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It is a conjecture due to M. E. Rossi that the Hilbert function of a one-dimensional Gorenstein local ring is non-decreasing. In this article, we show that the Hilbert function is non-decreasing for local Gorenstein rings with embedding dimension four associated to monomial curves, under some arithmetic assumptions on the generators of their defining ideals in the non-complete intersection case. In order to obtain this result, we determine the generators of their tangent cones explicitly by using standard basis computations under these arithmetic assumptions and show that the tangent cones are Cohen-Macaulay. In the complete intersection case, by characterizing certain families of complete intersection numerical semigroups, we give an inductive method to obtain large families of complete intersection local rings with arbitrary embedding dimension having non-decreasing Hilbert functions.
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Let be a Noetherian local ring with the maximal ideal and an ideal of Denote by the fiber cone of This paper characterizes the multiplicity and the Cohen-Macaulayness of fiber cones in terms of minimal reductions of ideals.
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Shiro Goto Yukihide Takayama 《Proceedings of the American Mathematical Society》2007,135(8):2355-2364
We introduce a class of Stanley-Reisner ideals called a generalized complete intersection, which is characterized by the property that all the residue class rings of powers of the ideal have FLC. We also give a combinatorial characterization of such ideals.
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《代数通讯》2013,41(7):3529-3546
Abstract For an ideal I of a Noetherian local ring (R, m ) we consider properties of I and its powers as reflected in the fiber cone F(I) of I. In particular,we examine behavior of the fiber cone under homomorphic image R → R/J = R′ as related to analytic spread and generators for the kernel of the induced map on fiber cones ψ J : F R (I) → F R′(IR′). We consider the structure of fiber cones F(I) for which ker ψ J ≠ 0 for each nonzero ideal J of R. If dim F(I) = d > 0,μ(I) = d + 1 and there exists a minimal reduction J of I generated by a regular sequence,we prove that if grade(G +(I)) ≥ d ? 1,then F(I) is Cohen-Macaulay and thus a hypersurface. 相似文献
6.
Given a simplicial complex, it is easy to construct a generic deformation of its Stanley-
Reisner ideal. The main question under investigation in this paper is how to characterize the
simplicial complexes such that their Stanley-Reisner ideals have Cohen-Macaulay generic deformations.
Algorithms are presented to construct such deformations for matroid complexes, shifted
complexes, and tree complexes.AMS Subject Classification: 13P10, 13C14, 52B20, 52B40, 52B22. 相似文献
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We study monomial curves, toric ideals and monomial algebras associated to 4-generated pseudo symmetric numerical semigroups. Namely, we determine indispensable binomials of these toric ideals, give a characterization for these monomial algebras to have strongly indispensable minimal graded free resolutions. We also characterize when the tangent cones of these monomial curves at the origin are Cohen–Macaulay. 相似文献
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Yuji Yoshino 《Proceedings of the American Mathematical Society》1996,124(9):2641-2647
Let be a Gorenstein complete local ring. Auslander's higher delta invariants are denoted by for each module and for each integer . We propose a conjecture asking if for any positive integers and . We prove that this is true provided the associated graded ring of has depth not less than . Furthermore we show that there are only finitely many possibilities for a pair of positive integers for which .
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We explicitly explain how to fix a wrong statement in a preliminary result of our previous paper on the conductor at a multiplanar singularity. 相似文献
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Joachim Apel 《Journal of Algebraic Combinatorics》2003,17(1):57-74
In 1982 Richard P. Stanley conjectured that any finitely generated
n
-graded module M over a finitely generated
n
-graded K-algebra R can be decomposed as a direct sum M = i = 1
t
i
S
i
of finitely many free modules
i
S
i
which have to satisfy some additional conditions. Besides homogeneity conditions the most important restriction is that the S
i
have to be subalgebras of R of dimension at least depth M.We will study this conjecture for modules M = R/I, where R is a polynomial ring and I a monomial ideal. In particular, we will prove that Stanley's Conjecture holds for the quotient modulo any generic monomial ideal, the quotient modulo any monomial ideal in at most three variables, and for any cogeneric Cohen-Macaulay ring. Finally, we will give an outlook to Stanley decompositions of arbitrary graded polynomial modules. In particular, we obtain a more general result in the 3-variate case. 相似文献
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Kohji Yanagawa 《Proceedings of the American Mathematical Society》1999,127(2):377-383
Let be a monomial ideal of . Bayer-Peeva-Sturmfels studied a subcomplex of the Taylor resolution, defined by a simplicial complex . They proved that if is generic (i.e., no variable appears with the same non-zero exponent in two distinct monomials which are minimal generators), then is the minimal free resolution of , where is the Scarf complex of . In this paper, we prove the following: for a generic (in the above sense) monomial ideal and each integer , there is an embedded prime of . Thus a generic monomial ideal with no embedded primes is Cohen-Macaulay (in this case, is shellable). We also study a non-generic monomial ideal whose minimal free resolution is for some . In particular, we prove that if all associated primes of have the same height, then is Cohen-Macaulay and is pure and strongly connected.
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A useful characterization of the cone of attainable directions is provided.This research was partially supported by the Ministerio de Ciencia y Tecnología (Spain), Project BFM2003-02194. 相似文献
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In this paper we determine the possible Hilbert functions ofa Cohen–Macaulay local ring of dimension d and multiplicitye, in the case where the embedding dimension v satisfies v =e + d – 3 and the Cohen–Macaulay type is less thanor equal to e – 3. 1991 Mathematics Subject Classification:primary 13D40; secondary 13P99. 相似文献
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In this paper, we give equivalent conditions for the factor rings ofthe polynomial ring $k[x,y]$ modulo monomial ideals to be Armendariz rings,where $k$ is a field. For an ideal $I$ with 2 or 3 monomial generators, or $n$ homogeneous monomial generators, such that $k[x,y]/I$ is an Armendariz ring, wecharacterize the minimal generator set $G(I)$ of $I.$ 相似文献
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We explicitly calculate the normal cones of all monomial primes which define the curves of the form , where . All of these normal cones are reduced and Cohen-Macaulay, and their reduction numbers are independent of the reduction. These monomial primes are new examples of integrally closed ideals for which the product with the maximal homogeneous ideal is also integrally closed.
Substantial use was made of the computer algebra packages Maple and Macaulay2.
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Takesi Kawasaki 《Transactions of the American Mathematical Society》2008,360(5):2709-2739
The notion of the Cousin complex of a module was given by Sharp in 1969. It wasn't known whether its cohomologies are finitely generated until recently. In 2001, Dibaei and Tousi showed that the Cousin cohomologies of a finitely generated -module are finitely generated if the base ring is local, has a dualizing complex, satisfies Serre's -condition and is equidimensional. In the present article, the author improves their result. He shows that the Cousin cohomologies of are finitely generated if is universally catenary, all the formal fibers of all the localizations of are Cohen-Macaulay, the Cohen-Macaulay locus of each finitely generated -algebra is open and all the localizations of are equidimensional. As a consequence of this, he gives a necessary and sufficient condition for a Noetherian ring to have an arithmetic Macaulayfication.
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本文用一种新的方法研究了有关一维Noether环的一个不等式.通过用重数替换长度进一步地把该不等式推广到二维的情形. 相似文献
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Gábor Horváth László Székelyhidi Bettina Wilkens 《Journal of Mathematical Analysis and Applications》2014
In 2004 a counterexample was given for a 1965 result of R.J. Elliott claiming that discrete spectral synthesis holds on every Abelian group. Here we present a ring-theoretical approach to this problem, and show that some varieties fail to have spectral synthesis. In particular, we give a new proof for the result of the second author that spectral synthesis does not hold on Abelian groups with infinite torsion free rank. 相似文献
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Several bounds on the number of generators of Cohen-Macaulay ideals known in the literature follow from a simple inequality which bounds the number of generators of such ideals in terms of mixed multiplicities. Results of Cohen and Akizuki, Abhyankar, Sally, Rees and Boratynski-Eisenbud-Rees are deduced very easily from this inequality.