We study the relation type question, raised by C. Huneke, which asks whether for a complete equidimensional local ring R there exists a uniform bound for the relation type of parameter ideals. Wang gave a positive answer to this question when the non-Cohen-Macaulay locus of R, denoted by NCM(R), has dimension zero. We first present an example, due to the first author, which gives a negative answer to the question when dim NCM(R) is at least 2. The major part of our work then is to investigate the remaining case, i.e., when dim NCM(R) = 1. We introduce the notion of homology multipliers and show that the question has a positive answer when R/A(R) is a domain, where A(R) is the ideal generated by all homology multipliers in R. In a more general context, we also discuss many interesting properties of homology multipliers.
Let X be a smooth variety and J, K two ideal sheaves on X. We prove the following formula relating the multiplier ideals of J, K and J+K: I(X, c(J+K))\subset \sum_{a+b=c} I(X, aJ)\cdot I(X,bK). An analogous formula holds for the asymptotic multiplier ideals of two graded systems of ideals. As a consequence, we show how to approximate at any given point arbitrary multiplier ideals by multiplier ideals of zero dimensional ideals. We apply this to study the invariance of multiplier ideals under different embeddings.
A simple formula computing the multiplier ideal of a monomial ideal on an arbitrary affine toric variety is given. Variants for the multiplier module and test ideals are also treated.
The generalized test ideals introduced in [HY] are related to multiplier ideals via reduction to characteristic p. In addition, they satisfy many of the subtle properties of the multiplier ideals, which in characteristic zero follow via vanishing theorems. In this note we give several examples to emphasize the different behavior of test ideals and multiplier ideals. Our main result is that every ideal in an F-finite regular local ring can be written as a generalized test ideal. We also prove the semicontinuity of $F$-pure thresholds (though the analogue of the Generic Restriction Theorem for m…
In recent years, multiplier ideals have found many applications in local and global algebraic geometry. Because of their importance, there has been some interest in the question of which ideals on a smooth complex variety can be realized as multiplier ideals. Other than integral closure no local obstructions have been known up to now, and in dimension two it was established by Favre-Jonsson and Lipman-Watanabe that any integrally closed ideal is locally a multiplier ideal. We prove the somewhat unexpected result that multiplier ideals in fact satisfy some rather strong algebraic properties inv…
By using Mather-Jacobian multiplier ideals, we first prove a formula on comparing Grauert-Riemenschneider canonical sheaf with canonical sheaf of a variety over an algebraically closed field of characteristic zero. Then we turn to study Mather-Jacobian multiplier ideals on algebraic curve, in which case the definition of Mather-Jacobian multiplier ideal can be extended to a ground field of any characteristic. We show that Mather-Jacobian multiplier ideal on curves is essentially the same as an integrally closed ideal. Finally by comparing conductor ideal with Mather-Jacobian multiplier ideal, …
In this note, we calculate the Koszul homology of the codimension 3 Gorenstein ideals. We find filtrations for the Koszul homology in terms of modules with pure free resolutions and completely describe these resolutions. We also consider the Huneke-Ulrich deviation 2 ideals.
It is well known that the multiplier ideal $\multr{I}$ of an ideal $I$ determines in a straightforward way the multiplier ideal $\multr{f}$ of a sufficiently general element $f$ of $I$. We give an explicit condition on a polynomial $f \in \CC[x_1,...,x_n]$ which guarantees that it is a sufficiently general element of the most natural associated monomial ideal, the ideal generated by its terms. This allows us to directly calculate the multiplier ideal $\multr{f}$ (for all $r$) of ``most'' polynomials $f$.
We compute the multiplier ideals of hyperplane arrangements via the interpretation of these ideals in terms of spaces of arcs, due to Ein, Lazarsfeld and the author.
We describe a new software package for computing multiplier ideals in certain cases, including monomial ideals, monomial curves, generic determinantal ideals, and hyperplane arrangements. In these cases we take advantage of combinatorial formulas for multiplier ideals given by results of Howald, Thompson, and Johnson. The package uses Normaliz. It is available as a library for Macaulay2.
We introduce a generalization of the notion of local homology module, which we call a local homology module with respect to a pair of ideals $\left(I,J\right)$, and study its various properties such as vanishing, co-support and co-associated. We also discuss its connection with ordinary local homology.
The aim of this paper is to study mixed multiplier ideals associated to a tuple of ideals in a two-dimensional local ring with a rational singularity. We are interested in the partition of the real positive orthant given by the regions where the mixed multiplier ideals are constant. In particular we reveal which information encoded in a mixed multiplier ideal determines its corresponding jumping wall and we provide an algorithm to compute all the constancy regions, and their corresponding mixed multiplier ideals, in any desired range.
Thompson (2014) exhibits a formula for the multiplier ideal with multiplier lambda of a monomial curve C with ideal I as an intersection of a term coming from the I-adic valuation, the multiplier ideal of the term ideal of I, and terms coming from certain specified auxiliary valuations. This short note shows it suffices to consider only one auxiliary valuation. This improvement is achieved through a more intrinsic approach, reduction to the toric case.
The aim of this paper is to study jumping numbers and multiplier ideals of any ideal in a two-dimensional local ring with a rational singularity. In particular we reveal which information encoded in a multiplier ideal determines the next jumping number. This leads to an algorithm to compute sequentially the jumping numbers and the whole chain of multiplier ideals in any desired range. As a consequence of our method we develop the notion of jumping divisor that allows to describe the jump between two consecutive multiplier ideals. In particular we find a unique minimal jumping divisor that is s…
Demailly, Ein and Lazarsfeld \cite{DEL} proved the subadditivity theorem for multiplier ideals, which states the multiplier ideal of the product of ideals is contained in the product of the individual multiplier ideals, on non-singular varieties. We prove that, in two-dimensional case, the subadditivity theorem holds on log-terminal singularities. However, in higher dimensional case, we have several counter-examples. We consider the subadditivity theorem for monomial ideals on toric rings, and construct a counter-example on a three-dimensional toric ring.
This article outgrew from an effort to understand our basic question: Are the annihilators of the non-zero Koszul homology modules $H_i$ of an unmixed ideal $I$ contained in the integral closure $\bar{I}$ of $I$? We also obtain some variations on a result of Burch, which continue the theme of the paper in that they deal with annihilators of homology and integrally closed ideals.
We show that the Koszul homology algebra of a quotient by the edge ideal of a forest is generated by the lowest linear strand. This provides a large class of Koszul algebras whose Koszul homology algebras satisfy this property. We obtain this result by constructing the minimal graded free resolution of a quotient by such an edge ideal via the so called iterated mapping cone construction and using the explicit bases of Koszul homology given by Herzog and Maleki. Using these methods we also recover a result of Roth and Van Tuyl on the graded Betti numbers of quotients of edge ideals of trees.
It was recently established by the first two authors that multiplier ideals on a smooth variety satisfy some special syzygetic properties. The purpose of this note is to show how some of these can be extended to the singular setting.
With a particular focus on explicit computations and applications of the Koszul homology and Betti numbers of monomial ideals, the main goals of this thesis are the following: Analyze the Koszul homology of monomial ideals and apply it to describe the structure of monomial ideals. Describe algorithms to perform efficient computations of the homological invariants of monomial ideals. Apply the theory and computations on monomial ideals to problems inside and outside mathematics The thesis introduces as a main tool Mayer-Vietoris trees of monomial ideals.
As a generalization of formulas given by Demailly-Ein-Lazarsfeld, Eisenstein and Takagi, we prove a subadditivity formula for multiplier ideals associated to log pairs.
We prove a generalization of Demailly-Ein-Lazarsfeld's subadditivity formula and Mustata's summation formula for multiplier ideals to the case of singular varieties, using characteristic $p$ methods. As an application of our formula, we improve Hochster-Huneke's result on the growth of symbolic powers of ideals in singular affine algebras.
We introduce the notion of strong test module and show that a large number of such modules appear in the tight closure theory of complete domains: the test ideal (this has already been known), the parameter test module, and the module of relative test elements. They also appear as certain multiplier ideals, a concept of interest in algebraic geometry.
The goal of this paper is to produce a formula for the multiplier ideals of monomial space curves in the spirit of Howald's formula for the multiplier ideals of monomial ideals. This is achieved by constructing a toric blowup of affine space in such a way that a log resolution of the monomial curve may be constructed from this toric variety in a well controlled manner.
We present an algebro-geometric perspective on some generalizations, due to S. Takagi, of the restriction theorem for multiplier ideals. The first version of the restriction theorem for multiplier ideals was discovered by Esnault and Viehweg. In a series of papers S. Takagi has discovered generalizations of the restriction theorem and some formulas for multiplier ideals that follow from the restriction theorem. He uses the technique of tight closure and reduction to positive characteristic. We are able to provide an algebro-geometric proof of generalizations of his restriction theorem and his …
We prove that each positive power of the maximal ideal of a commutative Noetherian local ring is Tor-rigid, and strongly-rigid. This gives new characterizations of regularity and, in particular, shows that such ideals satisfy the torsion condition of a long-standing conjecture of Huneke and Wiegand.
We prove the rationality of the Poincaré series of multiplier ideals in any dimension and thus extending the main results for surfaces of Galindo and Monserrat and Alberich-Carramiñana et al. Our results also hold for Poincaré series of test ideals. In order to do so, we introduce a theory of Hilbert functions indexed over $\mathbb{R}$ which gives an unified treatment of both cases.
There has arisen in recent years a substantial theory of "multiplier ideals'' in commutative rings. These are integrally closed ideals with properties that lend themselves to highly interesting applications. But how special are they among integrally closed ideals in general? We show that in a two-dimensional regular local ring with algebraically closed residue field, there is in fact no difference between "multiplier" and "integrally closed" (or "complete.") However, among multiplier ideals arising from an integer multiplying constant (also known as adjoint ideals) the only simple complete one…
Hara and Smith independently proved that in a normal $\mQ$-Gorenstein ring of characteristic $p \gg 0$, the test ideal coincides with the multiplier ideal associated to the trivial divisor. We extend this result for a pair $(R, Δ)$ of a normal ring $R$ and an effective $\mQ$-Weil divisor $Δ$ on $\Spec R$. As a corollary, we obtain the equivalence of strongly F-regular pairs and klt pairs.
We consider the following conjecture: if X is a smooth projective variety over a field of characteristic zero, then there is a dense set of reductions X_s of X to positive characteristic such that the action of the Frobenius morphism on the top Zariski cohomology of the structure sheaf of X_s is bijective. We also consider a conjecture relating certain invariants of singularities in characteristic zero (the multiplier ideals) with invariants in positive characteristic (the test ideals). We prove that the former conjecture implies the latter one in the case of ambient nonsingular varieties.
We give algorithms for computing multiplier ideals using Gröbner bases in Weyl algebras. The algorithms are based on a newly introduced notion which is a variant of Budur--Mustaţǎ--Saito's (generalized) Bernstein--Sato polynomial. We present several examples computed by our algorithms.
The goal of this note is to survey some recent results on the Grauert-Riemenschneider multiplier ideal sheaves on any (reduced) complex space of pure dimension. In particular, we obtain the Briançon-Skoda number for any Noetherian ring of weakly holomorphic functions with weakly rational singularities (\emph{not} essentially of finite type over $\mathbb{C}$ and Cohen-Macaulay local rings), which will partially answer a question of Huneke.
We construct a 3-dimensional complete intersection toric variety on which the subadditivity formula doesn't hold, answering negatively a question by Takagi and Watanabe. A combinatorial proof of the subadditivity formula on 2-dimensional normal toric varieties is also provided.
Koszul homology of monomial ideals provides a description of the structure of such ideals, not only from a homological point of view (free resolutions, Betti numbers, Hilbert series) but also from an algebraic viewpoint. In this paper we show that, in particular, the homology at degree (n-1), with n the number of indeterminates of the ring, plays an important role for this algebraic description in terms of Stanley and irreducible decompositions.
Using perfectoid algebras, we introduce a mixed characteristic analog of the multiplier ideal, respectively test ideal, from characteristic zero, respectively $p > 0$, in the case of a regular ambient ring. We prove several properties about this ideal such as subadditivity. We then use these techniques to derive a uniform bound on the growth of symbolic powers of radical ideals in all excellent regular rings. The analogous result was shown in equal characteristic by Ein-Lazarsfeld-Smith and Hochster-Huneke.
We consider the following conjecture: if X is a smooth projective variety over a field of characteristic zero, then there is a dense set of reductions X_s to positive characteristic such that the action of the Frobenius morphism on the top Zariski cohomology of the structure sheaf on X_s is bijective. We also consider the conjecture relating the multiplier ideals of an ideal J on a nonsingular variety in characteristic zero, and the test ideals of the reductions of J to positive characteristic. We prove that the latter conjecture implies the former one. The converse was proved in a joint paper…
We introduce a new variant of tight closure associated to any fixed ideal $\a$, which we call $\a$-tight closure, and study various properties thereof. In our theory, the annihilator ideal $τ(\a)$ of all $\a$-tight closure relations, which is a generalization of the test ideal in the usual tight closure theory, plays a particularly important role. We prove the correspondence of the ideal $τ(\a)$ and the multiplier ideal associated to $\a$ (or, the adjoint of $\a$ in Lipman's sense) in normal $\Q$-Gorenstein rings reduced from characteristic zero to characteristic $p \gg 0$. Also, in fixed prim…
We study the multiplicity of the jumping numbers of an $\mathfrak m$-primary ideal $\mathfrak a$ in a two-dimensional local ring with a rational singularity. The formula we provide for the multiplicities leads to a very simple and efficient method to detect whether a given rational number is a jumping number. We also give an explicit description of the Poincaré series of multiplier ideals associated to $\mathfrak a$ proving, in particular, that it is a rational function.
The Bernstein-Sato polynomial (or global b-function) is an important invariant in singularity theory, which can be computed using symbolic methods in the theory of D-modules. After surveying algorithms for computing the global b-function, we develop a new method to compute the local b-function for a single polynomial. We then develop algorithms that compute generalized Bernstein-Sato polynomials of Budur-Mustata-Saito and Shibuta for an arbitrary polynomial ideal. These lead to computations of log canonical thresholds, jumping coefficients, and multiplier ideals. Our algorithm for multiplier i…
The Koszul homology of modules of the polynomial ring $R$ is a central object in commutative algebra.It is strongly related with the minimal free resolution of these modules, and thus with regularity, Hilbert functions, etc. Here we consider the case of modules of the form $R/I$ where $I$ is a monomial ideal. So far, some good algorithms have been given in the literature and implemented in different Computer Algebra Systems (e.g. CoCoa, Singular, Macaulay), which compute minimal free resolutions of modules of the form $R/I$ with $I$ an ideal in $R$, which include the case of $I$ being a monomi…
Multiplier ideals, and the vanishing theorems they satisfy, have found many applications in recent years. In the global setting they have been used to study pluricanonical and other linear series on a projective variety. More recently, they have led to the discovery of some surprising uniform results in local algebra. The present notes aim to provide a gentle introduction to the algebraically-oriented local side of the theory. They follow closely a short course on multiplier ideals given in September 2002 at the Introductory Workshop of the program in commutative algebra at MSRI.
The literature in persistent homology often refers to a "structure theorem for finitely generated graded modules over a graded principal ideal domain". We clarify the nature of this structure theorem in this context.
We study Koszul homology over Gorenstein rings. If an ideal is strongly Cohen-Macaulay, the Koszul homology algebra satisfies Poincaré duality. We prove a version of this duality which holds for all ideals and allows us to give two criteria for an ideal to be strongly Cohen-Macaulay. The first can be compared to a result of Hartshorne and Ogus; the second is a generalization of a result of Herzog, Simis, and Vasconcelos using sliding depth.
It gives a class of $p$-Borel principal ideals of a polynomial algebra over a field $K$ for which the graded Betti numbers do not depend on the characteristic of $K$ and the Koszul homology modules have monomial cyclic basis. Also it shows that all principal $p$-Borel ideals have binomial cycle basis on Koszul homology modules.
Let (R,m) be an n-dimensional regular local ring, essentially of finite type over a field of characteristic zero. In this paper we study the relationship between the singularities of the scheme defined by an m-primary ideal I of R and the multiplier ideals J(I^c), with c varying among the positive rational numbers. In particular, we prove that, for every nonnegative integer k, the Samuel multiplicity of I is greater or equal to (n+k)^n/c^n whenever J(I^c) is contained in the (k+1)-th power of the maximal ideal m. This formula generalizes an inequality on log canonical thresholds previously obt…
We discuss a class of modules, which we call $\underline{g}$-weak complete intersection modules, inspired by the weak complete intersection ideals studied by Rahmati, Striuli, and Yang and we present explicit formulas for the generators of Koszul homology with coefficients in a $\underline{g}$-weak complete intersection module. This generalizes work of Herzog and of Corso, Goto, Huneke, Polini, and Ulrich. We use these explicit formulas to study connections between $\underline{g}$-weak complete intersection ideals and weak complete intersection ideals.
This paper gives an explicit formula for the multiplier ideals, and consequently for the log canonical thresholds, of any GL(V)xGL(W)-invariant ideal in the symmetric algebra S of the tensor product of V with the dual of W, where V and W are vector spaces over a field of characteristic 0. This characterization is done in terms of a polytope constructed from the set of Young diagrams corresponding to the Schur modules generating the ideal. Our approach consists in computing the test ideals of some invariant ideals of S in positive characteristic: Namely, we will compute the test ideals (and so …
Fix nonzero ideal sheaves a_1,...,a_r on a normal Q-Gorenstein complex variety X. Fix any positive real number c, and consider the multiplier ideal J of the sum a_1+...+a_r with weighting coefficient c. We construct an exact sequence resolving J by sheaves over X that are direct sums of multiplier ideals for products a_1^{v_1}...a_r^{v_r} for various real vectors v such that v_1+...+v_r = c. The resolution is cellular, in the sense that its boundary maps are encoded by the algebraic chain complex of a regular CW-complex. The CW-complex is naturally expressed as a triangulation T of the simplex…
We revisit a bound on symbolic powers found by Ein-Lazarsfeld-Smith and subsequently improved by Takagi-Yoshida. We show that the original argument of Ein-Lazarsfeld-Smith actually gives the same improvement. On the other hand, we show by examples that any further improvement based on the same technique appears unlikely. This is primarily an exposition; only some examples and remarks might be new.
This paper investigates the existence and properties of a Bernstein-Sato functional equation in nonregular settings. In particular, we construct $D$-modules in which such formal equations can be studied. The existence of the Bernstein-Sato polynomial for a direct summand of a polynomial over a field is proved in this context. It is observed that this polynomial can have zero as a root, or even positive roots. Moreover, a theory of $V$-filtrations is introduced for nonregular rings, and the existence of these objects is established for what we call differentially extensible summands. This famil…
We study in this paper some local invariants attached via multiplier ideals to an effective divisor or ideal sheaf on a smooth complex variety. First considered (at least implicitly) by Libgober and by Loeser and Vaquie, these jumping coefficients consist of an increasing sequence of rational numbers beginning with the log canonical threshold of the divisor or ideal in question. They encode interesting geometric and algebraic information, and we show that they arise naturally in several different contexts. Given a polynomial f having only isolated singularities, results of Varchenko, Loeser an…
Let $\mathfrak{a}$ be an ideal of a commutative noetherian ring $R$ and $M$ an $R$-module with Cosupport in $\mathrm{V}(\mathfrak{a})$. We show that $M$ is $\mathfrak{a}$-coartinian if and only if $\mathrm{Ext}_{R}^{i}(R/\mathfrak{a},M)$ is artinian for all $0\leq i\leq \mathrm{cd}(\mathfrak{a},M)$, which provides a computable finitely many steps to examine $\mathfrak{a}$-coartinianness. We also consider the duality of Hartshorne's questions: for which rings $R$ and ideals $\mathfrak{a}$ are the modules $\mathrm{H}^{\mathfrak{a}}_{i}(M)$ $\mathfrak{a}$-coartinian for all $i\geq 0$; whether the…
We deal with the complete-intersection property of maximally differential ideals. Also, we connect the Gorenstein homology of derivations to the Gorenstein property of the base rings. These equipped with some applications.
For a commutative ring R with an ideal I, generated by a finite regular sequence, we construct differential graded algebras which provide R-free resolutions of I^s and of R/I^s for s>0 and which generalise the Koszul resolution. We derive these from a certain multiplicative double complex. By means of a Cartan-Eilenberg spectral sequence we express Tor_*^R(R/I,R/I^s) and Tor_*^R(R/I, I^s) in terms of exact sequences and find that they are free as R/I-modules. Except for R/I, their product structure turns out to be trivial; instead, we consider an exterior product. The paper is based on ideas b…
