Let R be a commutative noetherian ring. $$\mathsf {H}(R)$$ the quotient of Grothendieck group finitely generated R-modules by subgroup pseudo-zero modules. Suppose that $$\mathbb {R}$$ -vector space {H}(R)_\mathbb {R}=\mathsf {H}(R)\otimes _\mathbb {Z}\mathbb has finite dimension. {C}(R)$$ (resp. {C}_r(R)$$ ) convex cone in spanned maximal Cohen–Macaulay rank r). We explore interior, closure an...

we consider a class of hypergraphs called hypercycles and we show that a hypercycle $c_n^{d,alpha}$ is shellable or sequentially the cohen--macaulay if and only if $nin{3,5}$. also, we characterize cohen--macaulay hypercycles. these results are hypergraph versions of results proved for cycles in graphs.

We characterize when the monomial maximal ideal of a simplicial affine semigroup ring has minimal reduction. When this is case, we study Cohen–Macaulay and Gorenstein properties associated graded provide several bounds for reduction number with respect to

‎we introduce a generalization of the notion of‎ depth of an ideal on a module by applying the concept of‎ local cohomology modules with respect to a pair‎ ‎of ideals‎. ‎we also introduce the concept of $(i,j)$-cohen--macaulay modules as a generalization of concept of cohen--macaulay modules‎. ‎these kind of modules are different from cohen--macaulay modules‎, as an example shows‎. ‎also an art...

Let (R,m) be a local Cohen-Macaulay ring whose m-adic completion R̂ has an isolated singularity. We verify the following conjecture of F.-O. Schreyer: R has finite Cohen-Macaulay type if and only if R̂ has finite Cohen-Macaulay type. We show also that the hypersurface k[[x0, . . . , xd]]/(f) has finite Cohen-Macaulay type if and only if k [[x0, . . . , xd]]/(f) has finite Cohen-Macaulay type, whe...

We reduce Hochster's Canonical Element Conjecture (theorem since 2016) to a localization problem in characteristic free way. prove the validity of new variant Theorem (CET) and explain how deduction from original CET would provide us with proof CET. also show that Balanced Big Cohen-Macaulay Module can be settled by if big Cohen-Macaulayness modification module deduced existence maximal complex

In this note we address the problem of determining the maximum number of points of intersection of two arithmetically Cohen-Macaulay curves in P. We give a sharp upper bound for the maximum number of points of intersection of two irreducible arithmetically Cohen-Macaulay curves Ct and Ct−r in P 3 defined by the maximal minors of a t× (t+ 1), resp. (t− r)× (t − r + 1), matrix with linear entries...

The Chern numbers of the title are the first coefficients (after the multiplicities) of the Hilbert functions of various filtrations of ideals of a local ring (R, m). For a Noetherian (good) filtration A of m-primary ideals, the positivity and bounds for e1(A) are well-studied if R is Cohen-Macaulay, or more broadly, if R is a Buchsbaum ring or mild generalizations thereof. For arbitrary geomet...