arXiv subjects
Publications and source records attributed to Xiao-Wu Chen.
In this article, we construct an explicit pre-triangulated category which is not a triangulated category. Its underlying additive category is the category of finitely generated projective modules of the type-$A_5$ preprojective algebra over $\mathbb F_2$, and the suspension is induced by the graph-reflection automorphism.
We propose the notion of Frobenius quotients between Frobenius exact categories. It turns out that any Frobenius quotient induces Frobenius quotients between the corresponding inflation categories. We obtain an explicit Frobenius quotient from the category of vector bundles on weighted projective lines with three weights to a certain category consisting of monomorphism grids.
We give a detailed proof of the following fundamental result: the singularity category of a ring is triangle equivalent to the stabilization of its stable module category. The result yields singular equivalences between rings of different nature. We use Leavitt rings to describe singularity categories of artinian rings.
We establish noncommutative Knörrer periodicity for projective-module factorizations over an arbitrary ring, using the equivariantization theory with respect to various actions by a cyclic group of order two. We obtain an explicit quasi-inverse of the periodicity. We compare the periodicity with a certain tensor functor between big singularity categories.
We investigate the stabilization $\mathcal{S}$ of the module category over an artinian ring $Λ$ by formally inverting the tensor endofunctor given by the bimodule of relative noncommutative differential $1$-forms. It turns out that $\mathcal{S}$ is a Frobenius abelian category, which is equivalent to the category of finitely presented modules over the zeroth component $L_0$ of the Leavitt ring $L$. It follows that $L_0$ is an FC ring in the sense of Damiano, which is usually not quasi-Frobenius. Moreover, the singularity category of $Λ$ is triangle equivalent to the stable module category over $L_0$.
For a regular normal element in an arbitrary ring, we study the category of its module factorizations. The cokernel functor relates module factorizations with Gorenstein projective components to Gorenstein projective modules over the quotient ring. The results are vast extensions of Eisenbud's matrix factorization theorem.
We introduce higher-dimensional module factorizations associated to a regular sequence. They include higher-dimensional matrix factorizations, which are commutative cubes consisting of free modules with edges being classical matrix factorizations. We characterize the stable category of maximal Cohen-Macaulay modules over a complete intersection via higher-dimensional matrix factorizations over the corresponding regular local ring. The result generalizes to noncommutative rings, including quantum complete intersections.
We characterize $τ$-tilting modules as $1$-tilting modules over quotient algebras satisfying a tensor-vanishing condition, and characterize $1$-tilting modules as $τ$-tilting modules satisfying a ${\rm Tor}^1$-vanishing condition. We use delooping levels to study \emph{Self-orthogonal $τ$-tilting Conjecture}: any self-orthogonal $τ$-tilting module is $1$-tilting. We confirm the conjecture when the endomorphism algebra of the module has finite global delooping level.
We give a direct proof of the following known result: the Grothendieck group of a triangulated category with a silting subcategory is isomorphic to the split Grothendieck group of the silting subcategory. Moreover, we obtain its cluster-tilting analogue.
Let $\mathbb{K}$ be a field of characteristic $p$ and $G$ be a cyclic $p$-group which acts on a finite acyclic quiver $Q$. The folding process associates a Cartan triple to the action. We establish a Morita equivalence between the skew group algebra of the preprojective algebra of $Q$ and the generalized preprojective algebra associated to the Cartan triple in the sense of Geiss, Leclerc and Schröer. The Morita equivalence induces an isomorphism between certain ideal monoids of these preprojective algebras, which is compatible with the embedding of Weyl groups appearing in the folding process.
The singularity category of a ring detects the homological singularity of the given ring, and appears in many different contexts. We describe two different dg enhancements of the singularity category, that is, the Vogel dg category and the singular Yoneda dg category. These two dg enhancements turn out to be quasi-equivalent. We report some progress on the Singular Presilting Conjecture.
We introduce the notion of pre-weight structure on a triangulated category and study the corresponding pseudo-identities. We propose the notion of canonical derived equivalence between algebras that are not necessarily flat, which is associated to a tilting complex. In the flat situation, canonical derived equivalences coincide with standard derived equivalences in the sense of Rickard. We prove that any derived equivalence starting from a hereditary algebra is canonical. The key tool is a general factorization theorem: any derived equivalence is uniquely factorized as a pseudo-identity followed by a canonical derived equivalence.
For any finite dimensional algebra $Λ$ given by a quiver with relations, we prove that its dg singularity category is quasi-equivalent to the perfect dg derived category of a dg Leavitt path algebra. The result might be viewed as a deformed version of the known description of the dg singularity category of a radical-square-zero algebra in terms of a Leavitt path algebra with trivial differential. The above result is achieved in two steps. We first introduce the singular Yoneda dg category of $Λ$, which is quasi-equivalent to the dg singularity category of $Λ$. The construction of this new dg category follows from a general operation for dg categories, namely an explicit dg localization inverting a natural transformation from the identity functor to a dg endofunctor. This localization turns out to be quasi-equivalent to a dg quotient category. Secondly, we prove that the endomorphism algebra of the quotient of $Λ$ modulo its Jacobson radical in the singular Yoneda dg category is isomorphic to the dg Leavitt path algebra. The appendix is devoted to an alternative proof of the result using Koszul-Moore duality and derived localizations.
We prove that a virtually periodic object in an abelian category gives rise to a non-vanishing result on certain Hom groups in the singularity category. Consequently, for any artin algebra with infinite global dimension, its singularity category has no silting subcategory, and the associated differential graded Leavitt algebra has a non-vanishing cohomology in each degree. We verify the Singular Presilting Conjecture for singularly-minimal algebras and ultimately-closed algebras. We obtain a trichotomy on the Hom-finiteness of the cohomology of differential graded Leavitt algebras.
This work presents results on the finiteness, and on the symmetry properties, of various homological dimensions associated to the Jacobson radical and its higher syzygies, of a semiperfect ring.
We give an informal introduction to model categories, and treat three important examples in some details: the category of small categories, the category of dg algebras, and the category of small dg categories.
We reformulate a result of Bernhard Keller on extensions of $t$-structures and give a detailed proof. In the study of hereditary $t$-structures, the notions of regular $t$-structures and global dimensions arise naturally.
For a noetherian ring $Λ$, the stabilization functor in the sense of Krause yields an embedding of the singularity category of $Λ$ into the homotopy category of acyclic complexes of injective $Λ$-modules. When $Λ$ contains a semisimple artinian subring $E$, we give an explicit description of the stabilization functor using the Hom complexes in the $E$-relative singular Yoneda dg category of $Λ$.