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We prove that the categories of coherent sheaves over weighted projective lines of tubular type are explicitly related to each other via the equivariantization with respect to certain cyclic group actions. [ABSTRACT FROM AUTHOR]

Abstract: For a positively graded artin algebra we introduce its Beilinson algebra . We prove that if A is well-graded self-injective, then the category of graded A-modules is equivalent to the category of graded modules over the trivial extension algebra . Consequently, there is a full exact embedding from the bounded derived category of into the stable category of graded modules over A; it is an equivalence if and only if the 0-th component algebra has finite global dimension. [Copyright &y& Elsevier]

Abstract: An artin algebra A is said to be CM-finite if there are only finitely many, up to isomorphisms, indecomposable finitely generated Gorenstein-projective A-modules. We prove that for a Gorenstein artin algebra, it is CM-finite if and only if every its Gorenstein-projective module is a direct sum of finitely generated Gorenstein-projective modules. This is an analogue of Auslander''s theorem on algebras of finite representation type [M. Auslander, A functorial approach to representation theory, in: Representations of Algebras, Workshop Notes of the Third Internat. Conference, in: Lecture Notes in Math., vol. 944, Springer-Verlag, Berlin, 1982, pp. 105–179; M. Auslander, Representation theory of artin algebras II, Comm. Algebra (1974) 269–310]. [Copyright &y& Elsevier]

Abstract: We prove that triangulated categories with bounded t-structures are Karoubian. Consequently, for an Ext-finite abelian category over a commutative noetherian complete local ring, its bounded derived category is Krull–Schmidt. [Copyright &y& Elsevier]

Let K be a field of characteristic 0 containing all roots of unity. We classified all the Hopf structures on monomial K-coalgebras, or, in dual version, on monomial K-algebras. [Copyright &y& Elsevier]