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GROTHENDIECK GROUPS OF TRIANGULATED CATEGORIES VIA CLUSTER TILTING SUBCATEGORIES

Published online by Cambridge University Press:  11 June 2020

FRANCESCA FEDELE*
Affiliation:
School of Mathematics, Statistics and Physics, Newcastle University, Newcastle upon TyneNE1 7RU, UK email [email protected]

Abstract

Let $k$ be a field, and let ${\mathcal{C}}$ be a $k$-linear, Hom-finite triangulated category with split idempotents. In this paper, we show that under suitable circumstances, the Grothendieck group of ${\mathcal{C}}$, denoted by $K_{0}({\mathcal{C}})$, can be expressed as a quotient of the split Grothendieck group of a higher cluster tilting subcategory of ${\mathcal{C}}$. The results we prove are higher versions of results on Grothendieck groups of triangulated categories by Xiao and Zhu and by Palu. Assume that $n\geqslant 2$ is an integer; ${\mathcal{C}}$ has a Serre functor $\mathbb{S}$ and an $n$-cluster tilting subcategory ${\mathcal{T}}$ such that $\operatorname{Ind}{\mathcal{T}}$ is locally bounded. Then, for every indecomposable $M$ in ${\mathcal{T}}$, there is an Auslander–Reiten $(n+2)$-angle in ${\mathcal{T}}$ of the form $\mathbb{S}\unicode[STIX]{x1D6F4}^{-n}(M)\rightarrow T_{n-1}\rightarrow \cdots \rightarrow T_{0}\rightarrow M$ and

$$\begin{eqnarray}K_{0}({\mathcal{C}})\cong K_{0}^{\text{sp}}({\mathcal{T}})\left/\left\langle -[M]+(-1)^{n}[\mathbb{S}\unicode[STIX]{x1D6F4}^{-n}(M)]+\left.\mathop{\sum }_{i=0}^{n-1}(-1)^{i}[T_{i}]\right|M\in \operatorname{Ind}{\mathcal{T}}\right\rangle .\right.\end{eqnarray}$$
Assume now that $d$ is a positive integer and ${\mathcal{C}}$ has a $d$-cluster tilting subcategory ${\mathcal{S}}$ closed under $d$-suspension. Then, ${\mathcal{S}}$ is a so-called $(d+2)$-angulated category whose Grothendieck group $K_{0}({\mathcal{S}})$ can be defined as a certain quotient of $K_{0}^{\text{sp}}({\mathcal{S}})$. We will show
$$\begin{eqnarray}K_{0}({\mathcal{C}})\cong K_{0}({\mathcal{S}}).\end{eqnarray}$$
Moreover, assume that $n=2d$, that all the above assumptions hold, and that ${\mathcal{T}}\subseteq {\mathcal{S}}$. Then our results can be combined to express $K_{0}({\mathcal{S}})$ as a quotient of $K_{0}^{\text{sp}}({\mathcal{T}})$.

Type
Article
Copyright
© 2020 Foundation Nagoya Mathematical Journal

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References

Auslander, M., Representation theory of Artin algebras II , Comm. Algebra 1 (1974), 269310.CrossRefGoogle Scholar
Auslander, M., Relations for Grothendieck groups of artin algebras , Proc. Amer. Math. Soc. 91 (1984), 336340.Google Scholar
Baur, K. and Marsh, R. J., A geometric description of m-cluster categories , Trans. Amer. Math. Soc. 360 (2008), 57895803.CrossRefGoogle Scholar
Bergh, P. T. and Thaule, M., The Grothendieck group of an n-angulated category , J. Pure Appl. Algebra 218 (2014), 354366.CrossRefGoogle Scholar
Butler, M. C. R., Grothendieck Groups and Almost Split Sequences, Lecture Notes in Mathematics 822 , Springer, Berlin, Heidelberg, 1981, 357368.Google Scholar
Gabriel, P. and Roiter, A. V., Representations of Finite-dimensional Algebras, Springer Science and Business Media 73 , 1997.CrossRefGoogle Scholar
Geiss, C., Keller, B. and Oppermann, S., n-angulated categories , J. Reine Angew. Math. 675 (2013), 101120.Google Scholar
Iyama, O., Cluster-tilting for higher Auslander algebras , Adv. Math. 226 (2011), 161.Google Scholar
Iyama, O. and Oppermann, S., Stable categories of higher preprojective algebras , Adv. Math. 244 (2013), 2368.CrossRefGoogle Scholar
Iyama, O. and Yoshino, Y., Mutation in triangulated categories and rigid Cohen–Macaulay modules , Invent. Math. 172 (2008), 117168.Google Scholar
Jasso, G. and Külshammer, J., Higher Nakayama algebras I: construction , Adv. Math. 351 (2019), 11391200.CrossRefGoogle Scholar
Jørgensen, P., Tropical friezes and the index in higher homological algebra, Math. Proc. Cambridge Philos. Soc. (to appear), doi:10.1017/S0305004120000031.CrossRefGoogle Scholar
Murphy, J., Derived equivalence classification of m-cluster tilted algebras of type A n , J. Algebra 323 (2010), 920965.CrossRefGoogle Scholar
Oppermann, S. and Thomas, H., Higher-dimensional cluster combinatorics and representation theory , J. Eur. Math. Soc. 14 (2012), 16791737.Google Scholar
Palu, Y., Grothendieck group and generalized mutation rule for 2-Calabi–Yau triangulated categories , J. Pure Appl. Algebra 213 (2009), 14381449.CrossRefGoogle Scholar
Pescod, D., Homological algebra and friezes, Ph.D. thesis, Newcastle University, 2017.Google Scholar
Thomas, H., Defining an m-cluster category , J. Algebra 318 (2007), 3746.CrossRefGoogle Scholar
Xiao, J. and Zhu, B., Relations for the Grothendieck groups of triangulated categories , J. Algebra 257 (2002), 3750.CrossRefGoogle Scholar
Zhou, P., Grothendieck groups and Auslander–Reiten $(d+2)$ -angles, preprint, 2019,arXiv:math.RT/1912.11397.Google Scholar