Clare D'Cruz, Tony J. Puthenpurakal
Let $(A,\m)$ be a Noetherian local ring with infinite residue field and let $I$ be an ideal in $A$ and let $F(I) = \oplus_{n \geq 0}I^n/\m I^n$ be the fiber-cone of $I$. We prove certain relations among the Hilbert coefficients of $F(I)$ when the $a$-invariant of the associated graded ring $G(I)$ is negative.
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We give bounds on the Castelnuovo-Mumford regularity of the associated graded module of an arbitrary good filtration and of its fiber cone. These bounds extend previous results of Rossi-Trung-Valla and Linh.
We describe the cone of Hilbert functions of artinian graded modules finitely generated in degree 0 over the polynomial ring R = k[x, y] with the non-standard grading deg(x) = 1 and deg(y) = n, where n is any natural number.
In this paper we study the Hilbert coefficients the fiber cone. We also compare the depths of the fiber cone and the associated graded ring.
In this article, we study the Castelnuovo-Mumford regularity and Gorenstein properties of the fiber cone. We obtain upper bounds for the Castelnuovo-Mumford regularity of the fiber cone and obtain sufficient conditions for the regularity of the fiber cone to be equal to that of the Rees algebra. We obtain a formula for the canonical module of the fiber cone and use it to study the Gorenstein property of the fiber cone.
In this paper we give a unified approach for several results concerning the fiber cone of ideals
The paper characterizes the multiplicity and the Cohen-Macaulayness of fiber cones of good filtrations in terms of minimal reductions of these filtrations.
Criteria are given in terms of certain Hilbert coefficients for the fiber cone F(I) of an m-primary ideal I in a Cohen-Macaulay local ring (R,m) so that it is Cohen-Macaulay or has depth at least dim(R)-1. A version of Huneke's fundamental lemma is proved for fiber cones. S. Goto's results concerning Cohen-Macaulay fiber cones of ideals with minimal multiplicity are obtained as consequences.
Foa a given local ring, we study the fiber cone of ideals with analytic spread one. In this case, the fiber cone has a strucure as a module over its Noether normalization which is a polynomial ring in one variable over the residue field. One may then apply the structure theorem for graded modules over a graded principal domain to get a complete descriptionof the fiber cone as a module. We analyze this structure in order to study and characterize in terms of the ideal itself the aritmetical properties and other numerical invariants of the fiber cone as multiplicity, reduction number or Castelnu…
We determine in an explicit way the depth of the fiber cone and its relation ideal for classes of monomial ideals in two variables. These classes include concave and convex ideals as well as symmetric ideals.
Let $R$ be a polynomial ring over a field. We describe the extremal rays and the facets of the cone of local cohomology tables of finitely generated graded $R$-modules of dimension at most two. Moreover, we show that any point inside the cone can be written as a finite linear combination, with positive rational coefficients, of points belonging to the extremal rays of the cone. We also provide algorithms to obtain decompositions in terms of extremal points and facets.
Cohen Macaulay property of fiber cones of ideals is characterized in terms of its Hilbert series. Hilbert series of fiber cones of ideals with minimal mixed multiplicity is calculated. It is proved that the fiber cone of an m-primary ideal I with minimal mixed multiplicity is Cohen-Macaulay if and only if its reduction number is atmost one. Hilbert series of fiber cones of ideals generated by quadratic sequences in standard graded rings is computed by deforming it to a face ring of a simplicial complex. Applications are given to defining ideals of monomial projective space curves lying on the …
Canonical traces of graded fiber products: applications to disconnected Stanley--Reisner rings
本文研究了不连通Stanley-Reisner环的规范迹，通过建立graded纤维积的显式公式并应用于不连通simplicial复形，最终给出了不连通复形的分类结果。
In this paper we show that there is a close relationship between the invariants characterizing the homogeneous vanishing of the local cohomology of the Rees algebra and the associated graded ring for the good filtrations case. We obtain relationships between the Castelnuovo-Mumford regularity of the fiber cone, associated graded ring, Rees algebra and reduction number for the good filtrations case.
We give a complete description of the cone of Betti diagrams over a standard graded hypersurface ring of the form k[x,y]/<q>, where q is a homogeneous quadric. We also provide a finite algorithm for decomposing Betti diagrams, including diagrams of infinite projective dimension, into pure diagrams. Boij--Soederberg theory completely describes the cone of Betti diagrams over a standard graded polynomial ring; our result provides the first example of another graded ring for which the cone of Betti diagrams is entirely understood.
Defining ideals of Cohen-Macaulay fiber cones
本文针对交换 Noether 局部环中的理想，给出了 Cohen-Macaulay 纤维锥的定义理想的构造方法。
We consider the fiber cone of monomial ideals. It is shown that for monomial ideals $I\subset K[x,y]$ of height $2$, generated by $3$ elements, the fiber cone $F(I)$ of $I$ is a hypersurface ring, and that $F(I)$ has positive depth for interesting classes of height $2$ monomial ideals $I\subset K[x,y]$, which are generated by $4$ elements. For these classes of ideals we also show that $F(I)$ is Cohen--Macaulay if and only if the defining ideal $J$ of $F(I)$ is generated by at most 3 elements. In all the cases a minimal set of generators of $J$ is determined.
Two formulas for the multiplicity of the fiber cone $F(I)=\oplus_{n=0}^{\infty} I^n/\m I^n$ of an $\m$-primary ideal of a $d$-dimensional Cohen-Macaulay local ring $(R,\m)$ are derived in terms of the mixed multiplicity $e_{d-1}(\m | I),$ the multiplicity $e(I)$ and superficial elements. As a consequence, the Cohen-Macaulay property of $F(I)$ when $I$ has minimal mixed multiplicity or almost minimal mixed multiplicity is characterized in terms of reduction number of $I$ and lengths of certain ideals. We also characterize Cohen-Macaulay and Gorenstein property of fiber cones of $\m$-primary ide…
Betti numbers for connected sums of graded Gorenstein artinian algebras
本文确定了分次阿廷Gorenstein代数连通和的分次Betti数，并证明加倍代数的连通和等价于纤维积环的加倍，从而将连通和构造与加倍构造联系起来。
Let $(R,\mathfrak{m})$ be a Cohen-Macaulay local ring, $I$ an $\mathfrak{m}$-primary ideal of $R$ and $J$ its minimal reduction. We study the depths of $F(I)$ under certain depth assumptions on $G(I)$ and length condition on quotients of powers of $I$ and $J$, namely $\sum_{n\geq0}λ(\mathfrak{m}I^{n+1}/\mathfrak{m}JI^n)$ and $\sum_{n\geq0}λ(\mathfrak{m}I^{n+1} \cap J/\mathfrak{m}JI^n)$.
Minimal values of multiplicities of ideals have a strong relation with the depth of blowup algebras. In this paper, we introduce the notion of Goto-minimal $j$-multiplicity for ideals of maximal analytic spread. In a Cohen-Macaulay ring, inspired by the work of S. Goto, A. Jayanthan, T. Puthenpurakal, and J. Verma, we study the interplay among this new notion, the notion of minimal $j$-multiplicity introduced by C. Polini and Y. Xie, and the Cohen-Macaulayness of the fiber cone of ideals satisfying certain residual assumptions. We are also able to provide a bound on the reduction number of ide…
Betti cones over fibre products
本文研究域上标准分次代数的纤维积$R$上有限生成分次模的合冲结构，证明了正则度有限且投射维数无限的模的存在性蕴含$R$是Koszul代数，并刻画了depth为1时Betti锥极射线由纯模Betti表张成的等价条件。
In this article, we show that the depths of the associated graded ring and fiber cone of a lex-segment ideal in $K[x,y]$ are equal.
Fiber cones of 0-dimensional ideals with almost minimal multiplicity in Cohen-Macaulay local rings are studied. Ratliff-Rush closure of filtration of ideals with respect to another ideal is introduced. This is used to find a bound on the reduction number with respect to an ideal. Rossi's bound on reduction number in terms of Hilbert coefficients is obtained as a consequence. Sufficient conditions are provided for the fiber cone of 0-dimensional ideals to have almost maximal depth. Hilbert series of such fiber cones are also computed.
We contribute a new algebraic method for computing the orthogonal projections of a point onto a rational algebraic surface embedded in the three dimensional projective space. This problem is first turned into the computation of the finite fibers of a generically finite dominant rational map: a congruence of normal lines to the rational surface. Then, an in-depth study of certain syzygy modules associated to such a congruence is presented and applied to build elimination matrices that provide universal representations of its finite fibers, under some genericity assumptions. These matrices depen…
Generic Bourbaki ideals were introduced by Simis, Ulrich and Vasconcelos to study the Cohen-Macaulay property of Rees algebras of modules. In this article we prove that the same technique can sometimes be used to investigate the Cohen-Macaulay property of fiber cones of modules and to study the defining ideal of Rees algebras. This is possible as long as the Rees algebra of a given module $E$ is a deformation of the Rees algebra of a generic Bourbaki ideal $I$ of $E$. Our main technical result provides a deformation condition that in fact extends the applicability of generic Bourbaki ideals to…
In this article, we give the explicit minimal free resolution of the associated graded ring of certain affine monomial curves in affine 4-space based on the standard basis theory. As a result, we give the minimal graded free resolution and compute the Hilbert function of the tangent cone of these families.
Let (R, m) be a Cohen-Macaulay local ring and I be an m-primary ideal. We introduce ideals of almost minimal mixed multiplicty which are analogues of ideals studied by J. Sally. The main theorem describes the Hilbert series of fiber cones of these ideals.
The paper answers a question by Jonathan Wahl,giving examples of regular surfaces S (so their canonical ring is a Gorenstein graded ring) having the following properties: 1) their canonical divisor K_S = rL is a positive multiple of an ample divisor L 2) the graded ring R := R (X,L ) associated to L is not Cohen-Macaulay. In the appendix Wahl shows how these examples lead to the existence of Cohen-Macaulay singularities with K_X Q -Cartier which are not Q -Gorenstein, since their index one cover is not Cohen- Macaulay.
Let $R$ be a finitely generated positively graded algebra over a Noetherian local ring $B$, and $\mathfrak{m} = [R]_+$ be the graded irrelevant ideal of $R$. We provide a local criterion characterizing the $B$-freeness of all the local cohomology modules $\text{H}_\mathfrak{m}^i(M)$ of a finitely generated graded $R$-module $M$. We show that fiber-full modules are exactly the ones that satisfy this criterion. When we change $B$ by an arbitrary Noetherian ring $A$, we study the fiber-full locus of a module in $\text{Spec}(A)$: we show that the fiber-full locus is always an open subset of $\text…
From a local ring to its associated graded algebra
本文引入一种同质化技术，将关联格雷德环与特殊纤维及局部环与通用纤维联系起来，证明了关于环连通性的尖锐结果，并构造了一类不满足阿比扬卡尔不等式的局部域，但当环在codimension一连通时成立。
In this short paper, we show that the fiber cones of rational normal scrolls are Cohen-Macaulay. As an application, we compute their Castelnuovo-Mumford regularities and $\mathbf{a}$-invariants, as well as the reduction number of the defining ideals of the rational normal scrolls. We also characterize the Gorensteinness of the fiber cone.
Lim Ulrich sequences and Boij-Söderberg cones
本文通过引入lim Ulrich序列，将Boij–Söderberg锥结构理论从多项式环推广到所有具有线性Noether正规化的有限生成分次环。
We describe the cone of Betti tables of all finitely generated graded modules over the homogeneous coordinate ring of three non-collinear points in the projective plane. We also describe the cone of Betti tables of all finite length modules.
On the behavior of $F$-thresholds with respect to the fibers of blow-ups and $F$-rationality
本文研究了 $F$-阈值在吹脱纤维下的行为，并证明了格雷德环的 $F$-理性性可传递到局部环。
Mapping cones of monomial ideals over exterior algebras
本文研究外代数上单项式理想的迭代映射锥，利用 Cartan 复形构造极小分次自由分解、显式描述微分，并给出包含强稳定理想的一类新理想的分次 Betti 数公式。
Cohen-Macaulayness of associated graded rings of Gorenstein monomial curves
本文研究 4 维空间中 Gorenstein 非完全交单项式曲线的切锥 Cohen-Macaulay 性，利用定义理想的极小生成元集给出判定准则，并刻画了相应理想极小生成元个数的可能取值。
Let $(A, \mathfrak{m})$ be a Noetherian local ring and $\mathfrak{F}=(I_{n})_{n\geq 0}$ a filtration. In this paper, we study the Gorenstein properties of the fiber cone $F(\mathfrak{F})$, where $\mathfrak{F}$ is a Hilbert filtration. Suppose that $F(\mathfrak{F})$ and $G(\mathfrak{F})$ are Cohen-Macaulay. If in addition, the associated graded ring $G(\mathfrak{F})$ is Gorenstein; similarly to the $I$-adic case, we obtain a necessary and sufficient condition, in terms of lengths and minimal number of generators of ideals, for Gorensteiness of the fiber cone. Moreover, we find a description of …
Boij-Söderberg Theory views the Betti diagrams of graded modules over polynomial rings as vectors in a rational vector space, and studies the cone that these vectors generate (called a 'Betti Cone'). The objects of study in this paper are the Betti cones generated by edge ideals; in particular this paper will present and prove a formula for the dimensions of these cones, and for the subcones generated by edge ideals of specific heights.
Let ${\cal R}_{\mathbb{K}}[H]$ be the Hibi ring over a field $\mathbb{K}$ on a finite distributive lattice $H$, $P$ the set of join-irreducible elements of $H$ and $ω$ the canonical ideal of ${\cal R}_{\mathbb{K}}[H]$. We show the powers $ω^{(n)}$ of $ω$ in the group of divisors $\mathrm{Div}({\cal R}_{\mathbb{K}}[H])$ is identical with the ordinal powers of $ω$, describe the $\mathbb{K}$-vector space basis of $ω^{(n)}$ for $n\in\mathbb{Z}$. Further, we show that the fiber cones $\bigoplus_{n\geq 0}ω^n/\mathfrak{m}ω^n$ and $\bigoplus_{n\geq0}(ω^{(-1)})^n/\mathfrak{m}(ω^{(-1)})^n$ of $ω$ and $ω…
Let $A$ be a Noetherian flat $K[t]$-algebra, $h$ an integer and let $N$ be a graded $K[t]$-module, we introduce and study "$N$-fiber-full up to $h$" $A$-modules. We prove that an $A$-module $M$ is $N$-fiber-full up to $h$ if and only if $\mathrm{Ext}^i_A(M, N)$ is flat over $K[t]$ for all $i\le h-1$. And we show some applications of this result extending the recent result on squarefree Gröbner degenerations by Conca and Varbaro.
DG Algebra Structures on Mapping Cones with an Application to Edge Ideals
本文推广了Herzog-Takayama的构造，为基于顶点覆盖悬垂的新图边理想给出最小自由分解，并探讨DG代数结构的保持条件。
Given any equigenerated monomial ideal $I$ with the property that the defining ideal $J$ of the fiber cone $ F(I)$ of $I$ is generated by quadratic binomials, we introduce a matrix such that the set of its binomial $2$-minors is a generating set of $J$. In this way, we characterize the fiber cone of sortable and Freiman ideals.
We first provide here a very short proof of a refinement of a theorem of Kodiyalam and Cutkosky, Herzog and Trung on the regularity of powers of ideals. This result implies a conjecture of Hà and generalizes a result of Eisenbud and Harris concerning the case of ideals primary for the graded maximal ideal in a standard graded algebra over a field. It also implies a new result on the regularities of powers of ideal sheaves. We then compare the cohomology of the stalks and the cohomology of the fibers of a projective morphism to the effect of comparing the maximum over fibers and over stalks of …
Flat morphisms with regular fibers do not preserve $F$-rationality
本文构造了特征为 $p$ 的域上的标准分次 $F$-有理环 $R$，证明其在代数闭包或张量积下不再保持 $F$-有理性，从而说明具有正则纤维的平坦态射不保持 $F$-有理性。
Given a numerical semigroup ring $R=k[\![S]\!]$, an ideal $E$ of $S$ and an odd element $b \in S$, the numerical duplication $S \! \Join^b \! E$ is a numerical semigroup, whose associated ring $k[\![S \! \Join^b \! E]\!]$ shares many properties with the Nagata's idealization and the amalgamated duplication of $R$ along the monomial ideal $I=(t^e \mid e\in E)$. In this paper we study the associated graded ring of the numerical duplication characterizing when it is Cohen-Macaulay, Gorenstein or complete intersection. We also study when it is a homogeneous numerical semigroup, a property that is …
Let $X$ be a reduced closed subscheme in $\mathbb P^n$. As a slight generalization of property $\textbf{N}_p$ due to Green-Lazarsfeld, we can say that $X$ satisfies property $\textbf{N}_{2,p}$ scheme-theoretically if there is an ideal $I$ generating the ideal sheaf $\mathcal I_{X/¶^n}$ such that $I$ is generated by quadrics and there are only linear syzygies up to $p$-th step (cf. \cite{EGHP1}, \cite{EGHP2}, \cite{V}). Recently, many algebraic and geometric results have been proved for projective varieties satisfying property $\textbf{N}_{2,p}$(cf. \cite{CKP}, \cite{EGHP1}, \cite{EGHP2} \cite …
We prove that the generalized Hilbert-Kunz function of a graded module $M$ over a two-dimensional standard graded normal $K$-domain over an algebraically closed field $K$ of prime characteristic $p$ has the form $gHK(M,q)=e_{gHK}(M)q^{2}+γ(q)$, with rational generalized Hilbert-Kunz multiplicity $e_{gHK}(M)$ and a bounded function $γ(q)$. Moreover we prove that if $R$ is a $\mathbb{Z}$-algebra, the limit for $p\rightarrow+\infty$ of the generalized Hilbert-Kunz multiplicity $e_{gHK}^{R_p}(M_p)$ over the fibers $R_p$ exists and it is a rational number.
Let $R$ be a standard graded Noetherian algebra over an Artinian local ring. Motivated by the work of Achilles and Manaresi in intersection theory, we first express the multiplicity of $R$ by means of local $j$-multiplicities of various hyperplane sections. When applied to a homogeneous inclusion $A\subseteq B$ of standard graded Noetherian algebras over an Artinian local ring, this formula yields the multiplicity of $A$ in terms of that of $B$ and of local $j$-multiplicities of hyperplane sections along ${\rm Proj}\,(B)$. Our formulas can be used to find the multiplicity of special fiber ring…
Let $(S, \mathfrak n) $ be a regular local ring and let $I \subseteq \mathfrak n^2 $ be a perfect ideal of $S. $ Sharp upper bounds on the minimal number of generators of $I$ are known in terms of the Hilbert function of $R=S/I. $ Starting from information on the ideal $I, $ for instance the minimal number of generators, a difficult task is to find good bounds on the minimal number of generators of the leading ideal $I^* $, which defines the tangent cone of $R$ or to give information on its graded structure. Motivated by papers of S.C. Kothari, S. Goto ${\it{et al.}}$ concerning the leading id…
We prove that the arithmetic degree of a graded or local ring is bounded above by the arithmetic degree of any of its associated graded rings with respect to ideals $I$ in $A$. In particular, if $Spec (A)$ is equidimensional and has an embedded component in dimension $i$, then the normal cone of $Spec (A)$ along $V(I)$ has an embedded component in dimension $i$ too.
In this paper we describe the structure of the tangent cone of a numerical semigroup ring $A=k[[S]] \subseteq k[[t]]$ with multiplicity $e$ (as a module over the Noether normalization determined by the fiber cone of the ideal generated by $t^e$) in terms of some classical invariants of the corresponding numerical semigroup. Explicit computations are also made by using the GAP system.
A concrete description of all graded maximal Cohen-Macaulay modules of rank one and two over the affine cone of the simple node (a non-isolated singularity) is given. For this purpose we construct an alghoritm that provides extensions of MCM modules over an arbitrary hypersurface.
We characterize the tight closure of graded primary ideals in a homogeneous coordinate ring over an elliptic curve by numerical conditions and we show that it is in positive characteristic the same as the plus closure.
In this paper, we are mainly concerned with the blow-up algebras of the secant varieties of balanced rational normal scrolls. In the first part, we give implicit defining equations of their associated Rees algebras and fiber cones. Consequently, we can tell that the fiber cones are Cohen--Macaulay normal domains. Meanwhile, these fiber cones have rational singularities in characteristic zero, and are $F$-rational in positive characteristic. The Gorensteinness of the fiber cones can also be characterized. In the second part, we compute the Castelnuovo--Mumford regularities and $\mathbf{a}$-inva…
Regularity is bounded on a quasi-excellent Noetherian scheme
本文证明拟优良诺特概形上切锥的梯度Betti表只有有限种可能，从而正则性有界。
We describe a method for doing computations with Orlov's equivalence between the bounded derived category of certain hypersurfaces and the stable category of graded matrix factorisations of the polynomials describing these hypersurfaces. In the case of a smooth elliptic curve over an algebraically closed field we describe the indecomposable graded matrix factorisations of rank one. Since every indecomposable Maximal Cohen-Macaulay module over the completion of a smooth cubic curve is gradable, we obtain explicit descriptions of all indecomposable rank one matrix factorisations of such potentia…
We study the closed convex hull of various collections of Hilbert functions. Working over a standard graded polynomial ring with modules that are generated in degree zero, we describe the supporting hyperplanes and extreme rays for the cones generated by the Hilbert functions of all modules, all modules with bounded a-invariant, and all modules with bounded Castelnuovo-Mumford regularity. The first of these cones is infinite-dimensional and simplicial, the second is finite-dimensional but neither simplicial nor polyhedral, and the third is finite-dimensional and simplicial.
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