The Jordan-Hölder property and Grothendieck monoids of exact categories

Enomoto, Haruhisa

doi:10.1016/j.aim.2021.108167

Cited by 8 publications

(15 citation statements)

References 23 publications

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“…Then our M(E) (as an extriangulated category) coincides with the Grothendieck monoid of an exact category which is studied in [BG, Eno]. For example, [Eno,Theorem 4.12] implies that M(E) is a free monoid if and only if E satisfies the Jordan-Hölder-like property. In particular, if E is an abelian length category with n simple objects, then M(E) ∼ = N n holds.…”

Section: The Grothendieck Monoid Of An Extriangulated Categorymentioning

confidence: 97%

“…Proof. Since A is an abelian length category and the Jordan-Hölder theorem holds in A, we have that is a free commutative monoid with the basis {[S] ∈ M(A) | [S] ∈ sim A}, see [Eno,Corollary 4.10]. Thus we have an isomorphism…”

Section: Classification Of Intermediate Subcategoriesmentioning

confidence: 99%

“…• C is an exact R-category over a commutative noetherian ring R such that C(A, B) ∈ mod R holds for any A, B ∈ C. • C is a noetherian exact category, that is for any X ∈ C, the poset of admissible subobjects of X satisfies the ascending chain condition (see [Eno,Section 2] for this poset).…”

Section: Questionsmentioning

confidence: 99%

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Grothendieck monoids of extriangulated categories

Enomoto¹,

Saito²

2022

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We study the Grothendieck monoid (a monoid version of the Grothendieck group) of an extriangulated category, and give some results which are new even for abelian categories. First, we classify Serre subcategories and dense 2-out-of-3 subcategories using the Grothendieck monoid. Second, in good situations, we show that the Grothendieck monoid of the localization of an extriangulated category is isomorphic to the natural quotient monoid of the original Grothendieck monoid. This includes the cases of the Serre quotient of an abelian category and the Verdier quotient of a triangulated category. As a concrete example, we introduce an intermediate subcategory of the derived category of an abelian category, which lies between the abelian category and its one shift. We show that intermediate subcategories bijectively correspond to torsionfree classes in the abelian category, and then compute the Grothendieck monoid of an intermediate subcategory. Contents 24 5. Intermediate subcategories of the derived category 25 5.1. Classification of intermediate subcategories 25 5.2. Grothendieck monoid of an intermediate subcategory 27 5.3. Serre subcategories of an intermediate subcategory 29 6. Questions 34 6.1. Invertible elements in the Grothendieck monoid 34 6.2. Quasi-split extriangulated categories and c-closed subcategories 35 Appendix A. Basics on commutative monoids 37 References 41

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Section: The Grothendieck Monoid Of An Extriangulated Categorymentioning

confidence: 97%

Section: Classification Of Intermediate Subcategoriesmentioning

confidence: 99%

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Grothendieck monoids of extriangulated categories

Enomoto¹,

Saito²

2022

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“…It follows that each semistable object has a finite composition series whose factors are stable objects. However, we do not know in general that the set of these stable factors, nor the multiplicity with which each occurs, are well-defined -see [14] for a discussion of when a quasi-abelian category satisfies the Jordan-Hölder Theorem. Let Slice(C) denote the space of locally finite slicings on C. This has a metric…”

Section: Slicingsmentioning

confidence: 99%

Partial compactification of stability manifolds by massless semistable objects

Broomhead¹,

Pauksztello²,

Ploog³

et al. 2022

Preprint

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We introduce two partial compactifications of the space of Bridgeland stability conditions of a triangulated category. First we consider lax stability conditions where semistable objects are allowed to have mass zero but still have a phase. The subcategory of massless objects is thick and there is an induced classical stability on the quotient category. We study deformations of lax stability conditions. Second we consider the space arising by identifying lax stability conditions which are deformation-equivalent with fixed charge. This second space is stratified by stability spaces of Verdier quotients of the triangulated category by thick subcategories of massless objects. We illustrate our results through examples in which the Grothendieck group has rank 2. For these, our partial compactification can be explicitly described and related to the wall-and-chamber structure of the stability space. Contents 1. Introduction 2. Notation and preliminaries 3. Restriction, descent and glueing of slicings 4. Lax stability conditions and quotient categories 5. The space of lax stability conditions 6. Deforming lax stability conditions 7. The topology of the space of lax stability conditions 8. The space of quotient stability conditions 9. Codimension one strata 10. Finite type components 11. Closures of G-orbits 12. Two-dimensional stability spaces 13. Comparisons with other constructions 14. Open questions References Ginzburg algebra C = D b (Γ 2 A 2 ) Bound path algebra C = D b (Λ 1,2,0 ) Examples of quotient stability spaces Stab Q (C) * up to C action; see Figure 2.

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“…In studying lengths of objects in exact categories, Brüstle, Hassoun, Langford and Roy [3,Exam. 6.9] showed that an analogue of the classic Jordan-Hölder property can fail for an arbitrary exact category; see also Enomoto [8]. Motivated partly by this, Brüstle, Hassoun and Tattar [4] have recently considered additive categories with a mix of intrinsic and extrinsic structures.…”

Section: Introductionmentioning

confidence: 99%

Examples and non-examples of integral categories and the admissible intersection property

Hassoun¹,

Shah²,

Wegner³

2020

Preprint

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Integral categories form a sub-class of pre-abelian categories whose systematic study was initiated by Rump in 2001. In the first part of this article we determine whether several categories of topological and bornological vector spaces are integral. Moreover, we establish that the class of integral categories is not contained in the class of quasi-abelian categories. In the last part of the article we show that a category is quasi-abelian if and only if it has admissible intersections, in the sense considered recently by Brüstle, Hassoun and Tattar. This exhibits that a rich class of non-abelian categories having this property arises naturally in functional analysis.A pre-abelian category is an additive category in which every morphism has a kernel and a cokernel. Within the class of pre-abelian categories, one defines the following notions. Semi-abelian categories are the ones in which the canonical morphism between the coimage and image is always both monic and epic, but not necessarily an isomorphism as one would expect in an abelian category. Quasi-abelian categories are the ones in which

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The Jordan-Hölder property and Grothendieck monoids of exact categories

Cited by 8 publications

References 23 publications

Grothendieck monoids of extriangulated categories

Grothendieck monoids of extriangulated categories

Partial compactification of stability manifolds by massless semistable objects

Examples and non-examples of integral categories and the admissible intersection property

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