We introduce and study the concept of $alpha $-semi short modules.Using this concept we extend some of the basic results of $alpha $-short modules to $alpha $-semi short modules.We observe that if $M$ is an $alpha $-semi short module then the dual perfect dimension of $M$ is $alpha $ or $alpha +1$.%In particular, if a semiprime ring $R$ is $alpha $-semi short as an $R$-module, then its Noetheri...

In this paper, we discuss a few simple methods for computing the local topological index and its various analogs vector fields differential forms given on complex varieties with singularities of different types. They are based properties regular meromorphic logarithmic forms, dualizing (canonical) module related constructions. particular, show how to compute Cohen–Macaulay, Gorenstein monomial ...

Let p be an odd prime number and let S be a finite set of prime numbers congruent to 1 modulo p. We prove that the group GS(Q)(p) has cohomological dimension 2 if the linking diagram attached to S and p satisfies a certain technical condition, and we show that GS(Q)(p) is a duality group in these cases. Furthermore, we investigate the decomposition behaviour of primes in the extension QS(p)/Q a...

Given a commutative noetherian non-positive DG-ring with bounded cohomology which has dualizing DG-module, we study its regular, Gorenstein and Cohen-Macaulay loci. We give sufficient condition for the regular locus to be open, show that is always open. However, both of these loci are often empty: no matter how nice H0(A) is, there examples where A empty. then DG-module contains dense open set....

Introduction. The aim of this article is to show how one may use interesting properties of the canonical module for computational applications. It seems that the ideas we develop here are not used in computational algebraic geometry today. We do not claim any major improvement on the theoretical complexity of the problems we address, however we have the feeling that this alternative approach fo...

A balanced Cohen-Macaulay algebra is a connected algebra A having a balanced dualizing complex ωA[d] in the sense of Yekutieli (1992) for some integer d and some graded A-A bimodule ωA. We study some homological properties of a balanced Cohen-Macaulay algebra. In particular, we will prove the following theorem: Theorem 0.1. Let A be a Noetherian balanced Cohen-Macaulay algebra, and M a nonzero ...

In 1966, Auslander introduced the notion of the G-dimension of a finitely generated module over a Cohen-Macaulay noetherian ring and found the basic properties of these dimensions. His results were valid over a local Cohen-Macaulay ring admitting a dualizing module (also see Auslander and Bridger (Mem. Amer. Math. Soc., vol. 94, 1969)). Enochs and Jenda attempted to dualize the notion of G-dime...

Let p be an odd prime number and let S be a finite set of prime numbers congruent to 1 modulo p. We prove that the group GS(Q)(p) has cohomological dimension 2 if the linking diagram attached to S and p satisfies a certain technical condition, and we show that GS(Q)(p) is a duality group in these cases. Furthermore, we investigate the decomposition behaviour of primes in the extension QS(p)/Q a...

1. The construction of T(A) is motivated by dualizing the fundamental group 7Ti(X) of a differentiable manifold X with a base point Xo. Let A be the -R-algebra of C functions on X equipped with the derivation d, which is the usual differentiation from A into the Amodule M = ÛA of C 1-forms on X. Recall that the shuffle algebra Sh(M) consists of the i£-module of the tensor algebra TR(M) and the ...

The notion of the Cousin complex of a module was given by Sharp in 1969. It wasn’t known whether its cohomologies are finitely generated until recently. In 2001, Dibaei and Tousi showed that the Cousin cohomologies of a finitely generated A-module M are finitely generated if the base ring A is local, has a dualizing complex, M satisfies Serre’s (S2)-condition and is equidimensional. In the pres...