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Register a new userAbelian hearts of twin cotorsion pairs on extriangulated categories
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It was shown recently that the heart of a twin cotorsion pair on an extriangulated category is semi-abelian. In this article, we consider a special kind of hearts of twin cotorsion pairs induced by $d$-cluster tilting subcategories in extriangulated categories. We give a necessary and sufficient condition for such hearts to be abelian. In particular, we also can see that such hearts are hereditary. As an application, this generalizes the work by Liu in an exact case, thereby providing new insights in a triangulated case.
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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.
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.
We introduce pre-silting and silting subcategories in extriangulated categories and generalize the silting theory in triangulated categories. We prove that the silting reduction $mathcal B/({rm thick}mathcal W)$ of an extriangulated category $mathcal B$ with respect to a pre-silting subcategory $mathcal W$ can be realized as a certain subfactor category of $mathcal B$. This generalizes the result by Iyama-Yang. In particular, for a Gorenstein algebra, we get the relative version of the description of the singularity category due to Happel and Chen-Zhang by this reduction.
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.
Balanced pairs appear naturally in the realm of Relative Homological Algebra associated to the balance of right derived functors of the $mathsf{Hom}$ functor. A natural source to get such pairs is by means of cotorsion triplets. In this paper we study the connection between balanced pairs and cotorsion triplets by using recent quiver representation techniques. In doing so, we find a new characterization of abelian categories having enough projectives and injectives in terms of the existence of complete hereditary cotorsion triplets. We also give a short proof of the lack of balance for derived functors of $mathsf{Hom}$ computed by using flat resolutions which extends the one showed by Enochs in the commutative case.
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