Triangulated subcategories of extensions, stable t-structures, and triangles of recollements

Jørgensen, Peter; Kato, Kiriko

doi:10.1016/j.jpaa.2015.05.029

Cited by 7 publications

(7 citation statements)

References 10 publications

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“…) is a stable t-structure in K(A[ǫ] n ). Furthermore, it is derived from [28] or [27,Lemma 1.6] that there exists a triangle equivalence K p (A[ǫ] n ) ≃ D(A[ǫ] n ).…”

Section: We Use Kmentioning

confidence: 99%

Higher differential objects in additive categories

Tang¹,

Huang²

2019

Preprint

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Given an additive category C and an integer n 2. We form a new additive category C[ǫ] n consisting of objects X in C equipped with an endomorphism ǫ X satisfying ǫ n X = 0. First, using the descriptions of projective and injective objects in C[ǫ] n , we not only establish a connection between Gorenstein flat modules over a ring R and R[t]/(t n ), but also prove that an Artinian algebra R satisfies some homological conjectures if and only if so does R[t]/(t n ). Then we show that the corresponding homotopy category K(C[ǫ] n ) is a triangulated category when C is an idempotent complete exact category. Moreover, under some conditions for an abelian category A, the natural quotient functor Q from K(A[ǫ] n ) to the derived category D(A[ǫ] n ) produces a recollement of triangulated categories. Finally, we prove that if A is an Ab4-category with a compact projective generator, then D(A[ǫ] n ) is a compactly generated triangulated category.

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“…) is a stable t-structure in K(A[ǫ] n ). Furthermore, it is derived from [28] or [27,Lemma 1.6] that there exists a triangle equivalence K p (A[ǫ] n ) ≃ D(A[ǫ] n ).…”

Section: We Use Kmentioning

confidence: 99%

Higher differential objects in additive categories

Tang¹,

Huang²

2019

Preprint

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“…One of our main tools is the following result due to Jørgensen and Kato [10]. For a triangulated category T and two subcategories X and Y of T let X * Y be the full subcategory of T having as objects those objects E of T such that there is a distinguished triangle X −→ E −→ Y −→ X [1] with X ∈ X and Y ∈ Y.…”

Section: Given a Short Exact Sequence Of Triangulated Categoriesmentioning

confidence: 99%

“…Theorem 1.3. [10] Let T be a triangulated category. Let X , Y be two thick subcategories of T , and denote…”

Section: Given a Short Exact Sequence Of Triangulated Categoriesmentioning

confidence: 99%

“…We do not show the equivalences of the categories directly, but rather prove the existence of stable t-structures of the quotients as introduced by [9] which imply then isomorphisms of quotients. Simplifying the work, a result of Jørgensen and Kato [10] reduces the verification of stable t-structures (X /(X ∩ Y), Y/(X ∩ Y)) in T /(X ∩ Y) to a verification of the ambient category T being the full subcategory of T of middle terms of distinguished triangles X * Y with precisely the end terms in X respectively Y. Quotients involving exact complexes are more subtle than those given by other boundedness conditions. The appropriate additional hypothesis to be able to deal with this situation is that of an abelian category, instead of just an additive subcategory of an abelian category.…”

Section: Introductionmentioning

confidence: 99%

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Verdier quotients of homotopy categories

Zhou¹,

Zimmermann²

2017

Preprint

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We study Verdier quotients of diverse homotopy categories of a full additive subcategory E of an abelian category. In particular, we consider the categories K x,y (E) for x ∈ {∞, +, −, b}, and y ∈ {∅, b, +, −, ∞} the homotopy categories of left, right, unbounded complexes with homology being 0, bounded, left or right bounded, or unbounded. Inclusion of these categories give a partially ordered set, and we study localisation sequences or recollement diagrams between the Verdier quotients, and prove that many pairs of quotient categories identify.

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“…Lemma 1.6. [21] Let T be a triangulated category and U, V be full triangulated subcategories. Then the following conditions are equivalent.…”

Section: Preliminariesmentioning

confidence: 99%

Derived categories of $N$-complexes

Iyama,

Kato,

Miyachi

2013

Preprint

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We study the homotopy category K N (B) of N -complexes of an additive category B and the derived category D N (A) of an abelian category A. First we show that both K N (B) and D N (A) have natural structures of triangulated categories. Then we establish a theory of projective (resp., injective) resolutions and derived functors. Finally, under some conditions of an abelian category A, we show that D N (A) is triangle equivalent to the ordinary derived category D(Mor N−2 (A)) where Mor N−2 (A) is the category of sequential N − 2 morphisms of A.

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Triangulated subcategories of extensions, stable t-structures, and triangles of recollements

Cited by 7 publications

References 10 publications

Higher differential objects in additive categories

Higher differential objects in additive categories

Verdier quotients of homotopy categories

Derived categories of $N$-complexes

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