Xiao-Wu Chen

Algebras of Derived Dimension Zero

We prove that a finite-dimensional algebra over an algebraically closed field is of derived dimension 0 if and only if it is an iterated tilted algebra of Dynkin type.

Algebras with radical square zero are either self-injective or CM-free

An artin algebra is called CM-free provided that all its finitely generated Gorenstein projective modules are projective. We show that a connected artin algebra with radical square zero is either self-injective or CM-free. As a consequence, we prove that a connected artin algebra with radical square zero is Gorenstein if and only if its valued quiver is either an oriented cycle with the trivial valuation or does not contain oriented cycles.

Three results on Frobenius categories

This paper consists of three results on Frobenius categories: (1) we give sufficient conditions on when a factor category of a Frobenius category is still a Frobenius category; (2) we show that any Frobenius category is equivalent to an extension-closed exact subcategory of the Frobenius category formed by Cohen–Macaulay modules over some additive category; this is an analogue of Gabriel–Quillen’s embedding theorem for Frobenius categories; (3) we show that under certain conditions an exact category with enough projective and enough injective objects allows a natural new exact structure, with which the given category becomes a Frobenius category. Several applications of the results are discussed.

Singular equivalences induced by homological epimorphisms

We prove that a certain homological epimorphism between two algebras induces a triangle equivalence between their singularity categories. Applying the result to a construction of matrix algebras, we describe the singularity categories of some non-Gorenstein algebras.

On Graded Bialgebra Deformations

We introduce the graded bialgebra deformations, which explain the lifting method of Andruskiewitsch and Schneider. We also relate these graded bialgebra deformations with the corresponding graded bialgebra cohomology groups, which is the graded version of the one due to Gerstenhaber and Schack.

A recollement of vector bundles

For a weighted projective line, the stable category of its vector bundles modulo line bundles has a natural triangulated structure. We prove that, for any positive integers p,q,r and r with r r, there is an explicit recollement of the stable category of vector bundles on a weighted projective line of weight type (p,q,r) relative to the ones on weighted projective lines of weight

A SHORT PROOF OF HRS-TILTING

We give a short proof to the following tilting theorem by Happel, Reiten and Smal{\o} via an explicit construction: given two abelian categories

Duality Between Quantum Symmetric Algebras

\mathcal{A}

SINGULAR EQUIVALENCES OF TRIVIAL EXTENSIONS

and

Equivariantization and Serre Duality I

\mathcal{B}