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