A resolution theorem for extriangulated categories with applications to the index

TY - JOUR

T1 - A resolution theorem for extriangulated categories with applications to the index

AU - Ogawa, Yasuaki

AU - Shah, Amit

PY - 2024/11/15

Y1 - 2024/11/15

N2 - 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 K0sp(X) 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)≅K0sp(X).

AB - 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 K0sp(X) 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)≅K0sp(X).

KW - Extriangulated category

KW - Grothendieck group

KW - Index

KW - Localization

KW - Relative theory

KW - Resolution

KW - Triangulated category

UR - http://www.scopus.com/inward/record.url?scp=85197253747&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2024.05.041

DO - 10.1016/j.jalgebra.2024.05.041

M3 - Journal article

SN - 0021-8693

VL - 658

SP - 450

EP - 485

JO - Journal of Algebra

JF - Journal of Algebra

ER -