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Grothendieck ring

Contents

Context

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Contents

Idea

Any abelian category CC gives rise to an abelian group K(C)K(C) called its Grothendieck group (see there for more). If we apply this construction to a monoidal abelian category or generally to a rig category, K(C)K(C) is a ring, called the Grothendieck ring.

If CC is a braided monoidal category, K(C)K(C) becomes a commutative ring.

If CC is a symmetric monoidal category, K(C)K(C) becomes a Λ\Lambda-ring.

Examples

  • The Grothendieck ring of the monoidal category of finite G-sets is called the Burnside ring of GG.

References

  • Peter May, Picard groups, Grothendieck rings,and Burnside rings of categories, Advances in Mathematics 163, 1–16 (2001), (pdf).

Last revised on June 15, 2021 at 05:49:04. See the history of this page for a list of all contributions to it.