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Register a new userFrom recollements of abelian categories to recollements of triangulated categories
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In this paper, we first provide an explicit procedure to glue complete hereditary cotorsion pairs along the recollement $(mathcal{A},mathcal{C},mathcal{B})$ of abelian categories with enough projective and injective objects. As a consequence, we investigate how to establish recollements of triangulated categories from recollements of abelian categories by using the theory of exact model structures. Finally, we give applications to contraderived categories, projective stable derived categories and stable categories of Gorenstein injective modules over an upper triangular matrix ring.
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In this article, we prove that if $(mathcal A ,mathcal B,mathcal C)$ is a recollement of extriangulated categories, then torsion pairs in $mathcal A$ and $mathcal C$ can induce torsion pairs in $mathcal B$, and the converse holds under natural assumptions. Besides, we give mild conditions on a cluster tilting subcategory on the middle category of a recollement of extriangulated categories, for the corresponding abelian quotients to form a recollement of abelian categories.
In this paper, let $(mathcal{A},mathcal{B},mathcal{C})$ be a recollement of extriangulated categories. We introduce the global dimension and extension dimension of extriangulated categories, and give some upper bounds of global dimensions (resp. extension dimensions) of the categories involved in $(mathcal{A},mathcal{B},mathcal{C})$, which give a simultaneous generalization of some results in the recollement of abelian categories and triangulated categories.
Given a bounded-above cochain complex of modules over a ring, it is standard to replace it by a projective resolution, and it is classical that doing so can be very useful. Recently, a modified version of this was introduced in triangulated categories other than the derived category of a ring. A triangulated category is emph{approximable} if this modified procedure is possible. Not surprisingly this has proved a powerful tool. For example: the fact that the derived category of a quasi compact, separated scheme is approximable has led to major improvements on old theorems due to Bondal, Van den Bergh and Rouquier. In this article we prove that, under weak hypotheses, the recollement of two approximable triangulated categories is approximable. In particular, this shows many of the triangulated categories that arise in noncommutative algebraic geometry are approximable.
Let $mathcal{C}$ be a triangulated category. We first introduce the notion of balanced pairs in $mathcal{C}$, and then establish the bijective correspondence between balanced pairs and proper classes $xi$ with enough $xi$-projectives and enough $xi$-injectives. Assume that $xi:=xi_{mathcal{X}}=xi^{mathcal{Y}}$ is the proper class induced by a balanced pair $(mathcal{X},mathcal{Y})$. We prove that $(mathcal{C}, mathbb{E}_xi, mathfrak{s}_xi)$ is an extriangulated category. Moreover, it is proved that $(mathcal{C}, mathbb{E}_xi, mathfrak{s}_xi)$ is a triangulated category if and only if $mathcal{X}=mathcal{Y}=0$; and that $(mathcal{C}, mathbb{E}_xi, mathfrak{s}_xi)$ is an exact category if and only if $mathcal{X}=mathcal{Y}=mathcal{C}$. As an application, we produce a large variety of examples of extriangulated categories which are neither exact nor triangulated.
A notion of balanced pairs in an extriangulated category with a negative first extension is defined in this article. We prove that there exists a bijective correspondence between balanced pairs and proper classes $xi$ with enough $xi$-projectives and enough $xi$-injectives. It can be regarded as a simultaneous generalization of Fu-Hu-Zhang-Zhu and Wang-Li-Huang. Besides, we show that if $(mathcal A ,mathcal B,mathcal C)$ is a recollement of extriangulated categories, then balanced pairs in $mathcal B$ can induce balanced pairs in $mathcal A$ and $mathcal C$ under natural assumptions. As a application, this result gengralizes a result by Fu-Hu-Yao in abelian categories. Moreover, it highlights a new phenomena when it applied to triangulated categories.
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