Koji Nishida, Bernd Ulrich
The j-multiplicity is an invariant that can be defined for any ideal in a Noetherian local ring $(R, m)$. It is equal to the Hilbert-Samuel multiplicity if the ideal is $m$-primary. In this paper we explore the computability of the j-multiplicity, giving another proof for the length formula and the additive formula.
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The notion of epsilon multiplicity was originally defined by Ulrich and Validashti as a limsup and they used it to detect integral dependence of modules. It is important to know if it can be realized as a limit. In this article we show that the relative epsilon multiplicity of reduced Noetherian graded algebras over an excellent local ring exists as a limit. An important special case of a result of Cutkosky concerning epsilon multiplicity, is obtained as a corollary of our main theorem. We also develop the notion of mixed epsilon multiplicity for monomial ideals.
We show how subintegral extensions of certain local Noetherian domains $S$ can be constructed with specified invariants including reduction number, Hilbert function, multiplicity and local cohomology. The construction behaves analytically like Nagata idealization but rather than a ring extension of $S$, it produces a subring $R$ of $S$ such that $R \subseteq S$ is subintegral.
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…
Some Formulas for Epsilon Multiplicity in Local Rings
本文证明诺特局部环中 epsilon 重数在幂零根非极大维数时存在，并将体积等于重数的公式推广至此情形。
In a formally unmixed Noetherian local ring, if the colength and multiplicity of an integrally closed ideal agree, then $R$ is regular. We deduce this using the relationship between multiplicity and various ideal closure operations.
Let S be a finitely generated standard multi-graded algebra over a Noetherian local ring A. This paper first expresses mixed multiplicities of S in term of Hilbert-Samuel multiplicity that explained the mixed multiplicities S as the Hilbert-Samuel multiplicities for quotient modules of S. As an application, we get formulas for the mixed multiplicities of ideals that covers the main result of Trung-Verma 2007 [TV].
Multiplicities of graded families of ideals on Noetherian local rings
本文研究了Noetherian局部环上分级理想族的乘数概念，推广了传统理想乘数，并证明了多个经典定理在该框架下的泛化，包括混合乘数、Rees定理和Minkowski不等式。
Generalized Hilbert-Kunz Multiplicity for Families of Ideals
本文系统研究正特征Noetherian局部环中理想族的广义Hilbert-Kunz重数，引入Amao型重数这一新渐近不变量，证明p-理想族的广义Hilbert-Kunz重数是Amao型重数的极限。
Let $R$ be a Noetherian local ring. We define the minimal $j$-multiplicity and almost minimal $j$-multiplicity of an arbitrary $R$-ideal on any finite $R$-module. For any ideal $I$ with minimal $j$-multiplicity or almost minimal $j$-multiplicity on a Cohen-Macaulay module $M$, we prove that under some residual assumptions, the associated graded module ${\rm gr}_I(M)$ is Cohen-Macaulay or almost Cohen-Macaulay, respectively. Our work generalizes the results for minimal multiplicity and almost minimal multiplicity obtained by Sally, Rossi, Valla, Wang, Huckaba, Elias, Corso, Polini, and VazPinto…
Let $(R, {\mathfrak m})$ be a Noetherian local ring and let $I$ be an ${\mathfrak m}$-primary ideal. In this paper we give sharp bounds on the multiplicity of the special fiber ring ${\mathcal F}$ of $I$ in terms of other well-known invariants of $I$. A special attention is then paid in studying when equality holds in these bounds, with a particular interest in the unmixedness or, better, the Cohen-Macaulayness of ${\mathcal F}$.
Let $J\subset I$ be ideals in a formally equidimensional local ring with $λ(I/J)<\infty.$ Rees proved that for all $n\gg0$, $λ(I^n/J^n)$ is a polynomial $P(I/J)(X)$ in $n$ of degree at most dim $R$ and $J$ is a reduction of $I$ if and only if deg $P(I/J)(X)\leq$ dim $R-1.$ We extend this result for all Noetherian filtrations of ideals in a formally equidimensional local ring and for (not necessarily Noetherian) filtrations of ideals in analytically irreducible rings. We provide certain classes of ideals such that deg $P(I/J)$ achieves its maximal degree. On the other hand, for ideals $J\subset…
Irreducible multiplicity of idealizations
本文研究诺特局部环及其理想化中准素理想的不可约指数与不可约重数之间的关系，推广了Goto等人的一个主要结果。
In this paper, we work with certain families of ideals called $p$-families in rings of prime characteristic. This family of ideals is present in the theories of tight closure, Hilbert-Kunz multiplicity, and $F$-signature. For each $p$-family of ideals, we attach a Euclidean object called $p$-body, which is analogous to the Newton Okounkov body associated with a graded family of ideals. Using the combinatorial properties of $p$-bodies and algebraic properties of the Hilbert-Kunz multiplicity, we establish in this paper a Volume = Multiplicity formula for $p$-families of $\mathfrak{m}_{R}$-prima…
Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $d>0$ with infinite residue field. Let $M$ be a finitely generated proper $R$-submodule of a free $R$-module $F$ with $\ell (F/M) < \infty$ and having rank $r$. In this article, we study the fiber multiplicity $f_0(M)$ of the module $M$. We prove that if $(R,\mathfrak{m})$ is a two dimensional Cohen-Macaulay local ring, then $f_0(M)\le br_1(M)-br_0(M)+\ell (F/M)+μ(M)-r$, where $br_i(M)$ denotes the $i^{th}$ Buchsbaum-Rim coefficient of $M$.
Hilbert-Kunz Multiplicity of Fiber Product Rings and Nagata Idealizations
本文为纤维积环和Nagata理想化推导出Hilbert-Kunz重数的显式公式，并给出新界与结构结果。
Let $R$ be an excellent Noetherian ring of prime characteristic. Consider an arbitrary nested pair of ideals (or more generally, a nested pair of submodules of a fixed finite module). We do \emph{not} assume that their quotient has finite length. In this paper, we develop various sufficient numerical criteria for when the tight closures of these ideals (or submodules) match. For some of the criteria we only prove sufficiency, while some are shown to be equivalent to the tight closures matching. We compare the various numerical measures (in some cases demonstrating that the different measures g…
Let $(R, \mathfrak m)$ be a $d$-dimensional Noetherian local ring and $E$ a finitely generated $R$-submodule of a free module $R^p.$ In this work we introduce a multiplicity sequence $c_k(E), k=0,..., d+p-1$ for $E$ that generalize the Buchsbaum-Rim multiplicity defined when $E$ has finite colength in $R^p$ as well as the Achilles-Manaresi multiplicity sequence that applies when $E\subseteq R$ is an ideal. Our main result is that the new multiplicity sequence can indeed be used to detect integral dependence of modules. Our proof is self-contained and implies known numerical criteria for integr…
Computing multiplicity sequences
本文介绍了Macaulay2的MultiplicitySequence包，用于计算分次理想的重数序列及相关不变量，并讨论了基于二元Hilbert多项式和一般元素技术的两种计算策略。
Improved Bounds for the s-multiplicity
该研究针对素特征诺特局部环，建立了纤维积与理想化环的h-函数和s-重数的关联公式，为Taylor-Miller问题和Watanabe-Yoshida猜想提供了新估计。
In the derived category of modules over a commutative noetherian ring a complex $G$ is said to generate a complex $X$ if the latter can be obtained from the former by taking summands and finitely many cones. The number of cones required in this process is the generation time of $X$. In this paper we present some local to global type results for computing this invariant, and discuss applications.
The present paper is devoted to an algebraic treatment of the joint spectral theory within the framework of Noetherian modules over an algebra finite extension of an algebraically closed field. We prove the spectral mapping theorem and analyze the index of tuples in purely algebraic case. The index function over tuples from the coordinate ring of a variety is naturally extended up to a numerical Tor-polynomial. Based on Serre's multiplicity formula, we deduce that Tor-polynomial is just the Samuel polynomial of the local algebra.
Koszul property and finite linearity defect over $g$-stretched local rings
该研究针对Herzog-Iyengar关于剩余域有限线性缺陷与环Koszul性的公开问题，在几乎极小重数且特征零剩余域的Cohen-Macaulay局部环情形给出肯定回答，引入g-拉伸局部环类并完成其一维完备情形刻画，同时给出g-拉伸Koszul环的数值刻画。
Colength, multiplicity, and ideal closure operations II
本文研究 Noetherian 局部环中 $\mathfrak{m}$-准素理想 colength 与重数关系的极值不变量，通过闭包限制引入新不变量版本并交叉结合 Hilbert--Samuel 与 Hilbert--Kunz 理论，证明了相关不变量的极值与类似物。
We introduce and study equivariant Hilbert series of ideals in polynomial rings in countably many variables that are invariant under a suitable action of a symmetric group or the monoid $Inc(\mathbb{N})$ of strictly increasing functions. Our first main result states that these series are rational functions in two variables. A key is to introduce also suitable submonoids of $Inc(\mathbb{N})$ and to compare invariant filtrations induced by their actions. Extending a result by Hillar and Sullivant, we show that any ideal that is invariant under these submonoids admits a Gröbner basis consisting o…
Over a Noetherian, local ring R of prime characteristic p, the Frobenius functor F induces a diagonalizable map on certain quotients of rational Grothendieck groups. This leads to an explicit formula for the Dutta multiplicity, and it is shown that a weaker version of Serre's vanishing conjecture holds if only chi(F(X)) = p^{dim R}chi(X) for all bounded complexes X of finitely generated, projective modules with finite length homology.
Given a local Noetherian ring $(R, {\mathfrak m})$ of dimension $d>0$ and infinite residue field, we study the invariants $($dimension and multiplicity$)$ of the Sally module $S_J(I)$ of any ${\mathfrak m}$-primary ideal $I$ with respect to a minimal reduction $J$. As a by-product we obtain an estimate for the Hilbert coefficients of ${\mathfrak m}$ that generalizes a bound established by J. Elias and G. Valla in a local Cohen-Macaulay setting. We also find sharp estimates for the multiplicity of the special fiber ring ${\mathcal F}(I)$, which recover previous bounds established by C. Polini, …
We provide suitable conditions under which the asymptotic limit of the Hilbert-Samuel coefficients of the Frobenius powers of an $\mathfrak{m}$-primary ideal exists in a Noetherian local ring $(R,\mathfrak{m})$ with prime characteristic $p>0.$ This, in turn, gives an expression of the Hilbert-Kunz multiplicity of powers of the ideal. We also prove that for a face ring $R$ of a simplicial complex and an ideal $J$ generated by pure powers of the variables, the generalized Hilbert-Kunz function $\ell(R/(J^{[q]})^k)$ is a polynomial for all $q,k$ and also give an expression of the generalized Hilb…
Let $(R, \mathfrak m)$ be a Noetherian local ring and $I$ a $\mathfrak m$-primary ideal. In this paper, we study an inequality involving the number of generators, the Loewy length and the multiplicity of $I$. There is strong evidence that the inequality holds for all integrally closed ideals of finite colength if and only if $R$ has sufficiently nice singularities. We verify the inequality for regular local rings in all dimensions, for rational singularity in dimension $2$, and cDV singularities in dimension $3$. In addition, we can classify when the inequality always hold for a Cohen-Macaulay…
For a Noetherian local domain $A$, there exists an upper bound $N_τ(A)$ on the minimal number of generators of any height two ideal $I$ for which $A/I$ is Cohen-Macaulay of type $τ$. More precisely, we may take $N_τ(A):=(τ+1)e_{\text{h}}(A)$, where $e_{\text{h}}(A)$ is the homological multiplicity of $A$.
Test properties of some Cohen-Macaulay modules and criteria for local rings via finite vanishing of Ext or Tor
本文通过Ext或Tor的有限消失性质，测试Cohen-Macaulay模的特性，并用于刻画局部环，验证了Auslander-Reiten猜想的扩展结果。
In the present paper, we define a notion of numerical equivalence on Chow groups or Grothendieck groups of Noetherian local rings, which is an analogue of that on smooth projective varieties. Under a mild condition, it is proved that the Chow group modulo numerical equivalence is a finite dimensional ${\Bbb Q}$-vector space, as in the case of smooth projective varieties. Numerical equivalence on local rings is deeply related to that on smooth projective varieties. For example, if Grothendieck's standard conjectures are true, then a vanishing of Chow group (of local rings) modulo numerical equi…
Noetherian operators are differential operators that encode primary components of a polynomial ideal. We develop a framework, as well as algorithms, for computing Noetherian operators with local dual spaces, both symbolically and numerically. For a primary ideal, such operators provide an alternative representation to one given by a set of generators. This description fits well with numerical algebraic geometry, taking a step toward the goal of numerical primary decomposition.
Let $M$ be a finite module and let $I$ be an arbitrary ideal over a Noetherian local ring. We define the generalized Hilbert function of $I$ on $M$ using the 0th local cohomology functor. We show that our definition re-conciliates with that of Ciuperc$\breve{\rm a}$. By generalizing Singh's formula (which holds in the case of $λ(M/IM)<\infty$), we prove that the generalized Hilbert coefficients $j_0,..., j_{d-2}$ are preserved under a general hyperplane section, where $d={\rm dim}\,M$. We also keep track of the behavior of $j_{d-1}$. Then we apply these results to study the generalized Hilbert…
Limits of Amao multiplicities and Epsilon multiplicity
本文研究Amao重数与Epsilon重数的极限，证明在域上有限型的任意局部环中，相关极限恒存在，完善了Epsilon重数的极限相关结论。
Computing in complete local equicharacteristic Noetherian rings via topological rewriting on commutative formal power series
本文通过拓扑重写理论，重新证明了形式幂级数中标准基的表征及广义连通性属性的等价性，扩展了Gröbner基理论与经典代数重写理论之间的类比。
This paper gives a necessary and sufficient condition for Gorensteinness in Rees algebras of the $d$-th power of parameter ideals in certain Noetherian local rings of dimension $d\ge 2$. The main result of this paper produces many Gorenstein Rees algebras over non-Cohen-Macaulay local rings. For example, the Rees algebra $\mathcal{R}(\mathfrak{q}^d)=\oplus_{i\ge 0}\mathfrak{q}^{di}$ is Gorenstein for every parameter ideal $\mathfrak{q}$ that is a reduction of the maximal ideal in a $d$-dimensional Buchsbaum local ring of depth 1 and multiplicity 2.
The $F$-threshold $c^J(\a)$ of an ideal $\a$ with respect to an ideal $J$ is a positive characteristic invariant obtained by comparing the powers of $\a$ with the Frobenius powers of $J$. We study a conjecture formulated in an earlier paper \cite{HMTW} by the same authors together with M. Mustaţă, which bounds $c^J(\a)$ in terms of the multiplicities $e(\a)$ and $e(J)$, when $\a$ and $J$ are zero-dimensional ideals and $J$ is generated by a system of parameters. We prove the conjecture when $\a$ and $J$ are generated by homogeneous systems of parameters in a Noetherian graded $k$-algebra. We a…
The aim of this note is to describe how to compute the intersection multiplicity defined by Jean Pierre Serre.
Let $M$ be a finitely generated module over a Noetherian local ring. This paper reports, for a given parameter ideal $Q$ for $M$, a criterion for the equality ${\mathrm{g}}_s(Q;M)=\operatorname{hdeg}_Q(M)-{\mathrm{e}}_Q^0(M)-{\mathrm{T}}_Q^1(M)$, where ${\mathrm{g}}_s(Q;M)$, ${\mathrm{e}}_Q^0(M)$, ${\mathrm{e}}_Q^1(M)$, and ${\mathrm{T}}_Q^1(M)$ respectively denote the sectional genus, the multiplicity, the first Hilbert coefficient, and the Homological torsion of $M$ with respect to $Q$.
Uniform bounds in excellent $\mathbf{F}_p$-algebras and applications to semi-continuity
针对未必F-有限的诺特Fₚ-代数R，研究其谱上的Hilbert-Kunz重数与F-签名，证明优环时定义这两个不变量的极限一致，进而推出F-签名下半连续、局部等维时Hilbert-Kunz重数上半连续，且结果可在比优性更弱条件下成立。
Let $(A, \frak m)$ be a noetherian local ring with maximal ideal $\frak{m}$ and infinite residue field $k = A/\frak{m}.$ Let $J$ be an $\frak m$-primary ideal, $I_1,...,I_s$ ideals of $A$, and $M$ a finitely generated $A$-module. In this paper, we interpret mixed multiplicities of $(I_1,..., I_s,J)$ with respect to $M$ as multiplicities of joint reductions of them. This generalizes the Rees's theorem on mixed multiplicity (Theorem 2.4). As an application we show that mixed multiplicities are also multiplicities of Rees's superficial sequences.
Generalized mixed sequence multiplicity
本文提出方向混合重数，细化任意理想的重数序列，兼容经典情形，并建立可加性、结合性、分解及 Rees 型定理，推广 Bhattacharya--Rees--Teissier 公式。
Computing epsilon multiplicities in graded algebras
本文研究了分次代数中 $\varepsilon$-重数的计算问题，证明了二维标准分次整环中齐次理想 $\varepsilon$-重数的有理性，给出了基于混合重数的计算公式，并提供了 Macaulay2 算法。
On the Hilbert Series and Multiplicity of Generalized Binomial Edge Ideals
本文计算了图的联结及完全r-部图的广义二项式边理想的希尔伯特级数，并给出顶点移除下系数不变的约简技术，进而计算几类图的维数和重数。
We continue the study of intersection algebras $\mathcal B = \mathcal B_R(I, J)$ of two ideals $I, J$ in a commutative Noetherian ring $R$. In particular, we exploit the semigroup ring and toric structures in order to calculate various invariants of the intersection algebra when $R$ is a polynomial ring over a field and $I,J$ are principal monomial ideals. Specifically, we calculate the $F$-signature, divisor class group, and Hilbert-Samuel and Hilbert-Kunz multiplicities, sometimes restricting to certain cases in order to obtain explicit formulæ. This provides a new class of rings where formu…
Algorithms for computing mixed multiplicities, mixed volumes and sectional Milnor numbers
本研究推出MixedMultiplicity程序包，通过推广Cox等人的成果获取理想多重Rees代数的定义方程，可计算诺特环理想混合重数，还能求解凸格点多面体混合体积与孤立奇点超曲面的截面Milnor数。
Multiplicity $=$ Volume formula and Newton non-degenerate ideals in regular local rings
本文将复分析中的牛顿非退化理想与牛顿多面体推广至正则局部环，利用极限体为“重数$=$体积”公式提供新理解，并证明了该公式成立的充要条件是理想族包含NND理想子族。
Criteria for finite injective dimension of modules over a local ring
本文在交换诺特局部环上，利用满足重数与型不等式的 Cohen--Macaulay 模及有限多个 Ext 模的消失来刻画模的有限内射维数，并由此恢复已有结果、给出新的充分条件。
The Chern numbers of the title are the first coefficients (after the multiplicities) of the Hilbert functions of various filtrations of ideals of a local ring $(R, \mathfrak{m})$. For a Noetherian (good) filtration $\mathcal{A}$ of $\mathfrak{m}$-primary ideals, the positivity and bounds for $e_1(\mathcal{A})$ are well-studied if $R$ is Cohen-Macaulay, or more broadly, if $R$ is a Buchsbaum ring or mild generalizations thereof. For arbitrary geometric local domains, we introduce techniques based on the theory of maximal Cohen-Macaulay modules and of extended multiplicity functions to establish…
A commutative ring $R$ is stable provided every ideal of $R$ containing a nonzerodivisor is projective as a module over its ring of endomorphisms. The class of stable rings includes the one-dimensional local Cohen-Macaulay rings of multiplicity at most $2$, as well as certain rings of higher multiplicity, necessarily analytically ramified. The former are important in the study of modules over Gorenstein rings, while the latter arise in a natural way from generic formal fibers and derivations. We characterize one-dimensional stable local rings in several ways. The characterizations involve the …
Mixed multiplicity and Converse of Rees' theorem for modules
本文研究模的 Rees 混合重数定理逆命题，在高度、根及局部 Buchsbaum-Rim 重数相等等条件下证明元素组构成联合约化，并建立相关性质。
Let $(A,\mathfrak{m},\Bbbk)$ denote a local Noetherian ring and $\mathfrak{q}$ an ideal such that $\ell_A(M/\mathfrak{q}M) < \infty$ for a finitely generated $A$-module $M$. Let $\au = a_1,\ldots,a_d$ denote a system of parameters of $M$ such that $a_i \in \mathfrak{q}^{c_i} \setminus \mathfrak{q}^{c_i+1}$ for $i=1,\ldots,d$. It follows that $ χ:= e_0(\au;M) - c \cdot e_0(\mathfrak{q};M) \geq 0$, where $c = c_1\cdot \ldots \cdot c_d$. The main results of the report are a discussion when $χ= 0$ resp. to describe the value of $χ$ in some particular cases. Applications concern results on the mult…
Resultants and Gröbner bases are crucial tools in studying polynomial elimination theory. We investigate relations between the variety of the resultant of two polynomials and the variety of the ideal they generate. Then we focus on the bivariate case, in which the elimination ideal is principal. We study - by means of elementary tools - the difference between the multiplicity of the factors of the generator of the elimination ideal and the multiplicity of the factors of the resultant.
Computing mixed multiplicities, mixed volumes, and sectional Milnor numbers
本文为综述性工作，介绍了基于推广Cox等人结果的多重Rees代数方程计算的Macaulay2新算法，可高效计算诺特环理想混合重数、凸格点多胞体混合体积与孤立奇点超曲面的截面Milnor数。
This paper presents a new methodology to count the number of numerical semigroups of given genus or Frobenius number. We apply generating function tools to the bounded polyhedron that classifies the semigroups with given genus (or Frobenius number) and multiplicity. First, we give theoretical results about the polynomial-time complexity of counting the number of these semigroups. We also illustrate the methodology analyzing the cases of multiplicity 3 and 4 where some formulas for the number of numerical semigroups for any genus and Frobenius number are obtained.
Let $R$ be a one-dimensional, local, Noetherian domain, $\R$ the integral closure of $R$ in its quotient field and $v(R)$ the value set defined by the usual valuation. The aim of the paper is to study the non-negative invariant $b:=(c-δ)r- δ$, where $c, δ, r$ denote the conductor, the length of $\R/R$ and the Cohen Macaulay type, respectively. In particular, the classification of the semigroups $v(R)$ for rings having $b\leq 2(r-1)$ is realized. This method of classification might be successfully utilized with similar arguments but more boring computations in the cases $b\leq q(r-1), $ for rea…
When are trace ideals finite?
本文研究具有有限个迹理想的诺特局部环，证明此类环维数至多为二，并在特定条件下维数至多为一，同时给出了一维情形下迹理想有限的必要条件及极小重数时的对应关系。
Asymptotic $\mathrm{v}$-number of graded families of ideals and the Newton-Okounkov region
本文研究了格雷德理想族的渐进 $\mathrm{v}$-数，证明其极限存在并给出计算方法，同时通过牛顿-奥科诺夫区域提供了组合解释，并探讨了渐进准线性性质及 $\mathrm{v}$ 与 $\operatorname{reg}$ 的关系。
Density functions for epsilon multiplicity and families of ideals
本文为齐次理想的滤过及ε重数构造密度函数，证明其连续性与可微性，并揭示对角子代数重标度重数的连续性。
The Goto number of a parameter ideal Q in a Noetherian local ring (R,m) is the largest integer q such that Q : m^q is integral over Q. The Goto numbers of the monomial parameter ideals of R = k[[x^{a_1}, x^{a_2},..., x_{a_ν}]] are characterized using the semigroup of R. This helps in computing them for classes of numerical semigroup rings, as well as on a case-by-case basis. The minimal Goto number of R and its connection to other invariants is explored. Necessary and sufficient conditions for the associated graded rings of R and R/x^{a_1}R to be Gorenstein are also given, again using the semi…
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