A resolution theorem for extriangulated categories with applications to the index

Yasuaki OgawaAmit Shah

Yasuaki OgawaAmit Shah

Nov 2023

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摘要原文

Quillen's Resolution Theorem in algebraic $K$-theory provides a powerful computational tool for calculating $K$-groups of exact categories. At the level of $K_0$, this result goes back to Grothendieck. In this article, we first establish an extriangulated version of Grothendieck's Resolution Theorem. Second, we use our Extriangulated Resolution Theorem to gain new insight into the index theory of triangulated categories. Indeed, we propose an index with respect to an extension-closed subcategory $\mathscr{N}$ of a triangulated category $\mathscr{C}$ and we prove an additivity formula with error term. Our index recovers the index with respect to a contravariantly finite, rigid subcategory $\mathscr{X}$ defined by J{\o}rgensen and the second author, as well as an isomorphism between $K_0^{\mathsf{sp}}(\mathscr{X})$ and the Grothendieck group of a relative extriangulated structure on $\mathscr{C}$ when $\mathscr{X}$ is cluster tilting. In addition, we generalize and enhance some results of Fedele. Our perspective allows us to remove certain restrictions and simplify some arguments.