Injective Dimension of Weyl Algebras over Affine Coefficient Rings

The concern of this paper is to derive formulas for the injective dimension of the n-th Weyl algebra A_n(R)in case k is a field of characteristic zero and R is a commutative affine k-algebra of finite injective dimension.For the case n=1 we prove a more general result from which the above result follows.Such formulas can be viewed as generalizations of the corresponding results given by J.C.McConnell in the case R has finite global dimension.

The Excellent Extensions of Rings and Copure Dimensions

Let R be a noetherian ring and S an excellent extension of R.cid(M) denotes the copure injective dimension of M and cfd(M) denotes the copure flat dimension of M.We prove that if M S is a right S-module then cid(M S)=cid(M R) and if S M is a left S-module then cfd(S M)=cfd(R M).Moreover,cid-D(S)=cid-D(R) and cfd-D(S)=cfdD(R).

On Ding homological dimensions

In this paper, we investigate Ding projective dimensions and Ding injective di- mensions of modules and rings. Let R be a ring with rDPD（R） = n 〈 ∞, and let YYl = {Mild（M） 〈 ∞}. We prove that （DP,W1） is a complete hereditary cotorsion pair such that a module M belongs to DD∩W1 if and only if M is projective, moreover, 1421 = （M[pd（M） 〈 ∞} = {MIfd（M） ≤ n} = {MIpd（M） ≤ n}. Then we introduce and inves- tigate Ding derived functor Dext^i（-, -）, and use it to characterize global Ding dimension. We show that if R is a Ding-Chen ring, or if R is a ring with rDPD（R） 〈≤ and rDID（R） 〈 ≤, then rDPD（R） 〈 n if and only if rDID（R） 〈 n if and only if Dext^n＋i（M,N） = 0 for all modules M and N and all integer i ≥ 1.

On Rings with Finite Global Gorenstein Dimensions

As a proper setting to study Gorenstein projective and injective dimensions of modules via vanishing of Gorenstein Ext-functors, a notion of a generalized Gorenstein ring is introduced, which is a non-trivial generalization of Gorenstein rings. Moreover, a new proof for Bennis and Mahdou's equality of global Gorenstein dimension is given.

k-Gorenstein Modules

Let A and F be artin algebras and ∧UГa paper, we first introduce the notion of k-Gorenstein faithfully balanced selforthogonal bimodule. In this modules with respect to ∧UГ and then characterize it in terms of the U-resolution dimension of some special injective modules and the property of the functors Ext^i （Ext^i （-, U）, U） preserving monomorphisms, which develops a classical result of Auslander. As an application, we study the properties of dual modules relative to Gorenstein bimodules. In addition, we give some properties of ∧UГwith finite left or right injective dimension.

Actions of Semisimple Weak Hopf Algebras

Let H be a semisimple weak Hopf algebra, A a left H-module algebra. We prove that the injective dimension of the regular A-module is equal to the one of A as the module over the smash product A#H,nd equal to the one of the regular A#H-module. Also, we give a necessary and sufficient condition for A being a Gorenstein algebra, in terms of the fixed subalgebra of A under the action of H on A.

Finiteness Conditions and Relative Singularity Categories

We introduce the n-pure projective(resp.,injective)dimension of complexes in n-pure derived categories,and give some criteria for computing these dimensions in terms of the n-pure projective(resp.,injective)resolutions(resp.,coresolutions)and n-pure derived functors.As a consequence,we get some equivalent characterizations for the finiteness of n-pure global dimension of rings.Finally,we study Verdier quotient of bounded n-pure derived category modulo the bounded homotopy category of n-pure projective modules,which is called an n-pure singularity category since it can reflect the finiteness of n-pure global dimension of rings.