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Bin Zhu - One of the best experts on this subject based on the ideXlab platform.
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Tilting Subcategories in extriangulated categories
Frontiers of Mathematics in China, 2020Co-Authors: Bin Zhu, Xiao ZhuangAbstract:Extriangulated category was introduced by H. Nakaoka and Y. Palu to give a unification of properties in exact categories and triangulated categories. A notion of tilting (resp., cotilting) Subcategories in an extriangulated category is defined in this paper. We give a Bazzoni characterization of tilting (resp., cotilting) Subcategories and obtain an Auslander-Reiten correspondence between tilting (resp., cotilting) Subcategories and coresolving covariantly (resp., resolving contravariantly) finite subcatgories which are closed under direct summands and satisfy some cogenerating (resp., generating) conditions. Applications of the results are given: we show that tilting (resp., cotilting) Subcategories defined here unify many previous works about tilting modules (Subcategories) in module categories of Artin algebras and in abelian categories admitting a cotorsion triples; we also show that the results work for the triangulated categories with a proper class of triangles introduced by A. Beligiannis.
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Two-term relative cluster tilting Subcategories, τ-tilting modules and silting Subcategories
Journal of Pure and Applied Algebra, 2020Co-Authors: Panyue Zhou, Bin ZhuAbstract:Abstract Let C be a triangulated category with shift functor [1] and R a rigid subcategory of C . We introduce the notions of two-term R [ 1 ] -rigid Subcategories, two-term (weak) R [ 1 ] -cluster tilting Subcategories and two-term maximal R [ 1 ] -rigid Subcategories. Our main result shows that there exists a bijection between the set of two-term R [ 1 ] -rigid Subcategories of C and the set of τ-rigid Subcategories of mod R , which induces a one-to-one correspondence between the set of two-term weak R [ 1 ] -cluster tilting Subcategories of C and the set of support τ-tilting Subcategories of mod R . This generalizes the main results in [15] where R is a cluster tilting subcategory. When R is a silting subcategory, we prove that the two-term weak R [ 1 ] -cluster tilting Subcategories are precisely two-term silting Subcategories in [9] . Thus the bijection above induces the bijection given by Iyama-Jorgensen-Yang in [9] .
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Triangulated categories with cluster-tilting Subcategories
Pacific Journal of Mathematics, 2019Co-Authors: Wuzhong Yang, Panyue Zhou, Bin ZhuAbstract:Let $\C$ be a triangulated category with a cluster tilting subcategory $\T$. We introduce the notion of $\T[1]$-cluster tilting Subcategories (also called ghost cluster tilting Subcategories) of $\C$, which are a generalization of cluster tilting Subcategories. We first develop a basic theory on ghost cluster tilting Subcategories. Secondly, we study links between ghost cluster tilting theory and $\tau$-tilting theory: Inspired by the work of Iyama, Jorgensen and Yang \cite{ijy}, we introduce the notion of $\tau$-tilting Subcategories and tilting Subcategories of $\mod\T$. We show that there exists a bijection between weak $\T[1]$-cluster tilting Subcategories of $\C$ and support $\tau$-tilting Subcategories of $\mod\T$. Moreover, we figure out the Subcategories of $\mod\T$ which correspond to cluster tilting Subcategories of $\C$. This generalizes and improves several results by Adachi-Iyama-Reiten \cite{AIR}, Beligiannis \cite{Be2}, and Yang-Zhu \cite{YZ}. Finally, we prove that the definition of ghost cluster tilting objects is equivalent to the definition of relative cluster tilting objects introduced by the first and the third author in \cite{YZ}.
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Tilting Subcategories in extriangulated categories
arXiv: Representation Theory, 2019Co-Authors: Bin Zhu, Xiao ZhuangAbstract:Extriangulated category was introduced by Nakaoka and Palu to give a unification of properties in exact categories and triangulated categories. A notion of tilting (or cotilting) Subcategories in an extriangulated category is defined in this paper. We give a Bazzoni characterization of tilting (or cotilting) Subcategories and obtain an Auslander-Reiten correspondence between tilting (cotilting) Subcategories and coresolving covariantly (resolving contravariantly, resp.) finite subcatgories which are closed under direct summands and satisfies some cogenerating (generating, resp.) conditons. Applications of the results are given: we show that tilting (cotilting) Subcategories defined here unify many previous works about tilting theory in module categories of Artin algebras and abelian categories admitting a cotorsion triples; we also show that the results work for triangulated categories with a proper class of triangles introduced by Beligiannis.
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Two-term relative cluster tilting Subcategories, $\tau-$tilting modules and silting Subcategories
arXiv: Representation Theory, 2018Co-Authors: Panyue Zhou, Bin ZhuAbstract:Let $\mathcal{C}$ be a triangulated category with shift functor $[1]$ and $\mathcal{R}$ a rigid subcategory of $\mathcal{C}$. We introduce the notions of two-term $\mathcal{R}[1]$-rigid Subcategories, two-term (weak) $\mathcal{R}[1]$-cluster tilting Subcategories and two-term maximal $\mathcal{R}[1]$-rigid Subcategories, and discuss relationship between them. Our main result shows that there exists a bijection between the set of two-term $\mathcal{R}[1]$-rigid Subcategories of $\mathcal{C}$ and the set of $\tau$-rigid Subcategories of $\mod\mathcal{R}$, which induces a one-to-one correspondence between the set of two-term weak $\mathcal{R}[1]$-cluster tilting Subcategories of $\mathcal{C}$ and the set of support $\tau$-tilting Subcategories of $\mod\mathcal{R}$. This generalizes the main results in \cite{YZZ} where $\mathcal{R}$ is a cluster tilting subcategory. When $\mathcal{R}$ is a silting subcategory, we prove that the two-term weak $\mathcal{R}[1]$-cluster tilting Subcategories are precisely two-term silting Subcategories in \cite{IJY}. Thus the bijection above induces the bijection given by Iyama-J{\o}rgensen-Yang in \cite{IJY}
Zhong Sheng Sun - One of the best experts on this subject based on the ideXlab platform.
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a comparative study of the genetic components of three Subcategories of autism spectrum disorder
Molecular Psychiatry, 2019Co-Authors: Kun Zhang, Leisheng Shi, Yi Zhang, Tingting Zhao, Lin Wang, Kun Xia, Chunyu Liu, Zhong Sheng SunAbstract:The fifth edition of the Diagnostic and Statistical Manual of Mental Disorders (DSM-5) controversially combined previously distinct Subcategories of autism spectrum disorder (ASD) into a single diagnostic category. However, genetic convergences and divergences between different ASD Subcategories are unclear. By retrieving 1725 exonic de novo mutations (DNMs) from 1628 subjects with autistic disorder (AD), 1873 from 1564 subjects with pervasive developmental disorder not otherwise specified (PDD-NOS), 276 from 247 subjects with Asperger's syndrome (AS), and 2077 from 2299 controls, we found that rates of putative functional DNMs (loss-of-function, predicted deleterious missense, and frameshift) in all three Subcategories were significantly higher than those in control. We then investigated the convergences and divergences of the three ASD Subcategories based on four genetic aspects: whether any two ASD Subcategories (1) shared significantly more genes with functional DNMs, (2) exhibited similar spatio-temporal expression patterns, (3) shared significantly more candidate genes, and (4) shared some ASD-associated functional pathways. It is revealed that AD and PDD-NOS were broadly convergent in terms of all four genetic aspects, suggesting these two ASD Subcategories may be genetically combined. AS was divergent to AD and PDD-NOS for aspects of functional DNMs and expression patterns, whereas AS and AD/PDD-NOS were convergent for aspects of candidate genes and functional pathways. Our results indicated that the three ASD Subcategories present more genetic convergences than divergences, favouring DSM-5's new classification. This study suggests that specifically defined genotypes and their corresponding phenotypes should be integrated analyzed for precise diagnosis of complex disorders, such as ASD.
Ryo Takahashi - One of the best experts on this subject based on the ideXlab platform.
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classifying resolving Subcategories over a cohen macaulay local ring
arXiv: Commutative Algebra, 2012Co-Authors: Ryo TakahashiAbstract:Let R be a Cohen-Macaulay local ring. Denote by mod R the category of finitely generated R-modules. In this paper, we consider the classification problem of resolving Subcategories of mod R in terms of specialization-closed subsets of Spec R. We give a classification of the resolving Subcategories closed under tensor products and transposes. Under restrictive hypotheses, we also give better classification results.
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classification of resolving Subcategories and grade consistent functions
arXiv: Commutative Algebra, 2012Co-Authors: Hailong Dao, Ryo TakahashiAbstract:We classify certain resolving Subcategories of finitely generated modules over a commutative noetherian ring R by using integer-valued functions on Spec R. As an application we give a complete classification of resolving Subcategories when R is a locally hypersurface ring. Our results also recover a "missing theorem" by Auslander.
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classifying thick Subcategories of the stable category of cohen macaulay modules
Advances in Mathematics, 2010Co-Authors: Ryo TakahashiAbstract:Abstract Various classification theorems of thick Subcategories of a triangulated category have been obtained in many areas of mathematics. In this paper, as a higher-dimensional version of the classification theorem of thick Subcategories of the stable category of finitely generated representations of a finite p-group due to Benson, Carlson and Rickard, we consider classifying thick Subcategories of the stable category of Cohen–Macaulay modules over a Gorenstein local ring. The main result of this paper yields a complete classification of the thick Subcategories of the stable category of Cohen–Macaulay modules over a local hypersurface in terms of specialization-closed subsets of the prime ideal spectrum of the ring which are contained in its singular locus.
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Contravariantly finite resolving Subcategories over commutative rings
arXiv: Commutative Algebra, 2010Co-Authors: Ryo TakahashiAbstract:Contravariantly finite resolving Subcategories of the category of finitely generated modules have been playing an important role in the representation theory of algebras. In this paper we study contravariantly finite resolving Subcategories over commutative rings. The main purpose of this paper is to classify contravariantly finite resolving Subcategories over a henselian Gorenstein local ring; in fact there exist only three ones. Our method to obtain this classification also recovers as a by-product the theorem of Christensen, Piepmeyer, Striuli and Takahashi concerning the relationship between the contravariant finiteness of the full subcategory of totally reflexive modules and the Gorenstein property of the base ring.
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classifying Subcategories of modules over a commutative noetherian ring
Journal of The London Mathematical Society-second Series, 2008Co-Authors: Ryo TakahashiAbstract:Let R be a quotient ring of a commutative coherent regular ring by a finitely generated ideal. Hovey gave a bijection between the set of coherent Subcategories of the category of finitely presented R-modules and the set of thick Subcategories of the derived category of perfect R-complexes. Using this bijection, he proved that every coherent subcategory of finitely presented R-modules is a Serre subcategory. In this paper, it is proved that this holds whenever R is a commutative noetherian ring. This paper also yields a module version of the bijection between the set of localizing Subcategories of the derived category of R-modules and the set of subsets of Spec R which was given by Neeman
Panyue Zhou - One of the best experts on this subject based on the ideXlab platform.
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Relative rigid Subcategories and $\tau$-tilting theory
arXiv: Representation Theory, 2020Co-Authors: Yu Liu, Panyue ZhouAbstract:Let $\mathcal B$ be an extriangulated category with enough projectives $\mathcal P$ and enough injectives $\mathcal I$, and let $\mathcal R$ be a contravariantly finite rigid subcategory of $\mathcal B$ which contains $\mathcal P$. We have an abelian quotient category $\mathcal H/\mathcal R\subseteq \mathcal B/\mathcal R$ which is equivalent ${\rm mod}(\mathcal R/\mathcal P)$. In this article, we find a one-to-one correspondence between support $\tau$-tilting (resp. $\tau$-rigid) Subcategories of $\mathcal H/\mathcal R$ and maximal relative rigid (resp. relative rigid) Subcategories of $\mathcal H$, and show that support tilting Subcategories in $\mathcal H/\mathcal R$ is a special kind of support $\tau$-tilting Subcategories. We also study the relation between tilting Subcategories of $\mathcal B/\mathcal R$ and cluster tilting Subcategories of $\mathcal B$ when $\mathcal R$ is cluster tilting.
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Two-term relative cluster tilting Subcategories, τ-tilting modules and silting Subcategories
Journal of Pure and Applied Algebra, 2020Co-Authors: Panyue Zhou, Bin ZhuAbstract:Abstract Let C be a triangulated category with shift functor [1] and R a rigid subcategory of C . We introduce the notions of two-term R [ 1 ] -rigid Subcategories, two-term (weak) R [ 1 ] -cluster tilting Subcategories and two-term maximal R [ 1 ] -rigid Subcategories. Our main result shows that there exists a bijection between the set of two-term R [ 1 ] -rigid Subcategories of C and the set of τ-rigid Subcategories of mod R , which induces a one-to-one correspondence between the set of two-term weak R [ 1 ] -cluster tilting Subcategories of C and the set of support τ-tilting Subcategories of mod R . This generalizes the main results in [15] where R is a cluster tilting subcategory. When R is a silting subcategory, we prove that the two-term weak R [ 1 ] -cluster tilting Subcategories are precisely two-term silting Subcategories in [9] . Thus the bijection above induces the bijection given by Iyama-Jorgensen-Yang in [9] .
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Triangulated categories with cluster-tilting Subcategories
Pacific Journal of Mathematics, 2019Co-Authors: Wuzhong Yang, Panyue Zhou, Bin ZhuAbstract:Let $\C$ be a triangulated category with a cluster tilting subcategory $\T$. We introduce the notion of $\T[1]$-cluster tilting Subcategories (also called ghost cluster tilting Subcategories) of $\C$, which are a generalization of cluster tilting Subcategories. We first develop a basic theory on ghost cluster tilting Subcategories. Secondly, we study links between ghost cluster tilting theory and $\tau$-tilting theory: Inspired by the work of Iyama, Jorgensen and Yang \cite{ijy}, we introduce the notion of $\tau$-tilting Subcategories and tilting Subcategories of $\mod\T$. We show that there exists a bijection between weak $\T[1]$-cluster tilting Subcategories of $\C$ and support $\tau$-tilting Subcategories of $\mod\T$. Moreover, we figure out the Subcategories of $\mod\T$ which correspond to cluster tilting Subcategories of $\C$. This generalizes and improves several results by Adachi-Iyama-Reiten \cite{AIR}, Beligiannis \cite{Be2}, and Yang-Zhu \cite{YZ}. Finally, we prove that the definition of ghost cluster tilting objects is equivalent to the definition of relative cluster tilting objects introduced by the first and the third author in \cite{YZ}.
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Two-term relative cluster tilting Subcategories, $\tau-$tilting modules and silting Subcategories
arXiv: Representation Theory, 2018Co-Authors: Panyue Zhou, Bin ZhuAbstract:Let $\mathcal{C}$ be a triangulated category with shift functor $[1]$ and $\mathcal{R}$ a rigid subcategory of $\mathcal{C}$. We introduce the notions of two-term $\mathcal{R}[1]$-rigid Subcategories, two-term (weak) $\mathcal{R}[1]$-cluster tilting Subcategories and two-term maximal $\mathcal{R}[1]$-rigid Subcategories, and discuss relationship between them. Our main result shows that there exists a bijection between the set of two-term $\mathcal{R}[1]$-rigid Subcategories of $\mathcal{C}$ and the set of $\tau$-rigid Subcategories of $\mod\mathcal{R}$, which induces a one-to-one correspondence between the set of two-term weak $\mathcal{R}[1]$-cluster tilting Subcategories of $\mathcal{C}$ and the set of support $\tau$-tilting Subcategories of $\mod\mathcal{R}$. This generalizes the main results in \cite{YZZ} where $\mathcal{R}$ is a cluster tilting subcategory. When $\mathcal{R}$ is a silting subcategory, we prove that the two-term weak $\mathcal{R}[1]$-cluster tilting Subcategories are precisely two-term silting Subcategories in \cite{IJY}. Thus the bijection above induces the bijection given by Iyama-J{\o}rgensen-Yang in \cite{IJY}
Chunyu Liu - One of the best experts on this subject based on the ideXlab platform.
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a comparative study of the genetic components of three Subcategories of autism spectrum disorder
Molecular Psychiatry, 2019Co-Authors: Kun Zhang, Leisheng Shi, Yi Zhang, Tingting Zhao, Lin Wang, Kun Xia, Chunyu Liu, Zhong Sheng SunAbstract:The fifth edition of the Diagnostic and Statistical Manual of Mental Disorders (DSM-5) controversially combined previously distinct Subcategories of autism spectrum disorder (ASD) into a single diagnostic category. However, genetic convergences and divergences between different ASD Subcategories are unclear. By retrieving 1725 exonic de novo mutations (DNMs) from 1628 subjects with autistic disorder (AD), 1873 from 1564 subjects with pervasive developmental disorder not otherwise specified (PDD-NOS), 276 from 247 subjects with Asperger's syndrome (AS), and 2077 from 2299 controls, we found that rates of putative functional DNMs (loss-of-function, predicted deleterious missense, and frameshift) in all three Subcategories were significantly higher than those in control. We then investigated the convergences and divergences of the three ASD Subcategories based on four genetic aspects: whether any two ASD Subcategories (1) shared significantly more genes with functional DNMs, (2) exhibited similar spatio-temporal expression patterns, (3) shared significantly more candidate genes, and (4) shared some ASD-associated functional pathways. It is revealed that AD and PDD-NOS were broadly convergent in terms of all four genetic aspects, suggesting these two ASD Subcategories may be genetically combined. AS was divergent to AD and PDD-NOS for aspects of functional DNMs and expression patterns, whereas AS and AD/PDD-NOS were convergent for aspects of candidate genes and functional pathways. Our results indicated that the three ASD Subcategories present more genetic convergences than divergences, favouring DSM-5's new classification. This study suggests that specifically defined genotypes and their corresponding phenotypes should be integrated analyzed for precise diagnosis of complex disorders, such as ASD.
