Grothendieck groups in extriangulated categories

Zhu, Bin; Zhuang, Xiao

doi:10.1016/j.jalgebra.2021.01.029

Cited by 23 publications

(11 citation statements)

References 21 publications

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“…In particular, C is locally homologically finite, i.e., for any objects X, Y ∈ C , i∈Z e i (X, Y ) < ∞, where e i (X, Y ) := dim k E i I (X, Y ) = dim k E i II (X, Y ) for each i ∈ Z. The Grothendieck group of the extriangulated category C , denoted by K(C ), has been introduced in [12], which is similar to those of triangulated categories and exact categories.…”

Section: Comparisons Of Two Hall Algebrasmentioning

confidence: 99%

Hall algebras of extriangulated categories

Wang¹,

Wei²,

Zhang³

2021

Preprint

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Section: Comparisons Of Two Hall Algebrasmentioning

confidence: 99%

Hall algebras of extriangulated categories

Wang¹,

Wei²,

Zhang³

2021

Preprint

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“…In classical settings, the notion of a Grothendieck group has been used previously to give connections between certain subcategories of a category and subgroups of the ambient category's Grothendieck group; see [39], [49]. More recently, some of these results have been extended to the nexangulated cases; see [9], [27], [52]. In contrast, we will look at how the Grothendieck group theory changes as we tweak the n-exangulated structure on a fixed category and we will also be considering certain quotients of split Grothendieck groups.…”

Section: 2mentioning

confidence: 99%

Grothendieck groups of $d$-exangulated categories and a modified Caldero-Chapoton map

Jørgensen¹,

Shah²

2021

Preprint

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A strong connection between cluster algebras and representation theory was established by the cluster category. Cluster characters, like the original Caldero-Chapoton map, are maps from certain triangulated categories to cluster algebras and they have generated much interest. Holm and Jørgensen constructed a modified Caldero-Chapoton map from a sufficiently nice triangulated category to a commutative ring, which is a generalised frieze under some conditions. In their construction, a quotient K sp 0 (T )/M of a Grothendieck group of a cluster tilting subcategory T is used. In this article, we show that this quotient is the Grothendieck group of a certain extriangulated category, thereby exposing the significance of it and the relevance of extriangulated structures. We use this to define another modified Caldero-Chapoton map that recovers the one of Holm-Jørgensen.We prove our results in a higher homological context. Suppose S is a (d+2)-angulated category with subcategories X ⊆ T ⊆ S, where X is functorially finite and T is 2d-cluster tilting, satisfying some mild conditions. We show there is an isomorphism between the Grothendieck group K 0 (S, E X , s X ) of the category S, equipped with the d-exangulated structure induced by X , and the quotient K sp 0 (T )/N , where N is the higher analogue of M above. When X = T the isomorphism is induced by the higher index with respect to T introduced recently by Jørgensen. Thus, in the general case, we can understand the map taking an object in S to its K 0 -class in K 0 (S, E X , s X ) as a higher index with respect to the rigid subcategory X .

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“…γ is an extriangle in (C, E T , s T ) ; (1.3) see [33], [59], or Definition 3.1 and Remark 3.2. For C ∈ C, we denote its class in K 0 (C, E T , s T ) by [C] T .…”

Section: Introductionmentioning

confidence: 99%

The index with respect to a rigid subcategory of a triangulated category

Jørgensen¹,

Shah²

2022

Preprint

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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 in 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 surjectionThus, 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.

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Grothendieck groups in extriangulated categories

Cited by 23 publications

References 21 publications

Hall algebras of extriangulated categories

Hall algebras of extriangulated categories

Grothendieck groups of $d$-exangulated categories and a modified Caldero-Chapoton map

The index with respect to a rigid subcategory of a triangulated category

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