TY - JOUR
T1 - The index with respect to a rigid subcategory of a triangulated category
AU - Jørgensen, Peter
AU - Shah, Amit
PY - 2024/2
Y1 - 2024/2
N2 - Palu defined the index with respect to a cluster tilting object in a suitable triangulated category, in order to better understand the Caldero–Chapoton map that exhibits the connection between cluster algebras and representation theory. We push this further by proposing an index with respect to a contravariantly finite, rigid subcategory, and we show this index behaves similarly to the classical index. Let C be a skeletally small triangulated category with split idempotents, which is thus an extriangulated category (C, E, s). Suppose X is a contravariantly finite, rigid subcategory of C. We define the index ind
X (C) of an object C ∈ C with respect to X as the K
0-class [C]
X in Grothendieck group K
0(C, E
X , s
X ) of the relative extriangulated category (C, E
X , s
X ). By analogy to the classical case, we give an additivity formula with error term for ind
X on triangles in C. In case X is contained in another suitable subcategory T of C, there is a surjection Q: K
0(C, E
T , s
T ) → K
0(C, E
X , s
X ). Thus, in order to describe K
0(C, E
X , s
X ), it suffices to determine K
0(C, E
T , s
T ) and Ker Q. We do this under certain assumptions.
AB - Palu defined the index with respect to a cluster tilting object in a suitable triangulated category, in order to better understand the Caldero–Chapoton map that exhibits the connection between cluster algebras and representation theory. We push this further by proposing an index with respect to a contravariantly finite, rigid subcategory, and we show this index behaves similarly to the classical index. Let C be a skeletally small triangulated category with split idempotents, which is thus an extriangulated category (C, E, s). Suppose X is a contravariantly finite, rigid subcategory of C. We define the index ind
X (C) of an object C ∈ C with respect to X as the K
0-class [C]
X in Grothendieck group K
0(C, E
X , s
X ) of the relative extriangulated category (C, E
X , s
X ). By analogy to the classical case, we give an additivity formula with error term for ind
X on triangles in C. In case X is contained in another suitable subcategory T of C, there is a surjection Q: K
0(C, E
T , s
T ) → K
0(C, E
X , s
X ). Thus, in order to describe K
0(C, E
X , s
X ), it suffices to determine K
0(C, E
T , s
T ) and Ker Q. We do this under certain assumptions.
UR - http://www.scopus.com/inward/record.url?scp=85186083508&partnerID=8YFLogxK
U2 - 10.1093/imrn/rnad130
DO - 10.1093/imrn/rnad130
M3 - Journal article
SN - 1073-7928
VL - 2024
SP - 3278
EP - 3309
JO - International Mathematics Research Notices
JF - International Mathematics Research Notices
IS - 4
ER -