We use the theory of resolutions for a given Hilbert function to investigate the multiplicity conjectures of Huneke and Srinivasan and Herzog and Srinivasan. To prove the conjectures for all modules with a particular Hilbert function, we show that it is enough to prove the statements only for elements at the bottom of the partially ordered set of resolutions with that Hilbert function. This enables us to test the conjectured upper bound for the multiplicity efficiently with the computer algebra system Macaulay 2, and we verify the upper bound for many Artinian modules in three variables with small socle degree. Moreover, with this approach, we show that though numerical techniques have been sufficient in several of the known special cases, they are insufficient to prove the conjectures in general. Finally, we apply a result of Herzog and Srinivasan on ideals with a quasipure resolution to prove the upper bound for Cohen-Macaulay quotients by ideals with generators in high degrees relative to the regularity.
A weaker form of the multiplicity conjecture of Herzog, Huneke, and Srinivasan is proven for two classes of monomial ideals: quadratic monomial ideals and squarefree monomial ideals with sufficiently many variables relative to the Krull dimension. It is also shown that tensor products, as well as Stanley-Reisner ideals of certain unions, satisfy the multiplicity conjecture if all the components do. Conditions under which the bounds are achieved are also studied.
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.
In this exposition of the equality and inequality of Minkowski for multiplicity of ideals, we provide simple algebraic and geometric proofs. Connections with mixed multiplicities of ideals are explained.
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…
We give bounds for the Hilbert-Kunz multiplicity of the product of two ideals, and we characterize the equality in terms of the tight closures of the ideals. Connections are drawn with $*$-spread and with ordinary length calculations.
Ulrich ideals in numerical semigroup rings of small multiplicity are studied. If the semigroups are three-generated but not symmetric, the semigroup rings are Golod, since the Betti numbers of the residue class fields of the semigroup rings form an arithmetic progression; therefore, these semigroup rings are G-regular, possessing no Ulrich ideals. Nevertheless, even in the three-generated case, the situation is different, when the semigroups are symmetric. We shall explore this phenomenon, describing an explicit system of generators, that is the normal form of generators, for the Ulrich ideals…
In this paper we give new lower bounds on the Hilbert-Kunz multiplicity of unmixed non-regular local rings, bounding them uniformly away from one. Our results improve previous work of Aberbach and Enescu.
We show that if $(R, \m)$ is a Cohen-Macaulay local ring and $I$ is an ideal of minimal mixed multiplicity, then $\depth G(I) \geq d- 1$ implies that $\depth F(I) \geq d-1$. We use this to show that if $I$ is a contracted ideal in a two dimensional regular local ring then $\depth R[It]-1= \depth G(I) = \depth F(I)$. We also give an infinite class of ideals where $R[It]$ is Cohen-Macaulay but $F(I)$ is not.
We define a function, called s-multiplicity, that interpolates between Hilbert-Samuel multiplicity and Hilbert-Kunz multiplicity by comparing powers of ideals to the Frobenius powers of ideals. The function is continuous in s, and its value is equal to Hilbert-Samuel multiplicity for small values of s and is equal to Hilbert-Kunz multiplicity for large values of s. We prove that it has an Associativity Formula generalizing the Associativity Formulas for Hilbert-Samuel and Hilbert-Kunz multiplicity. We also define a family of closures such that if two ideals have the same s-closure then they ha…
Let $K$ be a field, $V$ a $K$-vector space with basis $e_1,\ldots,e_n$, and $E$ the exterior algebra of $V$. To a given monomial ideal $I\subsetneq E$ we associate a special monomial ideal $J$ with generators in the same degrees as those of $I$ and such that the number of the minimal monomial generators in each degree of $I$ and $J$ coincide. We call $J$ the colexsegment ideal associated to $I$. We prove that the class of strongly stable ideals in $E$ generated in one degree satisfies the colex lower bound, that is, the total Betti numbers of the colexsegment ideal associated to a strongly sta…
We consider ideals of minors of a matrix, ideals of minors of a symmetric matrix, and ideals of Pfaffians of an alternating matrix. Assuming these ideals are of generic height, we characterize the condition $G_{s}$ for these ideals in terms of the heights of other ideals of minors or Pfaffians of the same matrix. We additionally obtain bounds on the generation and concentration degrees of the Rees rings of a subclass of such ideals via specialization of the Rees rings in the generic case. We do this by proving, given sufficient height conditions on ideals of minors or Pfaffians of the matrix, …
We extend a result by Huneke and Watanabe bounding the multiplicity of $F$-pure local rings of prime characteristic in terms of their dimension and embedding dimensions to the case of $F$-injective, generalized Cohen-Macaulay rings. We then produce an upper bound for the multiplicity of any local Cohen-Macaulay ring of prime characteritic in terms of their dimensions, embedding dimensions and HSL numbers. Finally, we extend the upper bounds for the multiplicity of generalized Cohen-Macaulay rings in characteristic zero which have dense $F$-injective type.
This paper gives new bounds on the first Hilbert coefficient of an ideal of finite colength in a Cohen-Macaulay local ring. The bound given is quadratic in the multiplicity of the ideal. We compare our bound to previously known bounds, and give examples to show that at least in some cases it is sharp. The techniques come largely from work of Elias, Rossi, Valla, and Vasconcelos.
We study conjectured generalizations of a formula of Lech which relates the multiplicity of a finite colength ideal in an equicharacteristic local ring to its colength, and prove one of these generalizations involving the multiplicity of the maximal ideal times the finite colength ideal. We also propose a Lech-type formula that relates multiplicity and the number of generators. We prove the conjecture in dimension three and establish a weaker result in full generality.
We introduce the concept of edgewise domination in clutters, and use it to provide an upper bound for the projective dimension of any squarefree monomial ideal. We then use a simple recursion to recover a formula for the projective dimension of a monomial ideal associated to a chordal clutter, as defined by Woodroofe in \cite{russ}. We also study a family of clutters associated to graphs, and show that these clutters are chordal if and only if the associated graph is. Finally, we compute domination parameters for certain classes of these clutters.
We study Hilbert-Kunz multiplicity of the powers of an ideal and establish existence of the second coefficient at the full level of generality, thus extending a recent result of Trivedi. We describe the second coefficient as the limit of the Hilbert coefficients of Frobenius powers and show that it is additive in short exact sequences and satisfies a Northcott-type inequality.
In this paper we consider multi-graded extended Rees algebras of zero dimensional ideals which are Cohen-Macaulay (CM) with minimal multiplicity. We show that the minimal multiplicity property can occur only for the ordinary extended Rees algebra and the bigraded extended Rees algebra. For the bigraded extended Rees algebra we find necessary conditions for it to be CM with minimal multiplicity. We also produce bigraded Rees algebras which are Cohen-Macaulay with minimal multiplicity.
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.
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…
We study ideals of Borel type, including $k$-Borel ideals and $t$-spread Veronese ideals. We determine their free resolutions and their homological shift ideals. The multiplicity and the analytic spread of equigenerated squarefree principal Borel ideals are computed. For the multiplicity, the result is given under an additional assumption which is always satisfied for squarefree principal Borel ideals. These results are used to analyze the behaviour of height, multiplicity and analytic spread of the homological shift ideals $HS_j(I)$ as functions of $j$, when $I$ is an equigenerated squarefree…
Let $A = K[X_1,...,X_n]$ and let $I$ be a graded ideal in $A$. We show that the upper bound of Multiplicity conjecture of Herzog, Huneke and Srinivasan holds asymptotically (i.e., for $I^k$ and all $k \gg 0$) if $I$ belongs to any of the following large classes of ideals: \begin{enumerate}[\rm (1)] \item radical ideals. \item monomial ideals with generators in different degrees. \item zero-dimensional ideals with generators in different degrees. \end{enumerate} Surprisingly, our proof uses local techniques like analyticity, reductions, equimultiplicity and local results like Rees's theorem on …
Let $G$ be a bipartite graph and $I(G)$ the toric ideal associated to the graph $G$. In this article we calculate Hilbert-Samuel multiplicity of the graph $G$ for which the toric ideal $I(G)$ is generated by a quadratic binomials and it satiesfies some conditions.
We prove that the Hilbert-Kunz function of the ideal $(I,It)$ of the Rees algebra $\mathcal{R}(I)$, where $I$ is an $\mathfrak{m}$-primary ideal of a $1$-dimensional local ring $(R,\mathfrak{m})$, is a quasi-polynomial in $e$, for large $e.$ For $s \in \mathbb{N}$, we calculate the Hilbert-Samuel function of the $R$-module $I^{[s]}$ and obtain an explicit description of the generalized Hilbert-Kunz function of the ideal $(I,It)\mathcal{R}(I)$ when $I$ is a parameter ideal in a Cohen-Macaulay local ring of dimension $d \geq 2$, proving that the generalized Hilbert-Kunz function is a piecewise p…
Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring in $n$ variables over a field $K$ and $I\subset S$ a squarefree monomial ideal. In the present paper we are interested in the monomials $u \in S$ belonging to the socle $\Soc(S/I^{k})$ of $S/I^{k}$, i.e., $u \not\in I^{k}$ and $ux_{i} \in I^{k}$ for $1 \leq i \leq n$. We prove that if a monomial $x_1^{a_1}\cdots x_n^{a_n}$ belongs to $\Soc(S/I^{k})$, then $a_i\leq k-1$ for all $1 \leq i \leq n$. We then discuss squarefree monomial ideals $I \subset S$ for which $x_{[n]}^{k-1} \in \Soc(S/I^{k})$, where $x_{[n]} = x_{1}x_{2}\cdots x_{n}$. Furtherm…
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…
We prove a characterization of the j-multiplicity of a monomial ideal as the normalized volume of a polytopal complex. Our result is an extension of Teissier's volume-theoretic interpretation of the Hilbert-Samuel multiplicity for m-primary monomial ideals. We also give a description of the epsilon-multiplicity of a monomial ideal in terms of the volume of a region.
Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $d>0$ and depth R$\geq d-1$. Let $Q$ be a parameter ideal of $R$. In this paper, we derive uniform lower and upper bounds for the Hilbert coefficient $e_i(Q)$ under certain assumptions on the depth of associated graded ring $G(Q)$. For $2\leq i\leq d $, we show that (1) $e_i(Q)\leq 0$ provided depth $G(Q)\geq d-2$ and (2) $e_i(Q)\geq -λ_R(H_{\mathfrak{m}}^{d-1}(R))$ provided depth $G(Q)\geq d-1$. It is proved that $e_3(Q)\leq 0$. Further, we obtain a necessary condition for the vanishing of the last coefficient $e_d(Q)$. As a conse…
Over a regular local ring of dimension two with maximal ideal m, we study the Buchsbaum-Rim multiplicity of a finitely generated module M of finite colength in a free module F. First, we investigate the colength of an m-primary ideal and its Hilbert-Samuel multiplicity using linkage theory. As applications, we establish several multiplicity formula that express the Buchsbaum-Rim multiplicity of M in terms of the Hilbert-Samuel multiplicities of ideals related to a certain minimal reduction U of M. Moreover, there exists m-primary Bourbaki ideals I and J of the modules F and M respectively such…
Let A be a Buchsbaum local ring with the maximal ideal m and let e(A) denote the multiplicity of A. Let Q be a parameter ideal in A and put I=Q:m. Then the equality I^2=QI holds true, if e(A)=2 and depthA>0. The assertion is no longer true, unless e(A)=2. Counterexamples are given.
This paper investigates the relationship between multiplicities and the degree sequence of ideals in graded algebras, gives multiplicity equations of graded rings via the degree sequence of ideals, and characterizes mixed multiplicities and multiplicities of Rees rings in terms of the degree sequence of ideals. As an application, the paper answered an open question on the multiplicity of Rees rings.
Let $R$ be a polynomial ring over a field. We prove an upper bound for the multiplicity of $R/I$ when $I$ is a homogeneous ideal of the form $I=J+(F)$, where $J$ is a Cohen-Macaulay ideal and $F\notin J$. The bound is given in terms of two invariants of $R/J$ and the degree of $F$. We show that ideals achieving this upper bound have high depth, and provide a purely numerical criterion for the Cohen-Macaulay property. Applications to quasi-Gorenstein rings and almost complete intersections are given.
The $j$-multiplicity plays an important role in the intersection theory of Stückrad-Vogel cycles, while recent developments confirm the connections between the $ε$-multiplicity and equisingularity theory. In this paper we establish, under some constraints, a relationship between the $j$-multiplicity of an ideal and the degree of its fiber cone. As a consequence, we are able to compute the $j$-multiplicity of all the ideals defining rational normal scrolls. By using the standard monomial theory, we can also compute the $j$- and $ε$-multiplicity of ideals defining determinantal varieties: The fo…
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…
Herzog, Huneke, and Srinivasan have conjectured that for any homogeneous $k$-algebra, the multiplicity is bounded above by a function of the maximal degrees of the syzygies and below by a function of the minimal degrees of the syzygies. The goal of this paper is to establish the multiplicity conjecture of Herzog, Huneke, and Srinivasan about the multiplicity of graded Cohen-Macaulay algebras over a field $k$ for $k$-algebras $k[x_1, ..., x_n]/I$ being $I$ a determinantal ideal of arbitrary codimension.
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.
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.
Let I be the toric ideal defined by a 2 x n matrix of integers, A = ((1 1 ... 1)(a_1 a_2 ... a_n)) with a_1<a_2<...<a_n. We give a combinatorial proof that I is generated by elements of degree at most the sum of the two largest differences a_i - a_(i-1). The novelty is in the method of proof: the result has already been shown by L'vovsky using cohomological arguments.
We prove the multiplicity bounds conjectured by Herzog-Huneke-Srinivasan and Herzog-Srinivasan in the following cases: the strong conjecture for edge ideals of bipartite graphs, and the weaker Taylor bound conjecture for all quadratic monomial ideals. We attach a directed graph to a bipartite graph with perfect matching, and describe operations on the directed graph that would reduce the problem to a Cohen-Macaulay bipartite graph. We determine when equality holds in the conjectured bound for edge ideals of bipartite graphs, and verify that when equality holds, the resolution is pure. We chara…
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 Hilb^p be the Hilbert scheme parametrizing the closed subschemes of P^n with Hilbert polynomial p \in Q[t] over a field K of characteristic zero. By bounding below the cohomological Hilbert functions of the points of Hilb^p we define locally closed subspaces of the Hilbert scheme. The aim of this thesis is to show that some of these subspaces are connected. For this we exploit the ideals constructed by D. Mall. It turns out that these ideals are sequentially Cohen-Macaulay and that their initial ideals and their generic initial ideals coincide for any admissible term order.
Let $R$ be a polynomial ring and $I \subset R$ be a perfect ideal of height two minimally generated by forms of the same degree. We provide a formula for the multiplicity of the saturated special fiber ring of $I$. Interestingly, this formula is equal to an elementary symmetric polynomial in terms of the degrees of the syzygies of $I$. Applying ideas introduced in arXiv:1805.05180, we obtain the value of the j-multiplicity of $I$ and an effective method for determining the degree and birationality of rational maps defined by homogeneous generators of $I$.
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…
Let Hilb^p be the Hilbert scheme parametrizing the closed subschemes of P^n with Hilbert polynomial p\in Q[t] over a field K of characteristic zero. By bounding below the cohomological Hilbert functions of the points of Hilb^p we define locally closed subspaces of the Hilbert scheme. The aim of this paper is to show that some of these subspaces are connected. For this we exploit the edge ideals constructed by D. Mall. It turns out that these ideals are sequentially Cohen-Macaulay and that their initial ideals with respect to the reverse lexicographic term order are generic initial ideals.
Let $R$ be a polynomial ring over a field and $I \subset R$ be a Gorenstein ideal of height three that is minimally generated by homogeneous polynomials of the same degree. We compute the multiplicity of the saturated special fiber ring of $I$. The obtained formula depends only on the number of variables of $R$, the minimal number of generators of $I$, and the degree of the syzygies of $I$. Applying results from arXiv:1805.05180, we get a formula for the $j$-multiplicity of $I$ and an effective method to study a rational map determined by a minimal set of generators of $I$.
