Categorical Krull-Remak-Schmidt for triangulated categories

TL;DR

This paper proves a categorical Krull-Remak-Schmidt theorem for triangulated categories, showing that decompositions into indecomposable components are essentially unique up to permutation, with examples illustrating these decompositions.

Contribution

It establishes a Krull-Remak-Schmidt type result for triangulated categories, demonstrating the uniqueness of decompositions into connected components under equivalences.

Findings

Decomposition of triangulated categories into indecomposables is essentially unique.

Existence of a bijection between component indices under equivalence.

Examples of categories with finite decompositions.

Abstract

Let $R$ be a commutative ring If $\mathcal{C}_1$ and $\mathcal{C}_2$ are $R$-linear triangulated categories then we can give an obvious triangulated structure on $\mathcal{C} = \mathcal{C}_1 \oplus \mathcal{C}_2$ where $Hom_\mathcal{C}(U, V) = 0$ if $U \in \mathcal{C}_i$ and $V \in \mathcal{C}_j$ with $i \neq j$. We say a $R$-linear triangulated category $\mathcal{C}$ is disconnected if $\mathcal{C} = \mathcal{C}_1 \oplus \mathcal{C}_2$ where $\mathcal{C}_i$ are non-zero triangulated subcategories of $\mathcal{C}$. Let $\mathcal{C}_i$ and $\mathcal{D}_j$ be connected triangulated $R$ categories with $i \in \Gamma$ and $j \in \Lambda$. Suppose there is an equivalence of triangulated $R$-categories \[ \Phi \colon \bigoplus_{i \in \Gamma}\mathcal{C}_i \xrightarrow{\cong} \bigoplus_{j \in \Lambda}\mathcal{D}_j \] Then we show that there is a bijective function $\pi \colon \Gamma \rightarrow \Lambda$ such that we have an equivalence $\mathcal{C}_i \cong \mathcal{D}_{\pi(i)} $ for all $i \in \Gamma$. We give several examples of connected triangulated categories and also of triangulated subcategories which decompose into utmost finitely many components.