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Shengyong Pan

Publications and source records attributed to Shengyong Pan.

At least 19 recordsLinked to original sources
arXiv2026

Centres and semisimple deformations of contraction algebras

Let $A_{\rm{con}}$ be the contraction algebra of a $3$-fold flop of an irreducible curve, with genus-zero Gopakumar-Vafa invariants $n_1, \dots, n_l$. We show that $\prod_j M_j(\mathbb{C})^{n_j}$ is the only semisimple algebra to which $A_{\rm{con}}$ deforms. Hence $A_{\rm{con}}$ encodes its Gopakumar-Vafa invariants intrinsically, answering a question of Wemyss. Hua and Toda showed that this deformation exists, and Jelisiejew and Smoktunowicz proved its uniqueness in types A and D. The key step in our proof is that the centre of $A_{\rm{con}}$ has dimension $n_1 + \dots + n_l$. To prove this, we lift the contraction algebra of the universal flop to the Cartan subalgebra, where it becomes a corner of a deformed preprojective algebra. We show that the universal contraction algebra and its zeroth Hochschild homology are free over a polynomial ring, with presentations whose determinants are products of roots. Pulling these presentations back along the arc that classifies the flop gives the count, and recovers Toda's formula for $\dim A_{\rm{con}}$. We also prove Hua and Toda's flatness conjecture, and determine which Gopakumar-Vafa invariants occur in lengths 3 to 6.

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arXiv2026

A counterexamples to the Polishchuk--Van den Bergh conjecture on curves

We construct an effective $S_3$ action on the smooth projective model of $y^2=t^{12h-5}+t$ for every $h\geq1$. Its quotient has genus $h$, and the fixed quotient for the three-cycle class consists of two points. We prove that the derived category of these two points admits no $\bbC$-linear exact fully faithful embedding into the equivariant derived category. This gives a counterexample to Conjecture A of Polishchuk and Van den Bergh. More generally, for a positive-genus quotient we compute the maximum number of completely orthogonal exceptional objects as $\Sm_i\lfloor e_i/2\rfloor$, where $e_i$ are the inertia orders. We also give a necessary and sufficient condition for conjugacy-class decompositions with embeddings linear over the quotient, and construct kernels on the natural fibre products. For genus-zero quotients we obtain Euler-form bounds and exact capacities for all-two ramification with at least four branch points and for types $(2,3,3)$, $(2,3,4)$ and $(2,3,5)$.

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arXiv2026

Compatible gradings on dg models and cluster equivalence

Let $\La_1$ and $\La_2$ be finite-dimensional $\ta_2$-finite complex algebras of global dimension at most two. We prove that, if they are cluster equivalent, then they admit $\bbZ$-gradings for which the bounded derived categories of graded modules are triangle equivalent. The equivalence is not required to commute with the internal grading shifts. The proof uses common multigraded dg models, relative gradings on two-term silting objects, and covering quotients. We also consider tilting lifts of cluster-tilting objects with endomorphism algebras of global dimension at most two. In acyclic cluster type, the existence of such a lift for every cluster-tilting object implies piecewise heredity. We construct a counterexample without this assumption using a surface algebra associated with a one-holed torus.

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arXiv2026

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

Let $A$ be a ring, let ${\bf T}$ be a finite-term big tilting complex over $A$, and choose a bounded projective representative $\cpx{P}$ of ${\bf T}$. Put $B=\End_{\D{A}}(\cpx{P})$. We prove that the kernel of the associated derived tensor functor ${\bf R}:\D{B}\to\D{A}$ is equivalent, through the derived inclusion, to $\D{\mathscr E}$, where $\mathscr E=\Ker({\bf R})\cap B\Modcat$ carries the inherited exact structure. Consequently, $\D{B}$ admits a recollement by $\D{\mathscr E}$ and $\D{A}$. The proof combines a dg double-centralizer argument with a functorial realisation of the kernel on unbounded complexes. The category $\mathscr E$ is definable and $d$-symmetric whenever $d$ is at least the projective amplitude of the underlying perfect right $B$-complex. We also characterise when the kernel is induced by a homological ring epimorphism: this holds precisely when $\mathscr E$ is an abelian subcategory, or equivalently when the standard $t$-structure restricts to the kernel. In amplitude at most one, this recovers the universal-localisation description for two-term big tilting complexes. Over semiperfect endomorphism rings, the amplitude is detected by reduction modulo the Jacobson radical. Finally, we construct non-compact examples of arbitrarily large minimal projective amplitude. One family yields algebraic Calkin quotients; another, over connected rings constructed from singular local hypersurfaces, has nonzero, non-abelian exact kernels and cannot be obtained from modules by derived equivalences between ordinary rings.

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arXiv2026

The $E_6$ normal forms: semisimple deformations, geometry, DT invariants, and the $E_{6,\infty}$ order

We study the completed Jacobi algebras of $x^3+xy^3+p(y)$ for $\operatorname{ord}_yp=n\ge4$. The explicit normal-form morsification, the filtered length and centre proofs, and the cyclic terminal identities give finite free semisimple deformations. For every parameter of finite order we construct a polynomial curve neighbourhood and a local projective small contraction; the resulting geometric GV and Donaldson--Thomas calculations identify the block multiplicities. The one-dimensional limit $E_{6,\infty}$ is studied separately as an order: we determine its normalization, conductor, radical-idealizer chain, and maximal overorders. The canonical chain yields a nine-dimensional finite quotient whose Ringel dual is isomorphic to its opposite. An expanded construction gives a choice-independent $22$-dimensional quotient and a $40$-dimensional Ringel dual; a corner quotient relates the two finite constructions. The length and centre formulas are proved by a filtered comparison with a homogeneous matrix order and by terminal identities at the exceptional endpoint.

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arXiv2026

Relative tilting, interlacing categories 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

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 Veléz-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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arXiv2026

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

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