Hilbert Samuel Multiplicity Research Articles

The multiplicity of a singularity in a vexillary Schubert variety

In a classical-type flag variety, we consider a Schubert variety associated to a vexillary (signed) permutation, and establish a combinatorial formula for the Hilbert-Samuel multiplicity of a point on such a Schubert variety. The formula is expressed in terms of excited Young diagrams, and extends results for Grassmannians due to Krattenthaler, Lakshmibai-Raghavan-Sankaran, and for the maximal isotropic (symplectic and orthogonal) Grassmannians to Ghorpade-Raghavan, Raghavan-Upadhyay, Kreiman, and Ikeda-Naruse. We also provide a new proof of a theorem of Li-Yong in the type A vexillary case.The main ingredient is an isomorphism between certain neighborhoods of fixed points, known as Kazhdan-Lusztig varieties, which, in turn, relies on a direct sum embedding previously used by Anderson-Fulton to relate vexillary loci to Grassmannian loci.

Buchsbaum-Rim multiplicity: A Koszul homology description

We explain the Buchsbaum-Rim multiplicity as the Euler-Poincaré characteristic of an “ordinary” Koszul complex. This provides a generalization of Serre's formula for the Hilbert-Samuel multiplicity in terms of the length of the Koszul homologies.

Complex Links and Hilbert–Samuel Multiplicities

We describe a framework for estimating Hilbert–Samuel multiplicities for pairs of projective varieties from finite point samples rather than defining equations. The first step involves proving that this multiplicity remains invariant under certain hyperplane sections which reduce to a point and to a curve . Next, we establish that equals the Euler characteristic (and hence the cardinality) of the complex link of in . Finally, we provide explicit bounds on the number of uniform point samples needed (in an annular neighborhood of in ) to determine this Euler characteristic with high confidence.

Lim Ulrich sequences and Lech’s conjecture

The long standing Lech’s conjecture in commutative algebra states that for a flat local extension \((R,\mathfrak {m})\rightarrow (S,\mathfrak {n})\) of Noetherian local rings, we have an inequality on the Hilbert–Samuel multiplicities: \(e(R)\le e(S)\). In general the conjecture is wide open when \(\dim R>3\), even in equal characteristic. In this paper, we prove Lech’s conjecture in all dimensions, provided \((R,\mathfrak {m})\) is a standard graded ring over a perfect field localized at the homogeneous maximal ideal. We introduce the notions of lim Ulrich and weakly lim Ulrich sequences. Roughly speaking these are sequences of finitely generated modules that are not necessarily Cohen–Macaulay, but asymptotically behave like Ulrich modules. We prove that the existence of these sequences imply Lech’s conjecture. Though the existence of Ulrich modules is known in very limited cases, we construct weakly lim Ulrich sequences for all standard graded domains over perfect fields of positive characteristic.

Lech’s inequality for the Buchsbaum-Rim multiplicity and mixed multiplicity

We generalize an improved Lech bound, due to Huneke, Smirnov, and Validashti, from the Hilbert-Samuel multiplicity to the Buchsbaum-Rim multiplicity and mixed multiplicity. We reduce the problem to the graded case and then to the polynomial ring case. There we use complete reductions, studied by Rees, to prove sharper bounds for the mixed multiplicity in low dimensions before proving the general case.

Asymptotic Lech's inequality

We explore the classical Lech's inequality relating the Hilbert–Samuel multiplicity and colength of an m-primary ideal in a Noetherian local ring (R,m). We prove optimal versions of Lech's inequality for sufficiently deep ideals in characteristic p>0, and we conjecture that they hold in all characteristics.Our main technical result shows that if (R,m) has characteristic p>0 and Rˆ is reduced, equidimensional, and has an isolated singularity, then for any sufficiently deep m-primary ideal I, the colength and Hilbert–Kunz multiplicity of I are sufficiently close to each other. More precisely, for all ε>0, there exists N≫0 such that for any I⊆R with ℓ(R/I)>N, we have (1−ε)ℓ(R/I)≤eHK(I)≤(1+ε)ℓ(R/I).

On Semicontinuity of Multiplicities in Families

This paper investigates the behavior of Hilbert-Samuel multiplicity and Hilbert-Kunz multiplicity in families of ideals. We show that Hilbert-Samuel multiplicity is upper semicontinuous and that Hi ...

On Semicontinuity of Multiplicities in Families

The paper investigates the behavior of Hilbert-Samuel and Hilbert-Kunz multiplicities in families of ideals. It is shown that Hilbert-Samuel multiplicity is upper semicontinuous almost generally and that Hilbert-Kunz multiplicity is upper semicontinuous in families of finite type. Surprisingly, our machinery can be applied for families over Z and yields a partial solution to the question about characteristic zero Hilbert-Kunz multiplicity posed by Brenner, Li, and Miller. Another application is that for an affine ring the infimum in the definition of F-rational signature, an invariant defined by Hochster and Yao, is attained.

Toward Mixed Multiplicities and Joint Reductions

In the direction towards the question when mixed multiplicities are equal to the Hilbert–Samuel multiplicity of joint reductions, this paper not only generalizes Viet et al. (Proc Am Math Soc 142:1861–1873, 2014, Theorem 3.1) that covers the Rees’s theorem Rees (J Lond Math Soc 29:397–414, 1984, Theorem 2.4), but also removes the hypothesis that joint reductions are systems of parameters in Viet et al. (2014, Theorem 3.1). The results of the paper seem to make the problem of expressing mixed multiplicities into the Hilbert–Samuel multiplicity of joint reductions become closer.

Hilbert–Samuel multiplicity and Northcott’s inequality relative to an Artinian module

Let [Formula: see text] be a commutative quasi-local ring (with identity [Formula: see text]), and let [Formula: see text] be an [Formula: see text]-ideal such that [Formula: see text]. For [Formula: see text] an Artinian [Formula: see text]-module of N-dimension [Formula: see text], we introduce the notion of Hilbert-coefficients of [Formula: see text] relative to [Formula: see text] and give several properties. When [Formula: see text] is a co-Cohen–Macaulay [Formula: see text]-module, we establish the Northcott’s inequality for Artinian modules. As applications, we show some formulas involving the Hilbert coefficients and we investigate the behavior of these multiplicities when the module is the local cohomology module.

A formula for the associated Buchsbaum–Rim multiplicities of a direct sum of cyclic modules II

The associated Buchsbaum–Rim multiplicities of a module are a descending sequence of non-negative integers. These invariants of a module are a generalization of the classical Hilbert–Samuel multiplicity of an ideal. In this article, we compute the associated Buchsbaum–Rim multiplicity of a direct sum of cyclic modules and give a formula for the second to last positive associated Buchsbaum–Rim multiplicity in terms of the ordinary Buchsbaum–Rim and Hilbert–Samuel multiplicities. This is a natural generalization of a formula given by Kirby and Rees.

An algorithm for computing the Hilbert–Samuel multiplicities and reductions of zero-dimensional ideals of Cohen–Macaulay local rings

In this paper, we present an algorithm for computing the minimal reductions of m-primary ideals of Cohen–Macaulay local rings. Using this algorithm, we are able to compute the Hilbert–Samuel multiplicities and solve the membership problem for the integral closure of m-primary ideals.

Homological criteria for minimal multiplicity

Lower bounds on Hilbert–Samuel multiplicity are known for several types of commutative noetherian local rings, and rings with multiplicities which achieve these lower bounds are said to have minimal multiplicity. The first part of this paper gives characterizations of rings of minimal multiplicity in terms of the Ext-algebra of the ring; in particular, we show that minimal multiplicity can be detected via an Ext-algebra which is Gorenstein or Koszul AS-regular. The second part of this paper characterizes rings of minimal multiplicity via a numerical homological invariant introduced by J. Herzog and S. B. Iyengar called linearity defect. Our characterizations allow us to answer in two special cases a question raised by Herzog and Iyengar.

Hilbert–Kunz multiplicities and F-thresholds

Motivated by relations between Hilbert–Samuel multiplicity and F-thresholds, we conjecture an inequality that relates F-thresholds with Hilbert–Kunz multiplicity. In this article, we present several results that support the conjecture. In particular, we prove it for hypersurfaces and we give several consequences of this inequality. In addition, we extend previous results for the Hilbert–Samuel multiplicity.

Interpolating between Hilbert–Samuel and Hilbert–Kunz multiplicity

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 have the same s-multiplicity, and the converse holds under mild conditions. We describe the s-multiplicity of monomial ideals in toric rings as a certain volume in real space.

A formula for the associated Buchsbaum–Rim multiplicities of a direct sum of cyclic modules

In this article, we compute the Buchsbaum–Rim function of two variables associated to a direct sum of cyclic modules and give a formula for the last positive associated Buchsbaum–Rim multiplicity in terms of the ordinary Hilbert–Samuel multiplicity of an ideal. This is a generalization of a formula for the last positive Buchsbaum–Rim multiplicity given by Kirby and Rees.

Lech's conjecture in dimension three

Let (R,m)→(S,n) be a flat local extension of local rings. Lech conjectured around 1960 that there should be a general inequality e(R)≤e(S) on the Hilbert–Samuel multiplicities [24]. This conjecture is known when the base ring R has dimension less than or equal to two [24], and remains open in higher dimensions. In this paper, we prove Lech's conjecture in dimension three when R has equal characteristic. In higher dimension, our method yields substantial partial estimate: e(R)≤(d!/2d)⋅e(S) where d=dim⁡R≥4, in equal characteristic.

Generalized multiplicities of edge ideals

We explore connections between the generalized multiplicities of square-free monomial ideals and the combinatorial structure of the underlying hypergraphs using methods of commutative algebra and polyhedral geometry. For instance, we show that the j-multiplicity is multiplicative over the connected components of a hypergraph, and we explicitly relate the j-multiplicity of the edge ideal of a properly connected uniform hypergraph to the Hilbert–Samuel multiplicity of its special fiber ring. In addition, we provide general bounds for the generalized multiplicities of the edge ideals and compute these invariants for classes of uniform hypergraphs.

Graph connectivity and binomial edge ideals

We relate homological properties of a binomial edge ideal $\mathcal{J}_G$ to invariants that measure the connectivity of a simple graph $G$. Specifically, we show if $R/\mathcal{J}_G$ is a Cohen-Macaulay ring, then graph toughness of $G$ is exactly $\frac{1}{2}$. We also give an inequality between the depth of $R/\mathcal{J}_G$ and the vertex-connectivity of $G$. In addition, we study the Hilbert-Samuel multiplicity, and the Hilbert-Kunz multiplicity of $R/\mathcal{J}_G$.

Koszul factorization and the Cohen–Gabber theorem

We present a sharpened version of the Cohen–Gabber theorem for equicharacteristic, complete local domains $(A,\mathfrak{m},k)$ with algebraically closed residue field and dimension $d>0$. Namely, we show that for any prime number $p$, $\operatorname{Spec}A$ admits a dominant, finite map to $\operatorname{Spec}k[[X_{1},\ldots,X_{d}]]$ with generic degree relatively prime to $p$. Our result follows from Gabber’s original theorem, elementary Hilbert–Samuel multiplicity theory, and a “factorization” of the map induced on the Grothendieck group $\mathbf{G}_{0}(A)$ by the Koszul complex.