‎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...

In this article we study the structure of residual intersections via constructing a finite complex which is acyclic under some sliding depth conditions on the cycles of the Koszul complex. This complex provides information on an ideal which coincides with the residual intersection in the case of geometric residual intersection; and is closely related to it in general. A new success obtained thr...

To reduce to resolving Cohen–Macaulay singularities, Faltings initiated the program of “Macaulayfying” a given Noetherian scheme X. For wide class X, Kawasaki built sought-for modifications, with crucial drawback that his blowups did not preserve locus CM(X)?X, where X is already Cohen–Macaulay. We extend Kawasaki’s methods show every quasi-excellent, has X˜ proper map X˜?X an isomorphism over ...

In this paper, we explore the almost Cohen-Macaulayness of associated graded ring stretched ${\mathfrak m}$ -primary ideals with small first Hilbert coefficient in a Cohen-Macaulay local $(A,{\mathfrak m})$ . particular, structure satisfying equality e1(I) = e0(I) − ℓA(A/I) + 4, where and denote multiplicity coefficient, respectively.

Two formulas for the multiplicity of the fiber cone F (I) = ⊕∞n=0I /mI of an m-primary ideal of a d-dimensional Cohen-Macaulay local ring (R,m) are derived in terms of the mixed multiplicity ed−1(m|I), the multiplicity e(I) and superficial elements. As a consequence, the Cohen-Macaulay property of F (I) when I has minimal mixed multiplicity or almost minimal mixed multiplicity is characterized ...

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....

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.

‎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...