The Index With Respect to a Rigid Subcategory of a Triangulated Category

Abstract Palu defined the index with respect to a cluster tilting object in suitable triangulated category, order better understand Caldero–Chapoton map that exhibits connection between algebras and representation theory. We push this further by proposing an contravariantly finite, rigid subcategory, we show behaves similarly classical index. Let ${\mathcal{C}}$ be skeletally small category split idempotents, which is thus extriangulated $({\mathcal{C}},{\mathbb{E}},{\mathfrak{s}})$. Suppose ${\mathcal{X}}$ subcategory of ${\mathcal{C}}$. define $ {\operatorname{\textrm{ind}}}{_{{\mathcal{X}}}}(C)$ $C\in{\mathcal{C}}$ as {K}{_{0}}$-class $[C {]}{_{{\mathcal{X}}}}$ Grothendieck group {K}{_{0}}({\mathcal{C}}, {{\mathbb{E}}}{_{{\mathcal{X}}}}, {{\mathfrak{s}}}{_{{\mathcal{X}}}})$ relative $({\mathcal{C}}, {{\mathfrak{s}}}{_{{\mathcal{X}}}})$. By analogy case, give additivity formula error term for {\operatorname{\textrm{ind}}}{_{{\mathcal{X}}}}$ on triangles In case contained another ${\mathcal{T}}$ ${\mathcal{C}}$, there surjection $Q\colon{K}{_{0}}({\mathcal{C}}, {{\mathbb{E}}}{_{{\mathcal{T}}}}, {{\mathfrak{s}}}{_{{\mathcal{T}}}}) \twoheadrightarrow{K}{_{0}}({\mathcal{C}}, Thus, describe {{\mathfrak{s}}}{_{{\mathcal{X}}}})$, it suffices determine {{\mathfrak{s}}}{_{{\mathcal{T}}}})$ $\operatorname{Ker}\nolimits Q$. do under certain assumptions.