Categorical Krull-Remak-Schmidt for triangulated categories

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.