We study Gorenstein dimension and grade of a module M over a filtered ring whose assosiated graded ring is a commutative Noetherian ring. An equality or an inequality between these invariants of a filtered module and its associated graded module is the most valuable property for an investigation of filtered rings. We prove an inequality G-dimM ≤ G-dimgrM and an equality gradeM = grade grM , whe...

We prove that over a commutative noetherian ring the three approaches to introducing depth for complexes: via Koszul homology, via Ext modules, and via local cohomology, all yield the same invariant. Using this result, we establish a far reaching generalization of the classical AuslanderBuchsbaum formula for the depth of finitely generated modules of finite projective dimension. We extend also ...

In this paper we will compare the connectivity dimension c(P/I) of an ideal I in a polynomial ring P with that of any initial ideal of I. Generalizing a theorem of Kalkbrener and Sturmfels [18], we prove that c(P/LT≺(I)) ≥ min{c(P/I), dim(P/I)−1} for each monomial order ≺. As a corollary we have that every initial complex of a Cohen-Macaulay ideal is strongly connected. Our approach is based on...

Let $M$ and $N$ be two finitely generated graded modules over a standard graded Noetherian ring $R=bigoplus_{ngeq 0} R_n$. In this paper we show that if $R_{0}$ is semi-local of dimension $leq 2$ then, the set $hbox{Ass}_{R_{0}}Big(H^{i}_{R_{+}}(M,N)_{n}Big)$ is asymptotically stable for $nrightarrow -infty$ in some special cases. Also, we study the torsion-freeness of graded generalized local ...

Given a finite, flat morphism between embeddable noetherian schemes of pure dimension 1, we define the notion direct and inverse image for generalized divisors line bundles. In case when deal with (possibly reducible, non-reduced) projective curves over field codomain curve is smooth, introduce compactified Jacobians parametrizing torsion-free rank-1 sheaves study Norm maps Jacobians. Finally, ...

Let A be a Koszul Artin-Schelter regular algebra, σ graded automorphism of and δ degree-one -derivation . We introduce an invariant for called the -divergence describe Nakayama Ore extension B = [ z ; , ] explicitly using construct twisted superpotential ω ˆ so that it is derivation quotient algebra defined by also determine all extensions noetherian algebras dimension 2 compute their automorph...

We study the cancellation property of projective modules rank $2$ with a trivial determinant over Noetherian rings dimension $\leq 4$. If $R$ is smooth affine algebra $4$ an algebraically closed field $k$ such that $6 \in k^{\times}$, then we prove stably free $R$-modules are if and only Hermitian $K$-theory group $\tilde{V}_{SL} (R)$ trivial.

It is proved that whenever P is a prime ideal in a commutative Noethe-rian ring such that the P-adic and the P-symbolic topologies are equivalent, then the two topologies are equivalent linearly. Several explicit examples are calculated, in particular for all prime ideals corresponding to non-torsion points on nonsingular elliptic cubic curves. There are many examples of prime ideals P in commu...

In 1966 [1], Auslander introduced a class of finitely generated modules having a certain complete resolution by projective modules. Then using these modules, he defined the G-dimension (G ostensibly for Gorenstein) of finitely generated modules. It seems appropriate then to call the modules of G-dimension 0 the Gorenstein projective modules. In [4], Gorenstein projective modules (whether finite...

(1) If $R$ is an affine algebra of dimension $d\geq 4$ over $\overline{\mathbb{F}}_{p}$ with $p>3$, then the group structure on ${\rm Um}_d(R)/{\rm E}_d(R)$ nice. (2) a commutative noetherian ring 2$ such that E}_{d+1}(R)$ acts transitively Um}_{d+1}(R),$ Um}_{d+1}(R[X])/{\rm E}_{d+1}(R[X])$