The index with respect to a contravariantly finite subcategory

Francesca Fedele, Peter Jorgensen, Amit Shah

Published 2024-01-17Version 1

The index in triangulated categories plays a key role in the categorification of cluster algebras; specifically, it categorifies the notion of $g$-vectors. For an object $C$ in a triangulated category $\mathcal{C}$, the index of $C$ was originally defined as $[X_1] - [X_0]$ when $X_1 \to X_0 \to C \to \Sigma X_1$ is a triangle with the $X_i$ in a cluster tilting subcategory $\mathcal{X}$ of $\mathcal{C}$. Using the theory of extriangulated categories, the index was later generalised to the case where $\mathcal{X}$ is a contravariantly finite subcategory which is rigid, that is, satisfies $\mathcal{C}( \mathcal{X},\Sigma \mathcal{X} ) = 0$. This paper generalises further by dropping the assumption that $\mathcal{X}$ is rigid, vastly increasing the potential choice of $\mathcal{X}$. We show that this version of the index still has the key property of being additive on triangles up to an error term.