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Houjun Zhang

Publications and source records attributed to Houjun Zhang.

9 recordsLinked to original sources
arXiv2025

Classification of silted algebras for two quivers of Dynkin type $\mathbb{D}_{n}$

Let $Q$ be the Dynkin quiver of type $\mathbb{D}_{n}$ with linear orientation and let $Q'$ be the quiver formed by reversing the arrow at the unique source in $Q$. In this paper, we present a complete classification of both silted algebras and strictly shod algebras associated with these two quivers. Based on the classification, we derive formulas for counting the number of silted algebras and strictly shod algebras. Furthermore, we establish that all strictly shod algebras corresponding to $Q$ and $Q'$ are string algebras. As an application, we provide a way to construct examples such that the realization functor which is induced from the $t$-structure does not extend to a derived equivalence.

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arXiv2025

Classification of silted algebras for two quivers of Dynkin type $\mathbb{A}_{n}$

In this paper, we give a complete classification of silted algebras for the quiver $\overrightarrow{\mathbb{A}}_{n}$ of type $\mathbb{A}_{n}$ with linear orientation and for the quiver obtained from $\overrightarrow{\mathbb{A}}_{n}$ by reversing the arrow at the unique source. Based on the classification, we also compute the number of silted algebras for these two quivers.

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arXiv2025

On the Gorensteiness of string algebras

In this paper, we give a description of the self-injective dimension of string algebras and obtain a necessary and sufficient condition for a string algebra to be Gorenstein.

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arXiv2023

Constructing projective resolution and taking cohomology for gentle algebras in the geometric model

The geometric models for the module category and derived category of any gentle algebra were introduced to realize the objects in module category and derived category by permissible curves and admissible curves respectively. The present paper firstly unifies these two realizations of objects in module category and derived category via same surface for any gentle algebra, by the rotation of permissible curves corresponding to the objects in the module category. Secondly, the geometric characterization of the cohomology of complexes over gentle algebras is established by the truncation of projective permissible curves. It is worth mentioning that the rotation of permissible curves and the truncation of projective permissible curves are mutually inverse processes to some extent. As applications, an alternative proof of ``no gaps" theorem as to cohomological length for the bounded derived categories of gentle algebras is provided in terms of the geometric characterization of the cohomology of complexes. Moreover, we obtain a geometric proof for the strong Nakayama conjecture for gentle algebras. Finally, we contain two examples to illustrate our results.

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arXiv2022

Classification silted algebras for a quiver of Dynkin type $\mathbb{A}_{n}$ via geometric models

Let $\Q$ be the quiver of Dynkin type $\mathbb{A}_n$ with linear orientation and $A_{n}=k\Q$. In this paper, we give a complete classification of the silted algebras of type $A_{n}$ by using the geometric models of gentle algebras. We show that any finite-dimensional algebra is a silted of type $A_{n}$ if and only if it is a tilted of type $A_{n}$ or a tilted algebra of type $A_{m}\times A_{n-m}$ for any positive integer $1\leq m\leq n-1$. Based on the classification, we obtain a formula for computing the number of silted algebras of type $A_{n}$.

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arXiv2022

Relative left Bongartz completions and their compatibility with mutations

In this paper, we introduce relative left Bongartz completions for a given basic $\tau$-rigid pair $(U,Q)$ in the module category of a finite dimensional algebra $A$. They give a family of basic $\tau$-tilting pairs containing $(U,Q)$ as a direct summand. We prove that relative left Bongartz completions have nice compatibility with mutations. Using this compatibility we are able to study the existence of maximal green sequences under $\tau$-tilting reduction. We also explain our construction and some of the results in the setting of silting theory.

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arXiv2021

A differential graded approach to the silting theorem

A silting theorem was established by Buan and Zhou as a generalisation of the classical tilting theorem of Brenner and Butler. In this paper, we give an alternative proof of the theorem by using differential graded algebras.

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arXiv2021

The ST correspondence for proper non-positive dg algebras

Let $A$ be a proper non-positive dg algebra over a field $k$. For a simple-minded collection of the finite-dimensional derived category $\mathcal{D}_{fd}(A)$, we construct a 'dual' silting object of the perfect derived category $\mathrm{per}(A)$ by using the Koszul duality for dg algebras. This induces a one-to-one correspondence between the equivalence classes of silting objects in $\mathrm{per}(A)$ and algebraic t-structures of $\mathcal{D}_{fd}(A)$.

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