2018/12/20 by Francesca Fedele, Fedele, Francesca · 2 citations
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1812.08493
openalex publication_date 2018/12/20 · openalex created_date 2022/08/01 · openalex updated_date 2026/07/28
Let k be a field and \C a k-linear, Hom-finite triangulated\ncategory with split idempotents. In this paper, we show that under suitable\ncircumstances, the Grothendieck group of \C, denoted\nK0(\C), can be expressed as a quotient of the split Grothendieck\ngroup of a higher-cluster tilting subcategory of \C.\n Assume that n\≥ 2 is an even integer, \C is n-Calabi Yau and\nhas an n-cluster tilting subcategory \T. Then, for every\nindecomposable M in \T, there is an Auslander-Reiten (n+2)-angle\nin \T of the form M\→ Tn-1\→\…\→\nT0\→ M and n K0(
mathcalC)
cong K0sp(
mathcalT)
big/
big
langle\n
sumi=0n-1(-1)i[Ti]
mid M
in
mathcalT
text indecomposable \n
big
rangle. Assume now that d is a positive integer and\n\C has a d-cluster tilting subcategory \S closed under\nd-suspension. Then \S is a so called (d+2)-angulated category\nwhose Grothendieck group K0(\S) can be defined as a certain\nquotient of K0sp(\S). We will show\n n K0(
mathcalC)
cong K0(
mathcalS).\n Moreover, assume that n=2d, that all the above assumptions\nhold, and that \T\⊆ \S. Then our results can be\ncombined to express K0(\S) as a quotient of\nK0sp(\T).\n