dress' definition of a Mackey functor, in the context of biset functors.
Symmetric cohomology of groups, defined by M. Staic in [2], is similar to the way one defines the cyclic cohomology for algebras. We show that there is a well-defined restriction, conjugation and transfer map in symmetric cohomology, which form a Mackey functor under a restriction. Some new properties for the symmetric cohomology group using normalized cochains are also given.
We prove that Hochschild cohomology of a certain class of fully group-graded algebras is a Mackey functor. We use the machinery of transfer maps between the Hochschild cohomology of symmetric algebras.
Let $k$ be a field of characteristic $p$. We construct a new inflation functor for cohomological Mackey functors for finite groups over $k$. Using this inflation functor, we give an explicit presentation of the graded algebra of self extensions of the simple functor $S_\un^G$, when $p$ is odd and $G$ is an elementary abelian $p$-group.
Let $G$ be a finite group. In [HTW], Hambleton, Taylor and Williams have considered the question of comparing Mackey functors for $G$ and biset functors defined on subgroups of $G$ and bifree bisets as morphisms. This paper proposes a different approach to this problem, from the point of view of various categories of $G$-sets. In particular, the category of fused $G$-sets is introduced, as well its category of spans. The fused Mackey functors for $G$ over a commutative ring $R$ are defined as $R$-linear functors from this ($R$-linearized) category of spans to $R$-modules. They form an abelian …
For any finite group G, we define a bivariant functor from the Dress category of finite G-sets to the conjugation biset category, whose objects are subgroups of G, and whose morphisms are generated by certain bifree bisets. Any additive functor from the conjugation biset category to abelian groups yields a Mackey functor by composition. We characterize the Mackey functors which arise in this way.
For groups with a uniform bound on the length of chains of finite subgroups, we study the relationship between the Bredon cohomological dimension for proper actions and the notions of cohomological dimension one obtains by restricting the coefficients of Bredon cohomology to (cohomological) Mackey functors or fixed point functors. We also investigate the closure properties of the class of groups with finite Bredon cohomological dimension for Mackey functors.
Let $k$ be a field of characteristic $p>0$. Call a finite group $G$ a poco group over $k$ if any finitely generated cohomological Mackey functor for $G$ over $k$ has polynomial growth. The main result of this paper is that $G$ is a poco group over $k$ if and only if the Sylow $p$-subgroups of $G$ are cyclic, when $p>2$, or have sectional rank at most 2, when $p=2$. A major step in the proof is the case where $G$ is an elementary abelian $p$-group. In particular, when $p=2$, all the extension groups between simple functors can be determined completely, using a presentation of the graded algebra…
Balmer and Dell'Ambrogio introduced the pseudo-functor $P$ from the bicategory of $k$-linear Mackey $2$-motives to the bicategory of $k$-linear cohomological Mackey $2$-motives over a commutative ring $k$. They showed that $P$ maps the general Mackey $2$-motives to the cohomological Mackey $2$-motives by using the ring homomorphism from the crossed Burnside ring of a finite group $G$ over $k$ to the center $ ZkG$ of group algebra $kG$ ([BD21, Theorem 5.3]). We study the behavior of motivic decomposition of cohomological Mackey $2$-motives as images by $P$ of motivic decomposition of Mackey $2$…
We establish a novel approach to computing $G$-equivariant cohomology for a finite group $G$, and demonstrate it in the case that $G = C_{p^n}$. For any commutative ring spectrum $R$, we prove a symmetric monoidal reconstruction theorem for genuine $G$-$R$-modules, which records them in terms of their geometric fixedpoints as well as gluing maps involving their Tate cohomologies. This reconstruction theorem follows from a symmetric monoidal stratification (in the sense of \cite{AMR-strat}); here we identify the gluing functors of this stratification in terms of Tate cohomology. Passing from ge…
Motivated by the work of Jürgen Neukirch and Ivan Fesenko we propose a general definition of an abelian class field theory from a purely group-theoretical and functorial point of view. This definition allows a modeling of abelian extensions of a field inside more general objects than the invariants of a discrete module over the absolute Galois group of the field. The main objects serving as such models are cohomological Mackey functors as they have enough structure to make several reduction theorems of classical approaches work in this generalized setting and, as observed by Fesenko, they even…
In the previous article 'A Mackey-functor theoretic interpretation of biset functors', we have constructed the 2-category $\mathbb{S}$ of finite sets with variable finite group actions, in which bicoproducts and bipullbacks exist. As shown in it, biset functors can be regarded as a special class of Mackey functors on $\mathbb{S}$. In this article, we equip $\mathbb{S}$ with a system of adjoint triplets, which satisfies properties analogous to a derivator. This system encodes the six operations for finite groups. As a corollary, we show that the associated Burnside rings satify analogous proper…
We extend the theory of Mackey 2-functors introduced in arXiv:1808.04902 by defining the appropriate notion of rings, namely Green 2-functors. After providing the first results of our theory and abundant examples, we show how all classical Green functors familiar from representation theory and topology arise by decategorification, in various ways, of some Green 2-functor occurring in Nature.
Let p be a prime number. This paper introduces the Roquette category R_p of finite p-groups, which is an additive tensor category containing all finite p-groups among its objects. In R_p, every finite p-group P admits a canonical direct summand, called the edge of P. Moreover P splits uniquely as a direct sum of edges of Roquette p-groups, and the tensor structure of R_p can be described in terms of such edges. The main motivation for considering this category is that the additive functors from R_p to abelian groups are exactly the rational p-biset functors. This yields in particular very effi…
Consider the Mackey functor assigning to each finite group G the Green ring of finitely generated kG-modules, where k is a field of characteristic p>0. Thevenaz foresaw in 1988 that the class of primordial groups for this functor is the family of k-Dress groups. In this paper we prove that this is true for the subfunctor defined by the Green ring of finitely generated kG-modules of trivial source.
Let $G$ be a finite group and $R$ be a commutative ring. The Mackey algebra $μ_{R}(G)$ shares a lot of properties with the group algebra $RG$ however, there are some differences. For example, the group algebra is a symmetric algebra and this is not always the case for the Mackey algebra. In this paper we present a systematic approach to the question of the symmetricity of the Mackey algebra, by producing symmetric associative bilinear forms for the Mackey algebra. The category of Mackey functors is a closed symmetric monoidal category, so using the formalism of J.P. May for these categories, S…
We survey several notions of Mackey functors and biset functors found in the literature and prove some old and new theorems comparing them. While little here will surprise the experts, we draw a conceptual and unified picture by making systematic use of finite groupoids. This provides a road map for the various approaches to the axiomatic representation theory of finite groups, as well as some details which are hard to find in writing.
In this paper, I give several characterizations of {\em rational biset functors over $p$-groups}, which are independent of the knowledge of genetic bases for $p$-groups. I also introduce a construction of new biset functors from known ones, which is similar to the Yoneda construction for representable functors, and to the Dress construction for Mackey functors, and I show that this construction preserves the class of rational $p$-biset functors.\par This leads to a characterization of rational $p$-biset functors as additive functors from a specific quotient category of the biset category to ab…
We define and study the Burnside quotient Green ring of a Mackey functor. Some refinements of Dress induction theory are presented, together with applications to computation results for $K$-theory and $L$-theory of finite and infinite groups.
We study the higher derived functors of the inverse limit of a functor F: D --> Z_{(p)}-mod, where D is one of the standard categories which arise when studying the homotopy theory of the classifying space of a finite group G, e.g., the orbit category or the Quillen category of G. These higher limits are of importance e.g., for the study of maps between classifying spaces as well as for group cohomology. We show that these higher limits can be identified with the G-equivariant Bredon cohomology of the subgroup complex of p-subgroups in G (i.e., the nerve of the poset of p-subgroups in G) with …
