Every local ring is dominated by a one-dimensional local ring

Let A be a one-dimensional reduced local ring with finite normalization. Let G(A) be the associated graded ring of A. In this paper we analyse the two conditions: Proj (G(A)) reduced and G(A) reduced together with their relations with the equality H(n)=HR (n), where H(n) and HR (n) are respectively the Hilbert function of the ring A and of the local ring R of G(A)red=G(A)/nil (G(A)) at its homogenous maximal ideal. As a consequence of our results we get a class of ordinary singularities with H(n) locally decreasing for any embedding dimension H(1) greater then 4.

Generalized Loewy length of Cohen-Macaulay local and graded rings

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The development of the theory of local rings has been greatly stimulated by the importance of the applications to algebraic geometry, but it is none the less true that this stimulus has produced a theory which, on aesthetic grounds, is somewhat unsatisfactory. In the first place, if a local ring Q arises in the ordinary way from a geometric problem, then Qwill have the same characteristic as its residue field. It is partly for this reason that our knowledge of equicharacteristic local rings is much more extensive than it is of those local rings which present the case of unequal characteristics. Again, in the geometric case, the integral closure of Q in its quotient field will be a finite Q-module. Here, once more, we have a special situation which it would be desirable to abandon from the point of view of a general abstract theory.

One-dimensional stable rings

Reiten has demonstrated that the trivial Hochschild extension of a Cohen-Macaulay local ring by a canonical module is a Gorenstein local ring. Here it is proved that any commutative extension of a Cohen-Macaulay local ring by a canonical module is a Gorenstein ring. Also Gorenstein extensions of a local Cohen-Macaulay ring by a module are studied.

On Residually Reducible Representations on Local Rings

Let R = ⨁ n ≥ 0 R n R = \bigoplus _{n \geq 0} R_n be a Noetherian homogeneous ring with one-dimensional local base ring ( R 0 , m 0 ) (R_0, {\mathfrak m}_0) . Let q 0 ⊆ R 0 {\mathfrak q}_0 \subseteq R_0 be an m 0 {\mathfrak m}_0 -primary ideal, let M M be a finitely generated graded R R -module and let i ∈ N 0 i \in {\mathbb N}_0 . Let H R + i ( M ) H^i_{R_+}(M) denote the i i -th local cohomology module of M M with respect to the irrelevant ideal R + := ⨁ n &gt; 0 R n R_+:= \bigoplus _{n &gt; 0} R_n of R R . We show that the first Hilbert-Samuel coefficient e 1 ( q 0 , H R + i ( M ) n ) e_1 \big ( {\mathfrak q}_0, H^i_{R_+}(M)_n \big ) of the n n -th graded component of H R + i ( M ) H^i_{R_+}(M) with respect to q 0 {\mathfrak q}_0 is antipolynomial of degree &gt; i &gt; i in n n . In addition, we prove that the postulation numbers of the components H R + i ( M ) n H^i_{R_+} (M)_n with respect to q 0 {\mathfrak q}_0 have a common upper bound.

Let (R, m) be a Noetherian, one-dimensional, local ring, with |R/m|=∞. We study when its associated graded ring G(m) is Buchsbaum; in particular, we give a theoretical characterization for G(m) to be Buchsbaum not Cohen–Macaulay. Finally, we consider the particular case of R being the semigroup ring associated to a numerical semigroup S: we introduce some invariants of S, and we use them in order to give a necessary and a sufficient condition for G(m) to be Buchsbaum.

A criterion is established, in terms of the Massey products structure carried by the homology of partial resolutions, for the Yoneda cohomology algebra ExtA(k, k) to be a free module over the universal envelope of a free graded Lie subalgebra.It is shown that several conjectures on the (co)homology of local rings, in particular on the asymptotic behaviour of the Betti numbers, follow from such a structure.For all rings with edim A -depths * 3, and for Gorenstein rings with edim A -depth A = 4, the following dichotomy is proved: Either A is a complete intersection, or xiA(k, k) contains a nonabelian free graded Lie subalgebra.0. Ten conjectures on the homology of local rings (in place of Introduction).In this paper (A, rrt, k) will denote a noetherian local ring A with maximal ideal m and residue field k.A well-known homological invariant of A is its Poincar series 00 ; = 0 where b (= b(A)) is the /'th Betti number of A, i.e. the rank of the /'th free module in the unique, up to noncanonical isomorphism, yl-free resolution X of k.Much of the work on the homology of local rings has been motivated by the following famous conjecture:C0. PA(t) represents a rational function.Recently this has been settled in the negative by D. Anick,1 who constructed for each k a A>algebra A requiring 5 generators for m and having m3 = 0, with transcendental Poincar series.On the other hand, more than 20 years of research have produced different kinds of conditions, which ensure the rationality of PA(t).Before collecting for further reference several of them in a theorem, we introduce some notation, to be kept for the rest of the paper.edim A = embedding dimension of A := dimt m/m2 (= bx(A)); dim A -Krull dimension of A ;depth A = maximal length of an ^-regular sequence in m; d(A) = complete intersection defect of A '-dim^ a/no.-(dim R -dim A), where (R, n) is a regular local ring such that R/a is isomorphic to the m-adic

An application of the almost purity theorem to the homological conjectures

In this paper, few weights linear codes over the local ring $$R={\mathbb {F}}_p+u{\mathbb {F}}_p+v{\mathbb {F}}_p+uv{\mathbb {F}}_p,$$ with $$u^2=v^2=0, uv=vu,$$ are constructed by using the trace function defined on an extension ring of degree m of R. These trace codes have the algebraic structure of abelian codes. Their weight distributions are evaluated explicitly by means of Gauss sums over finite fields. Two different defining sets are explored. Using a linear Gray map from R to $${\mathbb {F}}_p^4,$$ we obtain several families of p-ary codes from trace codes of dimension 4m. For two different defining sets: when m is even, or m is odd and $$p\equiv 3 ~(\mathrm{mod} ~4).$$ Thus we obtain two family of p-ary abelian two-weight codes, which are directly related to MacDonald codes. When m is even and under some special conditions, we obtain two classes of three-weight codes. In addition, we give the minimum distance of the dual code. Finally, applications of the p-ary image codes in secret sharing schemes are presented.

Extended plus closure in complete local rings

In this article, we prove a strong version of the local Bertini theorem for normality on local rings in mixed characteristic. The main result asserts that a generic hyperplane section of a normal, Cohen–Macaulay, and complete local domain of dimension at least 3 is normal. Applications include the study of characteristic ideals attached to torsion modules over normal domains, which is fundamental in the study of Euler system theory, Iwasawa's main conjectures, and the deformation theory of Galois representations.

It is shown that each commutative Artin local ring having each of its ideals generated by two elements is the homomorphic image of a one-dimensional local complete intersection ring which also has each of its ideals generated by two elements. It is indicated how this can be applied to show that the property that each ideal is projective over its endomorphism ring does not pass to homomorphic images, and in determining the commutative group rings with the two-generator property.