Formulas are obtained in terms of complete reductions for the bigraded components of local cohomology modules of bigraded Rees algebras of 0-dimensional ideals in 2-dimensional Cohen-Macaulay local rings. As a consequence, cohomological expressions for the coefficients of the Bhattacharya polynomial of such ideals are obtained.

This paper shows that Cohen-Macaulay algebras can be algebraically approximated in such a way their Cohen-Macaulayness and minimal Betti numbers are preserved. is achieved by showing finitely generated modules over power series rings manner preserves diagrams of initial exponents numbers. These results also applied to obtain an approximation result for flat homomorphisms from algebras.

Let $(R,\mathfrak m)$ be a commutative Noetherian local ring. There is variety of nice results about approximately Cohen-Macaulay rings. These were done by Goto. In this paper we prove some these for modules and generalize the concept rings to modules. It seen that when $M$ an module, any proper ideal $I$ have $grade(I,M) \geq \dim_R M -\dim_R M/IM -1$. Specially $R$ itself, obtain interval $gr...

1.1 Statement (Matijevic–Roberts type theorem (MRTT)). Let C be a class of noetherian local rings. Let R be a noetherian Z-graded ring, and P its prime ideal. Let P ∗ be the prime ideal generated by the all homogeneous elements of P . If RP ∗ ∈ C, then RP ∈ C. Clearly, the truth of the statement depends on the choice of C. Nagata conjectured the Matijevic–Roberts type theorem for the case that ...