A result of Watanabe and Yoshida says that an unmixed local ring of positive characteristic is regular if and only if its Hilbert-Kunz multiplicity is one. We show that, for fixed p and d, there exist a number ǫ(d, p) > 0 such that any nonregular unmixed ring R its Hilbert-Kunz multiplicity is at least 1+ ǫ(d, p). We also show that local rings with sufficiently small Hilbert-Kunz multiplicity a...

In this expository paper we survey results proved during the last fifty years that relate Hilbert coefficients e0(I) and e1(I) of an m-primary ideal I in a Cohen-Macaulay local ring (R, m) with depth of the associated graded ring G(I). Several results in this area follow from two theorems of S. Huckaba and T. Marley. These were proved using homological techniques. We provide simple proofs using...

This paper deals with the effective computation of the radical of certain polynomial ideals. Let k be a characteristic zero field, f1, . . . , fn−r ∈ k[X1, . . . , Xn] a regular sequence with d := maxj deg fj , = the generated ideal, √ = its radical, and suppose that the factor ring k[X1, . . . , Xn]/ √ = is a Cohen-Macaulay ring. Under these assumptions we exhibit a single exponential algorith...

In this paper, we discuss some necessary and sufficient conditions for a curve to be arithmetically Cohen-Macaulay, in terms of its general hyperplane section. We obtain a characterization of the degree matrices that can occur for points in the plane that are the general plane section of a non arithmetically Cohen-Macaulay curve of P. We prove that almost all the degree matrices with positive s...

We show that an excellent local domain of characteristic p has a separable big Cohen–Macaulay algebra. In the course of our work we prove that an element which is in the Frobenius closure of an ideal can be forced into the expansion of the ideal to a module-finite separable extension ring.

The so-called ’change-of-ring’ results are well-known expressions which present several connections between projective, injective and flat dimensions over the various base rings. In this note we extend these results to the Gorenstein dimensions over Cohen-Macaulay local rings.

A new \compressed straight" basis for the polynomial ring Z[ti;j ] is constructed. This basis is then employed to study the lattice of representations generated by the representations associated with row-convex shapes under the operations of intersection and linear span. Applications to ascertaining the Cohen-Macaulay property for rings associated to elements of this lattice are also given.

The so-called ’change-of-ring’ results are well-known expressions which present several connections between projective, injective and flat dimensions over the various base rings. In this note we extend these results to the Gorenstein dimensions over Cohen-Macaulay local rings.

R.G. and Perez proved that under certain conditions the test ideal of a module closure agrees with trace closure. We use this fact to compute ideals various rings respect closures coming from their indecomposable maximal Cohen–Macaulay modules. also give an easier way hypersurface ring in three variables particular type matrix factorization.