The index in $d$-exact categories

Abstract

Starting from its original definition in module categories with respect to projective modules, the index has played an important role in various aspects of homological algebra, categorification of cluster algebras and $K$-theory. In the last few years, the notion of index has been generalised to several different contexts in (higher) homological algebra, typically with respect to a (higher) cluster-tilting subcategory $\mathcal{X}$ of the relevant ambient category $\mathcal{C}$. The recent tools of extriangulated and higher-exangulated categories have permitted some conditions on the subcategory $\mathcal{X}$ to be relaxed. In this paper, we introduce the index with respect to a generating, contravariantly finite subcategory of a $d$-exact category that has $d$-kernels. We show that our index has the important property of being additive on $d$-exact sequences up to an error term.