Pinched Veronese rings are formed by removing an algebra generator from a subring of polynomial ring. We study the homological properties such rings, including Cohen-Macaulay, Gorenstein, and complete intersection properties. Greco Martino classified Cohen-Macaulayness pinched maximum entry exponent vector monomial; we re-prove their results with semigroup methods correct omission small class e...

The homological property of the associated graded ring an ideal is important problem in commutative algebra and algebraic geometry. In this paper we explore almost Cohen–Macaulayness stretched $\mathfrak{m}$-primary ideals case where reduction number attains minimal value a Cohen–Macaulay local $(A,\mathfrak{m})$. As application, present complete descriptions with small number.

The arithmetic degree, the smallest extended degree, and the homological degree are invariants that have been proposed as alternatives of the degree of a module if this module is not Cohen-Macaulay. We compare these degree functions and study their behavior when passing to the generic initial or the lexicographic submodule. This leads to various bounds and to counterexamples to a conjecture of ...

In this paper we use "ring changed'' Gorenstein homologicaldimensions to define Cohen-Macaulay injective, projective and flatdimensions. For doing this we use the amalgamated duplication of thebase ring with semi-dualizing ideals. Among other results, we prove that finiteness of these new dimensions characterizes Cohen-Macaulay rings with dualizing ideals.

We define very proper intersections of modules and projective subschemes. It turns out that equidimensional locally Cohen-Macaulay modules intersect very properly if and only if they intersect properly. We prove a Bezout theorem for modules which meet very properly. Furthermore, we show for equidimensional subschemes X and Y : If they intersect properly in an arithmetically Cohen-Macaulay subsc...

When studying a graded module $M$ over the Cox ring of smooth projective toric variety $X$, there are two standard types resolutions commonly used to glean information: free and vector bundle its sheafification. Each approach comes with own challenges. There is geometric information that fail encode, while can resist study using algebraic combinatorial techniques. Recently, Berkesch, Erman, Smi...

For a finitely generated A-module M we define the dimension filtration M = {Mi}0≤i≤d, d = dimA M, where Mi denotes the largest submodule of M of dimension ≤ i. Several properties of this filtration are investigated. In particular, in case the local ring (A,m) possesses a dualizing complex, then this filtration occurs as the filtration of a spectral sequence related to duality. Furthermore, we c...

For a Noetherian local domain A, there exists an upper bound Nτ (A) on the minimal number of generators of any height two ideal a for whichA/a is Cohen-Macaulay of type τ . If A contains an infinite field, then we may take Nτ (A) := (τ + 1)ehom(A), where ehom(A) is the homological multiplicity of A.

For a Noetherian local domain A, there exists an upper bound Nτ (A) on the minimal number of generators of any height two ideal a for which A/a is Cohen-Macaulay of type τ . More precisely, we may take Nτ (A) := (τ + 1)eh(A), where eh(A) is the homological multiplicity of A.