City Research Online

Abstract

Quillen's Resolution Theorem in algebraic K-theory provides a powerful computational tool for calculating K-groups of exact categories. At the level of K0, this result goes back to Grothendieck. In this article, we first establish an extriangulated version of Grothendieck's Resolution Theorem. Second, we use this 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 N of a triangulated category C and we prove an additivity formula with error term. Our index recovers the index with respect to a contravariantly finite, rigid subcategory X defined by Jørgensen and the second author, as well as an isomorphism between K0spX) and the Grothendieck group of a relative extriangulated structure CRX on C when X is n-cluster tilting. In addition, we generalize and enhance some results of Fedele. Our perspective allows us to remove certain restrictions and simplify some arguments. Third, as another application of our Extriangulated Resolution Theorem, we show that if X is n-cluster tilting in an abelian category, then the index introduced by Reid gives an isomorphism K0(CRX)≅K0spX).

Publication Type:	Article
Additional Information:	This is an open access article distributed under the terms of the Creative Commons CC-BY license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Publisher Keywords:	Extriangulated category, Grothendieck group, Index, Localization, Relative theory, Resolution, Triangulated category
Subjects:	Q Science > QA Mathematics
Departments:	School of Science & Technology School of Science & Technology > Department of Mathematics
SWORD Depositor:	Symplectic Administrator

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