GROTHENDIECK GROUPS OF TRIANGULATED CATEGORIES VIA CLUSTER TILTING SUBCATEGORIES

Let k be a field, and let C be a k-linear, Hom-finite triangulated category with split idempotents. In this paper, we show that under suitable circumstances, the Grothendieck group of C, denoted by K-0(C), can be expressed as a quotient of the split Grothendieck group of a higher cluster tilting subcategory of C. The results we prove are higher versions of results on Grothendieck groups of triangulated categories by Xiao and Zhu and by Palu. Assume that n >= 2 is an integer; C has a Serre functor S and an n-cluster tilting subcategory T such that Ind T is locally bounded. Then, for every indecomposable M in T, there is an Auslander{Reiten (n + 2)-angle in T of the form S Sigma (n)(M) -> Tn-1 -> ... -> T-0 -> M and K-0(C) congruent to K-0(sp)(T) /<-[M] + (-1)(n) [S Sigma(-n) (M)] + Sigma(n-1)(i=0)(-1)(i)[T-i]vertical bar M is an element of Ind T >. Assume now that d is a positive integer and C has a d-cluster tilting subcategory S closed under d-suspension. Then, S is a so-called (d + 2)-angulated category whose Grothendieck group K-0(S) can be defined as a certain quotient of K-0(sp)(S). We will show K-0(C) congruent to K-0 (S). Moreover, assume that n = 2d, that all the above assumptions hold, and that T subset of S. Then our results can be combined to express K-0 (S) as a quotient of K-0(sp)(T).