arXiv subjects

Tiwei Zhao

Publications and source records attributed to Tiwei Zhao.

At least 19 recordsLinked to original sources

arXiv2026

Semi-orthogonal and derived decompositions for gentle algebras

We study semi-orthogonal decompositions of perfect derived categories of gentle algebras via marked ribbon surfaces. We characterize such decompositions in terms of suitable disjoint union decompositions of full formal arc systems, and relate this description to good cuts of the corresponding surfaces. For gentle algebras, rotations of curves induce fully faithful functors from extension-closed subcategories of module categories to the components of the associated semi-orthogonal decompositions. Under additional Abelian and extension-comparison conditions, these constructions give derived decompositions of the module categories.

math.CT↗

arXiv2026

Tilting realizations of derived-equivalent matrix centralizer algebras

Let $A$ be the centralizer algebra of a matrix over an arbitrary field. We solve the fixed-source realization problem for matrix centralizers by proving that the matrix centralizer algebras derived equivalent to $A$ are precisely the opposite endomorphism algebras of tilting modules over $A$. We classify the basic tilting modules and determine their opposite endomorphism algebras. The tilting poset is a product of right weak orders on symmetric groups, with one factor for each primary block and degree equal to the number of distinct exponents in that block. Together with the center, this poset recovers the multiset of these numbers across all primary blocks, although it does not canonically match them with the local center factors. For each primary block, the target algebras are obtained by permuting the successive gaps between exponents, and their isomorphism classes are determined by the stabilizer of the gap word. Consequently, the quotient of the labeled mutation graph by target-algebra isomorphism is a Schreier multigraph. We also characterize when the weak-order orientation descends to its nonloop edges.

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arXiv2026

Support $\tau$-tilting posets and Hochschild reconstruction for matrix centralizer algebras

Let $R$ be a field and $A$ the endomorphism algebra of a finite direct sum of cyclic modules over a finite-dimensional commutative local principal ideal $R$-algebra. We construct a central quotient showing that the support $\tau$-tilting poset of $A$ is isomorphic to the poset of the symmetric group with the weak order. We show that the center of $A$ and degree-zero Hochschild homology, viewed as a module over the center, determine the truncated local algebra and the multiset of successive length gaps. For centralizer matrix algebras, the support $\tau$-tilting poset determines the multiset of distinct-exponent counts of the primary blocks. The corresponding algebra--module pair also recovers their local algebras and gap multisets. Combining this reconstruction with the known derived equivalence classification, we characterize derived equivalence by isomorphism of these algebra-module pairs. We apply the results to Morita reconstruction in the string and gentle classes.

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arXiv2023

On the generalizations of global dimensions and singularity categories

For each $n\in\mathbb{N}\cup\{\infty\}$, we introduce the notion of $n$-singularity category $\mathbf{D}_{n{\rm-}sg}(R)$ of a given ring $R$, which can be seen as a generalization of the classical singularity category. Moreover, the $n$-global dimension $n$-gldim$(R)$ of $R$ is investigated. We show that $\mathbf{D}_{n{\rm-}sg}(R)=0$ if and only if $n$-gldim$(R)$ is finite. Furthermore, we characterize the vanishing property of $n$-singularity categories in terms of recollements.

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arXiv2023

Little finitistic dimensions and generalized derived categories

In this paper, we introduced a generalization of the derived category, which is called the $n$-derived category and denoted by $\D_{n}(R)$, of a given ring $R$ for each $n\in\mathbb{N}\cup\{\infty\}$. The $n$-derived category of a ring is proved to be very closely connected with its left little finitistic dimension. We also introduce and investigate the notions of $n$-exact sequences, $n$-projective (resp., $n$-injective) modules and $n$-exact complexes. In particular, we characterize the left little finitistic dimensions in terms of all above notions. Finally, we build a connection of the classical derived categories and $n$-derived categories.

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arXiv2021

Resolving subcategories and dimensions in recollements of extriangulated categories

Recently, Wang, Wei and Zhang introduced the notion of recollements of extriangulated categories. In this paper, let $(\mathcal{A},\mathcal{B},\mathcal{C})$ be a recollement of extriangulated categories. We provide some methods to construct resolving subcategories in $(\mathcal{A},\mathcal{B},\mathcal{C})$. As applications of the Auslander-Reiten correspondence, we get the gluing of cotilting modules in a recollement of module categories for artin algebras. We also give some bounds of resolution dimensions of the categories involved in $(\mathcal{A},\mathcal{B},\mathcal{C})$ with respect to resolving subcategories, which generalize some known results.

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arXiv2020

Ideal approximation in $n$-angulated categories

In this paper, we study ideal approximation theory associated to almost $n$-exact structures in extension closed subcategories of $n$-angulated categories. For $n=3$, an $n$-angulated category is nothing but a classical triangulated category. Moreover, since every exact category can be embedded as an extension closed subcategory of a triangulated category, therefore, our approach extends the recent ideal approximations theories developed by Fu, Herzog et al. for exact categories and by Breaz and Modoi for triangulated categories.

math.RA↗

arXiv2020

Tilting pairs in extriangulated categories

Extriangulated categories were introduced by Nakaoka and Palu to give a unification of properties in exact categories and extension-closed subcategories of triangulated categories. A notion of tilting pairs in an extriangulated category is introduced in this paper. We give a Bazzoni characterization of tilting pairs in this setting. We also obtain Auslander-Reiten correspondence of tilting pairs which classifies finite $\mathcal{C}$-tilting subcategories for a certain self-orthogonal subcategory $\mathcal{C}$ with some assumptions. This generalizes the known results given by Wei and Xi for the categories of finitely generated modules over Artin algebras, thereby providing new insights in exact and triangulated categories.

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arXiv2020

Idempotent completion of extriangulated categories

Extriangulated categories were introduced by Nakaoka and Palu as a simultaneous generalization of exact categories and triangulated categories. In this paper, we show that the idempotent completion of an extriangulated category admits a natural extriangulated structure. As applications, we prove that cotorsion pairs in an extriangulated category induce cotorsion pairs in its idempotent completion under certain condition, and the idempotent completion of a recollement of extriangulated categories is still a recollement.

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arXiv2020

Recollements and tilting modules

Let $(\mbox{mod} \Lambda',\mbox{mod} \Lambda,\mbox{mod} \Lambda'')$ be a recollement of module categories for artin algebras $\Lambda'$, $\Lambda$ and $\Lambda''$. We provide a sufficient condition such that a glued torsion pair in $\mbox{mod} \Lambda$ is tilting when the given two torsion pairs are tilting in $\mbox{mod} \Lambda'$ and $\mbox{mod} \Lambda''$ respectively. Using this result, we give a construction of gluing of tilting modules in $\mbox{mod} \Lambda$ with respect to tilting modules in $\mbox{mod} \Lambda'$ and $\mbox{mod} \Lambda''$ respectively.

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arXiv2020

Duality Pairs Induced by One-Sided Gorenstein Subcategories

For a ring $R$ and an additive subcategory $\C$ of the category $\Mod R$ of left $R$-modules, under some conditions we prove that the right Gorenstein subcategory of $\Mod R$ and the left Gorenstein subcategory of $\Mod R^{op}$ relative to $\C$ form a coproduct-closed duality pair. Let $R,S$ be rings and $C$ a semidualizing ($R,S$)-bimodule. As applications of the above result, we get that if $S$ is right coherent and $C$ is faithfully semidualizing, then $(\mathcal{GF}_C(R),\mathcal{GI}_C(R^{op}))$ is a coproduct-closed duality pair and $\mathcal{GF}_C(R)$ is covering in $\Mod R$, where $\mathcal{G}\mathcal{F}_C(R)$ is the subcategory of $\Mod R$ consisting of $C$-Gorenstein flat modules and $\mathcal{G}\mathcal{I}_C(R^{op})$ is the subcategory of $\Mod R^{op}$ consisting of $C$-Gorenstein injective modules; we also get that if $S$ is right coherent, then $(\mathcal{A}_C(R^{op}),l\mathcal{G}(\mathcal{F}_C(R)))$ is a coproduct-closed and product-closed duality pair and $\mathcal{A}_C(R^{op})$ is covering and preenveloping in $\Mod R^{op}$, where $\mathcal{A}_C(R^{op})$ is the Auslander class in $\Mod R^{op}$ and $l\mathcal{G}(\mathcal{F}_C(R))$ is the left Gorenstein subcategory of $\Mod R$ relative to $C$-flat modules.

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arXiv2020

One-Sided Gorenstein Subcategories

We introduce the right (left) Gorenstein subcategory relative to an additive subcategory $\C$ of an abelian category $\A$, and prove that the right Gorenstein subcategory $r\mathcal{G}(\mathscr{C})$ is closed under extensions, kernels of epimorphisms, direct summands and finite direct sums. When $\C$ is self-orthogonal, we give a characterization for objects in $r\mathcal{G}(\mathscr{C})$, and prove that any object in $\A$ with finite $r\mathcal{G}(\C)$-projective dimension is isomorphic to a kernel (resp. a cokernel) of a morphism from an object in $\A$ with finite $\C$-projective dimension to an object in $r\mathcal{G}(\C)$. As an application, we obtain a weak Auslander-Buchweitz context related to the kernel of a hereditary cotorsion pair in $\A$ having enough injectives.

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arXiv2020

Avramov-Martsinkovsky type exact sequences for extriangulated categories

Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category with a proper class $\xi$ of $\mathbb{E}$-triangles. In this paper, we first introduce the $\xi$-Gorenstein cohomology in terms of $\xi$-$\mathcal{G}$projective resolutions and $\xi$-$\mathcal{G}$injective coresolutions, respectively, and then we get the balance of $\xi$-Gorenstein cohomology. Moreover, we study the interplay among $\xi$-cohomology, $\xi$-Gorenstein cohomology and $\xi$-complete cohomology, and obtain the Avramov-Martsinkovsky type exact sequences in this setting.

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arXiv2020

Almost Split Triangles and Morphisms Determined by Objects in Extriangulated Categories

Let $(\mathfrak{C},\mathbb{E},\mathfrak{s})$ be an Ext-finite, Krull-Schmidt and $k$-linear extriangulated category with $k$ a commutative artinian ring. We define an additive subcategory $\mathfrak{C}_r$ (respectively, $\mathfrak{C}_l$) of $\mathfrak{C}$ in terms of the representable functors from the stable category of $\mathfrak{C}$ modulo $\mathfrak{s}$-injectives (respectively, $\mathfrak{s}$-projectives) to $k$-modules, which consists of all $\mathfrak{s}$-projective (respectively, $\mathfrak{s}$-injective) objects and objects isomorphic to direct summands of finite direct sums of all third (respectively, first) terms of almost split $\mathfrak{s}$-triangles. We investigate the subcategories $\mathfrak{C}_r$ and $\mathfrak{C}_l$ in terms of morphisms determined by objects, and then give equivalent characterizations on the existence of almost split $\mathfrak{s}$-triangles.

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arXiv2020

Balance of complete cohomology in extriangulated categories

Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category with a proper class $\xi$ of $\mathbb{E}$-triangles. In this paper, we study the balance of complete cohomology in $(\mathcal{C},\mathbb{E},\mathfrak{s})$, which is motivated by a result of Nucinkis that complete cohomology of modules is not balanced in the way the absolute cohomology Ext is balanced. As an application, we give some criteria for identifying a triangulated catgory to be Gorenstein and an artin algebra to be $F$-Gorenstein.

math.CT↗

arXiv2020

Complete cohomology for extriangulated categories

Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category with a proper class $\xi$ of $\mathbb{E}$-triangles. In this paper, we study complete cohomology of objects in $(\mathcal{C},\mathbb{E},\mathfrak{s})$ by applying $\xi$-projective resolutions and $\xi$-injective coresolutions constructed in $(\mathcal{C},\mathbb{E},\mathfrak{s})$. Vanishing of complete cohomology detects objects with finite $\xi$-projective dimension and finite $\xi$-injective dimension. As a consequence, we obtain some criteria for the validity of the Wakamatsu Tilting Conjecture and give a necessary and sufficient condition for a virtually Gorenstein algebra to be Gorenstein. Moreover, we give a general technique for computing complete cohomology of objects with finite $\xi$-$\mathcal{G}$projective dimension. As an application, the relationships between $\xi$-projective dimensions and $\xi$-$\mathcal{G}$projective dimensions for objects in $(\mathcal{C},\mathbb{E},\mathfrak{s})$ are given.

math.RT↗

arXiv2018

Remarks on Gorenstein weak injective and weak flat modules

In this paper, we introduce the notions of Gorenstein weak injective and weak flat modules respectively in terms of weak injective and weak flat modules, which is larger than classical classes of Gorenstein injective and flat modules. In this new setting, we characterize rings over which all modules are Gorenstein weak injective. Moreover, we also discuss a relation between weak cosyzygy and Gorenstein weak cosyzygy of a module, and the stability of Gorenstein weak injective modules.

math.RA↗

arXiv2018

Support $\tau$-tilting modules and recollements

Let $(\mbox{mod} \Lambda',\mbox{mod} \Lambda,\mbox{mod} \Lambda'')$ be a recollement of abelian categories for artin algebras $\Lambda'$, $\Lambda$ and $\Lambda''$. Under certain conditions, we present an explicit construction of gluing of (support) $\tau$-tilting modules in $\mbox{mod} \Lambda$ with respect to (support) $\tau$-tilting modules in $\mbox{mod} \Lambda'$ and $\mbox{mod} \Lambda''$ respectively; conversely, we study the construction of (support) $\tau$-tilting modules in $\mbox{mod} \Lambda'$ and $\mbox{mod} \Lambda''$ obtained from (support) $\tau$-tilting modules in $\mbox{mod} \Lambda$.

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