On the graded algebra relative to a valuation (original) (raw)
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On the structure of the graded algebra associated to a valuation
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The main goal of this paper is to study the structure of the graded algebra associated to a valuation. More specifically, we prove that the associated graded algebra gr v (R) of a subring (R, m) of a valuation ring Ov, for which Kv := Ov/mv = R/m, is isomorphic to Kv t v(R) , where the multiplication is given by a twisting. We show that this twisted multiplication can be chosen to be the usual one in the cases where the value group is free or the residue field is closed by radicals. We also present an example that shows that the isomorphism (with the trivial twisting) does not have to exist.
International Electronic Journal of Algebra
Let GGG be an abelian group and SSS a given multiplicatively closed subset of a commutative GGG-graded ring AAA consisting of homogeneous elements. In this paper, we introduce and study GGG-graded SSS-Noetherian modules which are a generalization of SSS-Noetherian modules. We characterize GGG-graded SSS-Noetherian modules in terms of SSS-Noetherian modules. For instance, a GGG-graded AAA-module MMM is GGG-graded SSS-Noetherian if and only if MMM is SSS-Noetherian, provided GGG is finitely generated and SSS is countable. Also, we generalize some results on GGG-graded Noetherian rings and modules to GGG-graded SSS-Noetherian rings and modules.
On the associated graded ring of a semigroup ring
Journal of Commutative Algebra, 2010
Let (R, m) be a numerical semigroup ring. In this paper we study the properties of its associated graded ring G(m). In particular, we describe the H 0 M for G(m) (where M is the homogeneous maximal ideal of G(m)) and we characterize when G(m) is Buchsbaum. Furthermore, we find the length of H 0 M as a G(m)-module, when G(m) is Buchsbaum. In the 3-generated numerical semigroup case, we describe the H 0 M in term of the Apery set of the numerical semigroup associated to R. Finally, we improve two characterizations of the Cohen-Macaulayness and Gorensteinness of G(m) given in [2] and [3], respectively. MSC: 13A30; 13H10
Coinduction for semigroup-graded rings
Communications in Algebra, 1999
We describe the graded-simple modules over a semigroup-graded ring in terms of the simple modules over various component subrings.
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Lance Bryant noticed in his thesis [3], that there was a flaw in our paper [2]. It can be fixed by adding a condition, called the BF condition in [3]. We discuss some equivalent conditions, and show that they are fulfilled for some classes of rings, in particular for our motivating example of semigroup rings. Furthermore we discuss the connection to a similar result, stated in more generality, by Cortadella-Zarzuela in [4]. Finally we use our result to conclude when a semigroup ring in embedding dimension at most three has an associated graded which is a complete intersection. 2000 Mathematics Subject Classification: 13A30 If x ∈ R is an element of smallest positive value, i.e. v(x) = e, then xR is a minimal reduction of the maximal ideal, i.e. m n+1 = xm n , for n >> 0. Conversely each minimal reduction of the maximal ideal is a principal ideal generated by an element x of value e. The smallest integer n such that m n+1 = xm n is called the reduction number and we denote it by r. Observe that, if v(x) = e, then Ap e (S) = S \(e+S) = v(R)\v(xR), therefore w j / ∈ v(xR), for j = 0,. .. , e − 1. Consider the m-adic filtration m ⊃ m 2 ⊃ m 3 ⊃. .. . If a ∈ R, we set ord(a) := max{i | a ∈ m i }. If s ∈ S, we consider the semigroup filtration v(m) ⊃ v(m 2) ⊃. .. and set vord(s) := max{i | s ∈ v(m i)}. If a ∈ m i , then v(a) ∈ v(m i) and so ord(a) ≤ vord(v(a)). According to [3], we say that the m-adic filtration is essentially divisible with respect to the minimal reduction xR if, whenever u ∈ v(xR), then there is an a ∈ xR with v(a) = u and ord(a) = vord(u). The m-adic filtration is essentially divisible if there exists a minimal reduction xR such that it is essentially divisible with respect to xR. We fix for all the paper the following notation. Set, for j = 0,. .. , e − 1, b j = max{i|w j ∈ v(m i)}, and let c j = max{i|w j ∈ v(m i + xR)}. Note that the numbers b j 's do not depend on the minimal reduction xR, on the contrary the c j 's depend on xR. Lemma 1.1 If I and J are ideals of R, then v(I +J) = v(I)∪v(J) is equivalent to v(I ∩ J) = v(I) ∩ v(J).
Journal of the Mathematical Society of Japan, 1978
In this paper, we study a Noetherian graded ring RRR and the category of graded R-modules. We consider injective objects of this category and we define the graded Cousin complex of a graded R-module MMM . These concepts are essential in this paper
Valuation Derived from Graded Ring and Module and Krull Dimension Properties
In this paper we show if R is a graded ring then we can define a valuation on R induced by graded structure, and we prove some properties and relations for R. Later we show that if R is a graded ring and M a graded R-module then there exists a valuation on of M which is derived from graded structure and also we prove some properties and relations for R. In the following we give a new method for finding the Kurll dimension of a valuation ring.