Idempotents in nilpotent quotients and triangulated categories
Recall that an ideal \(\mathbb I\) of an additive category \(\mathbb A\) is a subbifunctor of the bifunctor \(\Hom_{\mathbb A}(-, -) \colon\mathbb A^{op} \times\mathbb A \to Ab\), where \(Ab\) is the category of abelian groups. The quotient category \(\mathbb A/\mathbb I\), which has the same objects as \(\mathbb A\), while morphisms in \(\mathbb A/\mathbb I\) are given by \(\Hom_{\mathbb A/\mathbb I}(A, B) := \Hom_{\mathbb A}(A, B)/\mathbb I(A, B)\).\N\NLet \(\mathbb I\) and \(\mathbb J\) be ideals of \(\mathbb A\), the product \(\mathbb I\mathbb J\) is defined by: For all objects \(A\) and \(B\), \( \mathbb I\mathbb J(A, B)\) is the set of all products \(fg\), where \(f \in \mathbb I(C, B)\) and \(g \in \mathbb J(A, C)\), for some \(C\). Am ideal \(\mathbb I\) is called nilpotent if \(\mathbb I^n=0\) for some \(n\in\mathbb N\).\N\NFor a nilpotent ideal \(\mathbb I\) of an additive category \(\mathbb A\) and \(e \colon A \to A\) an idempotent in \(\mathbb A\), the author proves that \(e\) splits iff \(F(e)\) splits, where \(F \colon \mathbb A \to \mathbb A/\mathbb I\) is the canonical functor. As an application to triangulated categories, the author gives a short proof of [\textit{J. Le} and \textit{X.-W. Chen}, J. Algebra 310, No. 1, 452--457 (2007; Zbl 1112.18009), Proposition 2.3], which is a key observation to prove Karoubianness of a triangulated category with bounded t-structure.
scientific article; zbMATH DE number 7905688