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Shengyong Pan

Publications and source records attributed to Shengyong Pan.

At least 19 recordsLinked to original sources
arXiv2026

Recollements of derived categories from $n$-term big tilting complexes

Let $A$ be a ring and let $\bf T$ be an $n$-term big tilting complex over $A$, represented by a bounded complex $\cpx{P}$ of projective $A$-modules. Set $B=\End_{\D{A}}(\cpx{P}), \Lambda:=\dotEnd_A(\cpx{P})$ and $\Delta:=\tau_{\leq 0}\Lambda$. The associated complex of $A$-$B$-bimodule $\cpx{T}=\cpx{P}\otimesL_{\Delta}B$ is given. We construct an extension-closed exact subcategory $\mathscr E$ of $B\Modcat$ and prove that the derived category $\D{B}$ admits a recollement by $\D{\mathscr E}$ and $\D{A}$. The proof passes through a dg double-centralizer description of a projective model of $\cpx{T}$ and an exact realisation theorem identifying $\D{\mathscr E}$ with the kernel of the derived tensor functor. We further show that $\mathscr E$ is $d$-symmetric for every $d$ not smaller than the amplitude of a perfect right $B$-model of $\cpx{T}$. Moreover, $\mathscr E$ is abelian if and only if the kernel is stable under the standard $t$-structure, equivalently, the recollement is induced by a homological ring epimorphism. In right $B$-amplitude at most one, the kernel is therefore abelian, and our construction specializes to the recollement arising from universal localisation in the two-term case. Thus the classical two-term picture extends to arbitrary finite-term big tilting complexes, with exact categories replacing abelian kernels in higher amplitude. Finally, we construct genuinely $n$-term non-compact big tilting complexes for every $n\geq2$, including an explicit three-term example whose left-hand term is determined by an algebraic Calkin-type quotient.

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arXiv2026

Relative interval tilting, higher Auslander staircase corners and rational Dyck posets

We construct an explicit tilting equivalence between the incidence algebra of every rational Dyck staircase and a canonical idempotent corner of a higher Auslander algebra of type~$A$. In the coprime case, this corner identifies with the algebra $B_0$ introduced by Xing. The resulting Dyck-corner equivalence supplies the missing link in the previously known chain of equivalences and thereby proves the Chapoton-Ladkani-Rognerud conjecture for coprime positive integers. The Dyck-corner equivalence itself requires no coprimality hypothesis and is compatible with replicated algebras. Our main tool is a linear-categorical extension of the interval-tilting mechanism of Chapoton-Ladkani-Rognerud. The relative theorem applies to finite $\kk$-linear categories under finite-global-dimension assumptions on the total category and its fibers. In contrast with the incidence-category setting, it allows arbitrary finite-dimensional $\Hom$ spaces and zero composites of nonzero morphisms, and it does not require the diagonal endomorphism algebras to be semisimple. The tilting object is constructed from exact right Kan extensions of fiberwise representables. We compute its opposite indexed endomorphism category, including all forced-zero compositions, and hence its opposite endomorphism algebra. Iterating this construction one coordinate at a time yields an explicit derived equivalence between the incidence algebra of every finite coordinate staircase and an idempotent corner of a higher Auslander algebra of type~$A$. We further realize the resulting staircase derived categories as triangulated subcategories generated by product Lagrangians in partially wrapped Fukaya categories of stopped-disk symmetric products and, in the coprime Dyck case, as Fukaya-Seidel categories of symmetric Brieskorn--Pham singularities.

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arXiv2026

On universal deformation rings and stable equivalences of Gorenstein-projective modules

Let $\mathbf{k}$ be a field and let $\Lambda$ and $\Gamma$ finite dimensional $\mathbf{k}$-algebras. Assume that ${_\Gamma}X_\Lambda$ and ${_\Lambda}Y_\Gamma$ are bimodules that define a singular equivalence of Morita type with level (in the sense of Z. Wang) between $\Lambda$ and $\Gamma$ and which also induce an equivalence between the stable categories of finitely generated Gorenstein-projective modules $\Lambda$-$\underline{\text{Gproj}}$ and $\Gamma$-$\underline{\text{Gproj}}$. We prove that if $V$ is an indecomposable object in $\Lambda$-$\underline{\text{Gproj}}$ with $\underline{\mathrm{End}}_\Lambda(V)\cong \mathbf{k}$, then $X\otimes_\Lambda V$ is an object in $\Gamma$-$\underline{\text{Gproj}}$ such that $\underline{\mathrm{End}}_\Gamma(X\otimes_\Lambda V)\cong \mathbf{k}$ and the universal deformation rings (in the sense of F.M. Bleher and the second author) $R(\Lambda,V)$ and $R(\Gamma, X\otimes_\Lambda V)$ are isomorphic. This result generalizes the one obtained by the second author assuming that $\Lambda$ and $\Gamma$ are Gorenstein $\mathbf{k}$-algebras.

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arXiv2026

Cohen-Montgomery duality for bimodules and singular equivalences of Morita type

Let $G$ be a group and $\Bbbk$ a commutative ring. All categories and functors are assumed to be $\Bbbk$-linear. We define a $G$-invariant bimodule ${}_SM_R$ over $G$-categories $R, S$ and a $G$-graded bimodule ${}_BN_A$ over $G$-graded categories $A, B$, and introduce the orbit bimodule $M/G$ and the smash product bimodule $N\# G$. We will show that these constructions are inverses to each other. This will be applied to Morita equivalences, stable equivalences of Morita type, singular equivalences of Morita type, and singular equivalences of Morita type with level to show that the orbit (resp. smash product) bimodule construction transforms an equivalent pair of $G$-categories (resp. $G$-graded categories) of each type to an equivalent pair of $G$-graded categories (resp. $G$-categories) of the same type.

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arXiv2026

The Smith normal form of the Q-walk matrix of the Dynkin graph $A_n$

In this paper, we give an explicit formula for the rank of the $Q$-walk matrix of the Dynkin graph $A_n$. Moreover, we prove that its Smith normal form is $$ \mathrm{diag}\left( \underset{r=\lceil \frac{n}{2} \rceil}{\underbrace{1,2,2,...,2}},0,...,0 \right), $$ where $r$ is the rank of the $Q$-walk matrix $W_Q\left( A_n \right) $ of the Dynkin graph $A_n$.

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arXiv2026

Characterizations of standard derived equivalences of diagrams of dg categories and their gluings

A diagram consisting of differential graded (dg for short) categories and dg functors is formulated in this paper as a colax functor $X$ from a small category $I$ to the 2-category $\mathbf{k}$-dgCat of small dg categories, dg functors and dg natural transformations over a fixed commutative ring $\mathbf{k}$. If $I$ is a group regarded as a category with only one object $*$, then $X$ is nothing but a colax action of the group $I$ on the dg category $X(*)$. In this sense, this $X$ can be regarded as a generalization of a dg category with a colax action of a group. We define a notion of standard derived equivalence between such colax functors by generalizing the corresponding notion between dg categories with a group action. Our first main result gives some characterizations of this notion, one of which is given in terms of generalized versions of a tilting object and a quasi-equivalence. On the other hand, for such a colax functor $X$, the dg categories $X(i)$ with $i$ objects of $I$ can be glued together to have a single dg category $\int_I X$, called the Grothendieck construction of $X$. Our second main result asserts that for such colax functors $X$ and $X'$, the Grothendieck construction $\int_I X'$ is derived equivalent to $\int_I X$ if there exists a standard derived equivalence from $X'$ to $X$. These results generalize the first-named author's results to the dg case, respectively. Even for dg categories with group actions, these results are new. In particular, the second result gives a new tool to show the derived equivalence between the orbit categories of dg categories with group actions, which will be illustrated in some examples.

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arXiv2025

Relative derived equivalences and relative Igusa-Todorov dimensions

Let $A$ be an Artin algebra and $F$ a non-zero subfunctor of $\Ext_A^{1}(-,-)$. In this paper, we characterize the relative $ϕ$-dimension of $A$ by the bi-functor $\Ext_F^1(-,-)$. Furthermore, we show that the finiteness of relative $ϕ$-dimension of an Artin algebra is invariant under relative derived equivalence. More precisely, for an Artin algebra $A$, assume that $F$ has enough projectives and injectives, such that there exists $G\in \modcat{A}$ such that $\add G=\mathcal {P}(F)$, where $\mathcal {P}(F)$ is the category of all $F$-projecitve $A$-modules. If $\cpx{T}$ is a relative tilting complex over $A$ with term length $t(\cpx{T})$ such that $B=\End(\cpx{T})$, then we have $\phd_{F}(A)-t(T^{\bullet})\leq \phd(B)\leq\phd_{F}(A)+t(T^{\bullet})+2$.

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arXiv2025

Universal deformation rings and derived equivalences

In this paper, we show that stable functors of derived equivalences preserve the isomorphism classes of versal deformation rings of finitely generated Gorenstein-projective modules over finite dimensional $k$-algebras. Then we generalize Vel\'ez-Marulanda's result \cite{V} in the case of singular equivalences of Morita type with levels for Gorenstein algebras. Moreover, we also prove that stable equivalences of Morita type preserve the isomorphism classes of versal deformation rings of finitely generated Gorenstein-projective modules over finite dimensional $k$-algebras.

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arXiv2023

Transfer of derived equivalences from subalgebras to endomorphism algebras II

We investigate derived equivalences between subalgebras of some $Φ$-Auslander-Yoneda algebras from a class of $n$-angles in weakly $n$-angulated categories. The derived equivalences are obtained by transferring subalgebras induced by $n$-angles to endomorphism algebras induced by approximation sequences. Then we extend our constructions \cite{BP} to $n$-angle cases. Finally, we give an explicit example to illustrate our result.

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arXiv2022

Symmetric approximation sequences, Beilinson-Green algebras and derived equivalences

In this paper, we will consider a class of locally $Φ$-Beilinson-Green algebras, where $Φ$ is an infinite admissible set of the integers, and show that symmetric approximation sequences in $n$-exangulated categories give rise to derived equivalences between quotient algebras of locally $Φ$-Beilinson-Green algebras in the principal diagonals modulo some factorizable ghost and coghost ideals by the locally finite tilting family. Then we get a class of derived equivalent algebras that have not been obtained by using previous techniques. From higher exact sequences, we obtain derived equivalences between subalgebras of endomorphism algebras by constructing tilting complexes, which generalizes Chen and Xi's result for exact sequences. From a given derived equivalence, we get derived equivalences between locally $Φ$-Beilinson-Green algebras of semi-Gorenstein modules. Finally, from given graded derived equivalences of group graded algebras, we get derived equivalences between associated Beilinson-Green algebras of group graded algebras.

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arXiv2019

Stable equivalences of Morita type for $Φ$-Beilinson-Green algebras

In this paper, we present a method to construct new stable equivalences of Morita type. Suppose that a stable equivalence of Morita type between finite dimensional algebras $A$ and $B$ is defined by a $B$-$A$-bimodule $N$. Then, for any finite admissible set $Φ$ of natural numbers and any generator $X$ of the $A$-module category, the $Φ$-Beilinson-Green algebras $\scr G^Φ_A(X)$ and $\scr G^Φ_B(N\otimes_AX)$ are stably equivalent of Morita type. In particular, if $Φ=\{0\}$, we get a known result in literature. As another consequence, we construct an infinite family of derived equivalent algebras of the same dimension and of the same dominant dimension such that they are pairwise not stably equivalent of Morita type. Finally, we will prove that, if there is a graded stable equivalence of Morita type between graded algebras, then we can get a stable equivalence of Morita type between Beilinson-Green algebras associated with graded algebras

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arXiv2018

Cohomology rings, differential graded algebras and derived equivalences

In this paper, we will consider derived equivalences for differential graded endomorphism algebras by Keller's approaches. First we construct derived equivalences of differential graded algebras which are endomorphism algebras of the objects from a triangle in the homotopy category of differential graded algebras. We also obtain derived equivalences of differential graded endomorphism algebras from a standard derived equivalence of finite dimensional algebras. Moreover, under some conditions, the cohomology rings of these differential graded endomorphism algebras are also derived equivalent. Then we give an affirmative answer to a problem of Dugas \cite{Dugas2015} in some special case.

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arXiv2015

Stable functors of derived equivalences and Gorenstein projective modules

From certain triangle functors, called non-negative functors, between the bounded derived categories of abelian categories with enough projective objects, we introduce their stable functors which are certain additive functors between the stable categories of the abelian categories. The construction generalizes a previous work by Hu and Xi. We show that the stable functors of non-negative functors have nice exactness property and are compatible with composition of functors. This allows us to compare conveniently the homological properties of objects linked by the stable functors. Particularly, we prove that the stable functor of a derived equivalence between two arbitrary rings provides an explicit triangle equivalence between the stable categories of Gorenstein projective modules. This generalizes a result of Y. Kato. Our results can also be applied to provide shorter proofs of some known results on homological conjectures.

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arXiv2014

Relative derived equivalences and relative homological dimensions

Let $\mathscr{A}$ be a small abelian category. For a closed subbifunctor $F$ of $\Ext_{\mathscr{A}}^{1}(-,-)$, Buan has generalized the construction of the Verdier's quotient category to get a relative derived category, where he localized with respect to $F$-acyclic complexes. In this paper, the homological properties of relative derived categories are discussed, and the relation with derived categories is given. For Artin algebras, using relatively derived categories, we give a relative version on derived equivalences induced by $F$-tilting complexes. We discuss the relationships between relative homological dimensions and relative derived equivalences.

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arXiv2010

Inductions and restrictions for stable equivalences of Morita type

In this paper, we present two methods, induction and restriction procedures, to construct new stable equivalences of Morita type. Suppose that a stable equivalence of Morita type between two algebras $A$ and $B$ is defined by a $B$-$A$-bimodule $N$. Then, for any finite admissible set $Φ$ and any generator $X$ of the $A$-module category, the $Φ$-Auslander-Yoneda algebras of $X$ and $N\otimes_AX$ are stably equivalent of Morita type. Moreover, under certain conditions, we transfer stable equivalences of Morita type between $A$ and $B$ to ones between $eAe$ and $fBf$, where $e$ and $f$ are idempotent elements in $A$ and $B$, respectively. Consequently, for self-injective algebras $A$ and $B$ over a field without semisimple direct summands, and for any $A$-module $X$ and $B$-module $Y$, if the $Φ$-Auslander-Yoneda algebras of $A\oplus X$ and $B\oplus Y$ are stably equivalent of Morita type for one finite admissible set $Φ$, then so are the $Ψ$-Auslander-Yoneda algebras of $A\oplus X$ and $B\oplus Y$ for {\it every} finite admissible set $Ψ$. Moreover, two representation-finite algebras over a field without semisimple direct summands are stably equivalent of Morita type if and only if so are their Auslander algebras. As another consequence, we construct an infinite family of algebras of the same dimension and the same dominant dimension such that they are pairwise derived equivalent, but not stably equivalent of Morita type. This answers a question by Thorsten Holm.

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