In this paper, an algebraic theory for local rings of finite embedding dimension is developed. Several extensions of (Krull) dimension are proposed, which are then used to generalize singularity notions from commutative algebra. Finally, variants of the homological theorems are shown to hold in equal characteristic. This theory is then applied to Noetherian local rings in order to get: (i) over...

We introduce the notion of residual intersections modules and prove their existence. show that projective dimension one have Cohen-Macaulay intersections, namely they satisfy relevant Artin-Nagata property. then establish a formula for core orientable satisfying certain homological conditions, extending previous results Corso, Polini, Ulrich on modules. Finally, we provide examples classes our ...

Let $(R,underline{m})$ be a commutative Noetherian local ring and $M$ be a non-zero finitely generated $R$-module. We show that if $R$ is almost Cohen-Macaulay and $M$ is perfect with finite projective dimension, then $M$ is an almost Cohen-Macaulay module. Also, we give some necessary and sufficient condition on $M$ to be an almost Cohen-Macaulay module, by using $Ext$ functors.

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 ...

We prove a theorem unifying three results from combinatorial homological and commutative algebra, characterizing the Koszul property for incidence algebras of posets and affine semigroup rings, and characterizing linear resolutions of squarefree monomial ideals. The characterization in the graded setting is via the Cohen-Macaulay property of certain posets or simplicial complexes, and in the mo...

The aim of this paper is to investigate properties of the local cohomology of rings of mixed characteristic that are analogous to Segre products of rings de ned over a eld. The main question is whether the local cohomology can be almost killed in a nite extension (we de ne what this means below). There are two reasons for considering this type of ring. First, there are special properties of the...