[Paper Review] The Grothendieck group of an n-angulated category

Petter Andreas Bergh, Marius Thaule|arXiv (Cornell University)|May 25, 2012

Algebraic structures and combinatorial models1 references4 citations

TL;DR

This paper introduces the Grothendieck group for $n$-angulated categories and establishes a bijective correspondence between subgroups of the Grothendieck group and complete, dense $n$-angulated subcategories when $n$ is odd. For tensor $n$-angulated categories, the Grothendieck group becomes a ring whose ideals classify complete, dense $n$-angulated tensor ideals, generalizing Thomason's classification in triangulated categories.

ABSTRACT

We define the Grothendieck group of an n-angulated category and show that for odd n its properties are as in the special case of n=3, i.e. the triangulated case. In particular, its subgroups classify the dense and complete n-angulated subcategories via a bijective correspondence. For a tensor n-angulated category, the Grothendieck group becomes a ring, whose ideals classify the dense and complete n-angulated tensor ideals of the category.

Motivation & Objective

• To define and study the Grothendieck group of an $n$-angulated category, generalizing the classical construction for triangulated categories.
• To establish a bijective correspondence between subgroups of the Grothendieck group and complete, dense $n$-angulated subcategories when $n$ is odd.
• To extend the classification to tensor $n$-angulated categories by showing that ideals in the Grothendieck ring classify $n$-angulated tensor ideals.
• To clarify the structural role of the Grothendieck group in higher-dimensional homological algebra, particularly in cluster tilting contexts.

Proposed method

• Define the Grothendieck group $K_0(\mathcal{C})$ as the free abelian group on isomorphism classes of objects modulo Euler relations from $n$-angles.
• Prove that for odd $n$, the additive inverse of $[A]$ in $K_0(\mathcal{C})$ is $[\Sigma A]$, enabling the classification of subcategories via subgroups.
• Construct a surjective homomorphism from $K_0(\mathcal{C})$ to $K_0(\mathcal{T})$ when $\mathcal{C}$ arises from a cluster tilting subcategory of a triangulated category $\mathcal{T}$.
• Define tensor $n$-angulated categories with a symmetric monoidal structure compatible with $n$-angles, turning $K_0(\mathcal{C})$ into a Grothendieck ring.
• Establish a correspondence between ideals in $K_0(\mathcal{C})$ and complete, dense $n$-angulated tensor ideals of $\mathcal{C}$, with prime ideals corresponding to prime tensor ideals.
• Use the structure of $n$-angles and the behavior of the suspension functor $\Sigma$ to prove that the correspondence is bijective for odd $n$.

Experimental results

Research questions

• RQ1 Does the Grothendieck group of an $n$-angulated category classify complete and dense $n$-angulated subcategories when $n$ is odd?
• RQ2 How does the Grothendieck group of an $n$-angulated category relate to that of the underlying triangulated category, especially in cluster tilting contexts?
• RQ3 Can the Grothendieck group of a tensor $n$-angulated category be endowed with a ring structure that classifies tensor ideals?
• RQ4 Why does the classification fail for even $n$, and what structural property breaks down?

Key findings

• For odd $n$, there is a bijective correspondence between subgroups of the Grothendieck group $K_0(\mathcal{C})$ and complete, dense $n$-angulated subcategories of $\mathcal{C}$.
• The correspondence recovers Thomason's classification theorem in the case $n=3$, where all triangulated subcategories are complete.
• When $\mathcal{C}$ is a tensor $n$-angulated category, $K_0(\mathcal{C})$ becomes a ring, and ideals in this ring classify complete, dense $n$-angulated tensor ideals.
• Prime ideals in $K_0(\mathcal{C})$ correspond bijectively to complete, dense $n$-angulated tensor prime ideals of $\mathcal{C}$.
• The proof relies crucially on the fact that for odd $n$, the inverse of $[A]$ in $K_0(\mathcal{C})$ is $[\Sigma A]$, which fails for even $n$.
• There is a natural surjective homomorphism from $K_0(\mathcal{C})$ to $K_0(\mathcal{T})$ when $\mathcal{C}$ arises from a cluster tilting subcategory of a triangulated category $\mathcal{T}$.