# Green equivalences in equivariant mathematics

@article{Balmer2021GreenEI, title={Green equivalences in equivariant mathematics}, author={Paul Balmer and Ivo Dell'Ambrogio}, journal={Mathematische Annalen}, year={2021} }

We establish Green equivalences for all Mackey 2-functors, without assuming Krull-Schmidt. By running through the examples of Mackey 2-functors, we recover all variants of the Green equivalence and Green correspondence known in representation theory and obtain new ones in several other contexts. Such applications include equivariant stable homotopy theory in topology and equivariant sheaves in geometry.

## 3 Citations

Clifford's theorem for orbit categories

- Mathematics
- 2022

Clifford theory relates the representation theory of finite groups to those of a fixed normal subgroup by means of induction and restriction, which is an adjoint pair of functors. We generalize this…

Green 2-functors

- Mathematics
- 2021

We extend the theory of Mackey 2-functors [BD20] by defining the appropriate notion of rings, namely Green 2-functors. After providing the first results of our theory and abundant examples, we show…

Cohomological Mackey 2-functors

- Mathematics
- 2021

. We show that the bicategory of ﬁnite groupoids and right-free permutation bimodules is a quotient of the bicategory of Mackey 2-motives introduced in [BD20], obtained by modding out the so-called…

## References

SHOWING 1-10 OF 33 REFERENCES

Equivariant E-Theory for C*-Algebras

- Mathematics
- 2000

Introduction Asymptotic morphisms The homotopy category of asymptotic morphisms Functors on the homotopy category Tensor products and descent $C^\ast$-algebra extensions $E$-theory Cohomological…

Equivariant Kasparov theory of finite groups via Mackey functors

- Mathematics
- 2011

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…

Green correspondence and relative projectivity for pairs of adjoint functors between triangulated categories

- Mathematics
- 2020

Auslander and Kleiner proved in 1994 an abstract version of Green correspondence for pairs of adjoint functors between three categories. They produce additive quotients of certain subcategories…

The Green Correspondence for Infinitely Generated Modules

- Mathematics
- 2001

If G is a finite group, then the usual version of the Green correspondence applies to finitely generated kG‐modules when k is a field of characteristic p > 0 or a p‐adic ring. The paper presents a…

Mackey 2-Functors and Mackey 2-Motives

- Mathematics
- 2020

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…

Homotopy limits in triangulated categories

- Mathematics
- 1993

© Foundation Compositio Mathematica, 1993, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions…