Gorenstein Dimension and AS-Gorenstein Algebras

The purpose of this paper is to connect the notion of Gorenstein dimension with AS-Gorenstein algebras. In particular, we show that a noetherian connected graded algebra having a balanced dualizing complex is AS-Gorenstein if the balanced dualizing complex has finite Gorenstein dimension. As a preparation, we generalize the Auslander–Bridger formula to the class of noncommutative noetherian connected graded algebras having balanced dualizing complexes.     

Noetherian PI Hopf algebras are Gorenstein

We prove that every noetherian affine PI Hopf algebra has finite injective dimension, which answers a question of Brown (1998).

     

Noetherian Connected Graded Algebras of Global Dimension 3

Let A be a noetherian connected graded algebra of global dimension 3. We show that A is regular in the sense of Artin and Schelter if one of the following conditions holds: (1) A is generated by two elements; (2) the graded simple module has a standard resolution; (3) the degrees of minimal relations are the same (this includes the quadratic case). Some general properties of A are studied without assuming regularity.     

Hopf algebras with rigid dualizing complexes

Let H be a Hopf algebra over a base field. If H has an ℕ-filtration such that the associated graded ring is connected graded noetherian and has enough normal elements, then H is Gorenstein. This gives a partial solution to a question of Brown and Brown-Goodearl. As a consequence, every quotient Hopf algebra of a generic quantized coordinate ring of a connected semisimple Lie group is Auslander-Gorenstein and Cohen-Macaulay. The last statement answers a question of Goodearl-Zhang.     

On graded characterizations of finite dimensionality for algebraic algebras

We observe that a finitely generated algebraic algebra R (over a field) is finite dimensional if and only if the associated graded ring \({{\text {gr}}}{R}\) is right noetherian, if and only if \({{\text {gr}}}{R}\) has right Krull dimension, if and only if \({{\text {gr}}}{R}\) satisfies a polynomial identity.     

Gorenstein Projective Precovers

We prove that the class of Gorenstein projective modules is special precovering over any left GF-closed ring such that every Gorenstein projective module is Gorenstein flat and every Gorenstein flat module has finite Gorenstein projective dimension. This class of rings includes (strictly) Gorenstein rings, commutative noetherian rings of finite Krull dimension, as well as right coherent and left n-perfect rings. In Sect. 4 we give examples of left GF-closed rings that have the desired properties (every Gorenstein projective module is Gorenstein flat and every Gorenstein flat has finite Gorenstein projective dimension) and that are not right coherent.     

Generalizations of the Sweedler Dual

As left adjoint to the dual algebra functor, Sweedler’s finite dual construction is an important tool in the theory of Hopf algebras over a field. We show in this note that the left adjoint to the dual algebra functor, which exists over arbitrary rings, shares a number of properties with the finite dual. Nonetheless the requirement that it should map Hopf algebras to Hopf algebras needs the extra assumption that this left adjoint should map an algebra into its linear dual. We identify a condition guaranteeing that Sweedler’s construction works when generalized to noetherian commutative rings. We establish the following two apparently previously unnoticed dual adjunctions: For every commutative ring R the left adjoint of the dual algebra functor on the category of R-bialgebras has a right adjoint. This dual adjunction can be restricted to a dual adjunction on the category of Hopf R-algebras, provided that R is noetherian and absolutely flat.     

Bounded complexes of cotorsion sheaves

We provide a necessary and su?cient condition which ensure that every flat quasi-coherent sheaf has finite cotorsion dimension. Also, we will show that every locally noetherian scheme with a dualizing complex has this requirement.     

Non-noetherian regular rings of dimension 2

We study connected, not necessarily noetherian, regular rings of global dimension 2.

     

Projectives of Large Uniform-Rank, in Krull Dimension 1

It is shown that for every positive integer r there is a (leftand right) noetherian domain, of Krull dimension 1, that hasan indecomposable projective module of uniform-rank r. Direct-sumdecompositions of free modules over this domain need not satisfyuniqueness of the number of indecomposable summands. If desired,the domain can be taken to be an order over a discrete valuationring, in a finite-dimensional division algebra over a globalfield.     

关于扭Hopf代数的一些注记

In this paper,we get some properties of the antipode of a twisted Hopf algebra.We proved that the graded global dimension of a twisted Hopf algebra coincides with the graded projective dimension of its trivial module k,which is also equal to the projective dimension of k.     

The GK dimension of relatively free algebras of PI-algebras

We prove a strict relation between the Gelfand–Kirillov (GK) dimension of the relatively free (graded) algebra of a PI-algebra and its (graded) exponent. As a consequence we show a Bahturin–Zaicev type result relating the GK dimension of the relatively free algebra of a graded PI-algebra and the one of its neutral part. We also get that the growth of the relatively free graded algebra of a matrix algebra is maximal when the grading is fine. Finally we compute the graded GK dimension of the matrix algebra with any grading and the graded GK dimension of any verbally prime algebra endowed with an elementary grading.     

Stable Projective Homotopy Theory of Modules,Tails, and Koszul Duality

A contravariant functor is constructed from the stable projective homotopy theory of finitely generated graded modules over a finite-dimensional algebra to the derived category of its Yoneda algebra modulo finite complexes of modules of finite length. If the algebra is Koszul with a noetherian Yoneda algebra, then the constructed functor is a duality between triangulated categories. If the algebra is self-injective, then stable homotopy theory specializes trivially to stable module theory. In particular, for an exterior algebra the constructed duality specializes to (a contravariant analog of) the Bernstein–Gelfand–Gelfand correspondence.     

Actions of hopf algebras on fully bounded noetherian rings

Let H be a finite dimensional Hopf algebra acting on a right noetherian algebra A and assume that the trace function [tcirc] : A → A ^H is surjective. Then A is right PBN if and only if so is A ^H . This extends the result of García-del Río for group actions which answered a question of Fisher-Osterburg, and the result of Nǎstǎsescu-Dǎscǎlescu for group graded algebras.     

Graded lie algebras of depth one

To each simply connected topological space is associated a graded Lie algebra; the rational homotopy Lie algebra. The Avramov-Felix conjecture says that for a space of finite Ljusternik-Schnirelmann category this Lie algebra contains a free Lie subalgebra on two generators. We prove the conjecture in the case when the Lie algebra has depth one.     

Mixed algebras and quivers related to cell complexes

In this paper we study finite dimensional algebras arising from categories of perverse sheaves on finite regular cell complexes (cellular perverse algebras). We prove that such algebras are quasi-hereditary and have finite global dimension. We discuss some restrictions, under which cellular perverse algebras are Koszul. We also study the relationship between Koszul duality functors in the derived categories of categories of graded and non-graded modules over an algebra and its quadratic dual.     

Double ore extensions

A double Ore extension is a natural generalization of the Ore extension. We prove that a connected graded double Ore extension of an Artin-Schelter regular algebra is Artin-Schelter regular. Some other basic properties such as the determinant of the DE-data are studied. Using the double Ore extension, we construct 26 families of Artin-Schelter regular algebras of global dimension four in a sequel paper.     

Compact DG modules and Gorenstein DG algebras

When the base connected cochain DG algebra is cohomologically bounded, it is proved that the difference between the amplitude of a compact DG module and that of the DG algebra is just the projective dimension of that module. This yields the unboundedness of the cohomology of non-trivial regular DG algebras. When A is a regular DG algebra such that H(A) is a Koszul graded algebra, H(A) is proved to have the finite global dimension. And we give an example to illustrate that the global dimension of H(A) may be infinite, if the condition that H(A) is Koszul is weakened to the condition that A is a Koszul DG algebra. For a general regular DG algebra A, we give some equivalent conditions for the Gorensteiness. For a finite connected DG algebra A, we prove that D^c(A) and D^c(A ^op) admit Auslander-Reiten triangles if and only if A and A ^op are Gorenstein DG algebras. When A is a non-trivial regular DG algebra such that H(A) is locally finite, D^c(A) does not admit Auslander-Reiten triangles. We turn to study the existence of Auslander-Reiten triangles in D_lf^b(A) and D_lf^b (A ^op) instead, when A is a regular DG algebra. This work was supported by the National Natural Science Foundation of China (Grant No. 10731070) and the Doctorate Foundation of Ministry of Education of China (Grant No. 20060246003)     

Algebras without noetherian filtrations

We provide examples of finitely generated noetherian PI algebras for which there is no finite dimensional filtration with a noetherian associated graded ring; thus we answer negatively a question of Lorenz (1988).