6 research outputs found
Cluster tilting modules and noncommutative projective schemes
In this paper, we study the relationship between equivalences of
noncommutative projective schemes and cluster tilting modules. In particular,
we prove the following result. Let be an AS-Gorenstein algebra of dimension
and the noncommutative projective scheme
associated to . If and
has a -cluster tilting module satisfying that its graded
endomorphism algebra is -graded, then the graded endomorphism
algebra of a basic -cluster tilting submodule of is a two-sided
noetherian -graded AS-regular algebra over of global dimension
such that is equivalent to .Comment: 16 page
Auslander's Theorem for permutation actions on noncommutative algebras
When and is a small subgroup of
, Auslander's Theorem says that the skew group
algebra is isomorphic to as graded
algebras. We prove a generalization of Auslander's Theorem for permutation
actions on -skew polynomial rings, -quantum Weyl algebras,
three-dimensional Sklyanin algebras, and a certain graded down-up algebra. We
also show that certain fixed rings are graded isolated singularities in
the sense of Ueyama
Ample Group Action on AS-regular Algebras and Noncommutative Graded Isolated Singularities
In this paper, we introduce a notion of ampleness of a group action on a
right noetherian graded algebra , and show that it is strongly related to
the notion of to be a graded isolated singularity introduced by the
second author of this paper. Moreover, if is a noetherian AS-regular
algebra and is a finite ample group acting on , then we will show that
where is the Beilinson algebra of . We will also
explicitly calculate a quiver such that when is of dimension 2.Comment: 25 page
Stable Categories of Graded Maximal Cohen-Macaulay Modules over Noncommutative Quotient Singularities
Tilting objects play a key role in the study of triangulated categories. A
famous result due to Iyama and Takahashi asserts that the stable categories of
graded maximal Cohen-Macaulay modules over quotient singularities have tilting
objects. This paper proves a noncommutative generalization of Iyama and
Takahashi's theorem using noncommutative algebraic geometry. Namely, if is
a noetherian AS-regular Koszul algebra and is a finite group acting on
such that is a "Gorenstein isolated singularity", then the stable
category of graded maximal
Cohen-Macaulay modules has a tilting object. In particular, the category
is triangle equivalent to the
derived category of a finite dimensional algebra.Comment: 28 pages, an error in the previous version has been correcte
Local cohomology associated to the radical of a group action on a noetherian algebra
An arbitrary group action on an algebra results in an ideal
of . This ideal fits into the classical
radical theory, and will be called the radical of the group action. If is a
noetherian algebra with finite GK-dimension and is a finite group, then the
difference between the GK-dimensionsof and that of is
called the pertinency of the group action. We provide some methods to find
elements of the radical, which helps to calculate the pertinency of some
special group actions. The -adic local cohomology of is
related to the singularities of the invariant subalgebra . We establish an
equivalence between the quotient category of the invariant and that of
the skew group ring through the torsion theory associated to the radical
. With the help of the equivalence, we show that the invariant
subalgebra will inherit certain Cohen-Macaulay property from .Comment: 26 page
Pertinency of Hopf actions and quotient categories of Cohen-Macaulay algebras
We study invariants and quotient categories of fixed subrings of
Artin-Schelter regular algebras under Hopf algebra actions.Comment: to appear at JNC