Related papers: On primordial groups for the Green ring

Our aim is to transfer several foundational results from the modular representation theory of finite groups to the wider context of profinite groups. We are thus interested in profinite modules over the completed group algebra k[[G]] of a…

Let $G$ be a finite group and $k$ a field of characteristic $p$. We conjecture that if $M$ is a $kG$-module with $H^*(G,M)$ finitely generated as a module over $H^*(G,k)$ then as an element of the stable module category…

We prove an induction theorem for the higher algebraic K-groups of group algebras $kG$ of finite groups $G$ over characteristic $p$ finite fields $k$. For a certain class of finite groups, which we call $p$-isolated, this reduces…

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…

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…

Let $p$ be a prime, let $G$ be a finite group of order divisible by $p$, and let $k$ be a field of characteristic $p$. An endotrivial $kG$-module is a finitely generated $kG$-module $M$ such that its endomorphism algebra…

Let G be a finite group. We systematically exploit general homological methods in order to reduce the computation of G-equivariant KK-theory to topological equivariant K-theory. The key observation is that the functor assigning to a…

Classifying endotrivial kG-modules, i.e., elements of the Picard group of the stable module category for an arbitrary finite group G, has been a long-running quest. By deep work of Dade, Alperin, Carlson, Thevenaz, and others, it has been…

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…

Let $\hat\Z_p$ be the ring of $p$-adic integers. We prove in the present paper that the category of polynomial functors from finitely generated free abelian groups to $\hat \Z_p$-modules of degree at most $p$ is equivalent to the category…

A Mackey type decomposition for group actions on abelian categories is described. This allows us to define new Mackey functors which associates to any subgroup the $K$-theory of the corresponding equivariantized abelian category. In the…

In this paper we develop computational tools to study the higher algebraic $K$-theory of Green functors. We construct a spectral sequence converging to the algebraic $\mathbb{G}$-theory of any $G$-Green functor, for $G$ a cyclic $p$-group.…

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…

In equivariant topology, Greenlees and May used Mackey functors to show that, rationally, the stable homotopy category of $G$-spectra over a finite group $G$ splits as a product of simpler module categories. We extend the algebraic part…

In Finite Group Modular Representation Theory, the basic objects are the indecomposable and simple modules. This paper offers a new classification of these objects that refines the Green Theory Classification of indecomposable and simple…

We study collections of additive categories $\mathcal{M}(G)$, indexed by finite groups $G$ and related by induction and restriction in a way that categorifies usual Mackey functors. We call them `Mackey 2-functors'. We provide a large…

In this paper, we study the K-theory on higher modules in spectral algebraic geometry. We relate the K-theory of an $\infty$-category of finitely generated projective modules on certain $\mathbb{E}_{\infty}$-rings with the K-theory of an…

Let G be a finite group and let k be a field of characteristic p. Given a finitely generated indecomposable non-projective kG-module M, we conjecture that if the Tate cohomology $\HHHH^*(G, M)$ of G with coefficients in M is finitely…

These notes provide an informal introduction to a type of Mackey functor that arises naturally in algebraic topology in connection with Morava $K$-theory of classifying spaces of finite groups. The main aim is to identify key algebraic…

A finite group $G$ is called $k$-factorizable if for every ordered factorization $|G|=a_1\cdots a_k$ into integers each greater than $1$ there exist subsets $A_1,\dots,A_k\subseteq G$ such that $|A_i|=a_i$ for each $i$ and $G=A_1\cdots…