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Huy Tai Ha

Publications and source records attributed to Huy Tai Ha.

At least 55 records · Page 3Linked to original sources
arXiv2009

Embedded Associated Primes of Powers of Square-free Monomial Ideals

An ideal I in a Noetherian ring R is normally torsion-free if Ass(R/I^t)=Ass(R/I) for all natural numbers t. We develop a technique to inductively study normally torsion-free square-free monomial ideals. In particular, we show that if a square-free monomial ideal I is minimally not normally torsion-free then the least power t such that I^t has embedded primes is bigger than beta_1, where beta_1 is the monomial grade of I, which is equal to the matching number of the hypergraph H(I) associated to I. If in addition I fails to have the packing property, then embedded primes of I^t do occur when t=beta_1 +1. As an application, we investigate how these results relate to a conjecture of Conforti and Cornuéjols.

math.AC↗
arXiv2009

Splittings of monomial ideals

We provide some new conditions under which the graded Betti numbers of a monomial ideal can be computed in terms of the graded Betti numbers of smaller ideals, thus complementing Eliahou and Kervaire's splitting approach. As applications, we show that edge ideals of graphs are splittable, and we provide an iterative method for computing the Betti numbers of the cover ideals of Cohen-Macaulay bipartite graphs. Finally, we consider the frequency with which one can find particular splittings of monomial ideals and raise questions about ideals whose resolutions are characteristic-dependent.

math.AC↗
arXiv2008

Whiskers and sequentially Cohen-Macaulay graphs

Let G be a simple (i.e., no loops and no multiple edges) graph. We investigate the question of how to modify G combinatorially to obtain a sequentially Cohen-Macaulay graph. We focus on modifications given by adding configurations of whiskers to G, where to add a whisker one adds a new vertex and an edge connecting this vertex to an existing vertex in G. We give various sufficient conditions and necessary conditions on a subset S of the vertices of G so that the graph G \cup W(S), obtained from G by adding a whisker to each vertex in S, is a sequentially Cohen-Macaulay graph. For instance, we show that if S is a vertex cover of G, then G \cup W(S) is a sequentially Cohen-Macaulay graph. On the other hand, we show that if G \backslash S is not sequentially Cohen-Macaulay, then G \cup W(S) is not a sequentially Cohen-Macaulay graph. Our work is inspired by and generalizes a result of Villarreal on the use of whiskers to get Cohen-Macaulay graphs.

math.AC↗
arXiv2008

Cohen-Macaulay admissible clutters

There is a one-to-one correspondence between square-free monomial ideals and clutters, which are also known as simple hypergraphs. It was conjectured that unmixed admissible clutters are Cohen-Macaulay. We prove the conjecture for uniform admissible clutters of heights 2 and 3. For admissible clutters of greater heights, we give a family of examples to show that the conjecture may fail. When the height is 4, we give an additional condition under which unmixed admissible clutters are Cohen-Macaulay.

math.AC↗
arXiv2007

Adjoint line bundles and syzygies of projective varieties

Let X be a smooth projective variety and let K be the canonical divisor of X. In this paper, we study embeddings of X given by adjoint line bundles of the form K+L, where L is an ample line bundle. When X is a regular surface (i.e. H^1(X, O_X) = 0), we obtain a numerical criterion for K+L to have property Np. When X is a regular variety of arbitrary dimension, under a mild condition, we give an explicit calculation for the regularity of the ideal sheaves of such embeddings.

math.AG↗
arXiv2007

Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers

We use the correspondence between hypergraphs and their associated edge ideals to study the minimal graded free resolution of squarefree monomial ideals. The theme of this paper is to understand how the combinatorial structure of a hypergraph H appears within the resolution of its edge ideal I(H). We discuss when recursive formulas to compute the graded Betti numbers of I(H) in terms of its sub-hypergraphs can be obtained; these results generalize our previous work on the edge ideals of simple graphs. We introduce a class of hypergraphs, which we call properly-connected, that naturally generalizes simple graphs from the point of view that distances between intersecting edges are ``well behaved''. For such a hypergraph H (and thus, for any simple graph), we give a lower bound for the regularity of I(H) via combinatorial information describing H, and an upper bound for the regularity when H = G is a simple graph. We also introduce triangulated hypergraphs, a properly-connected hypergraph which is a generalization of chordal graphs. When H is a triangulated hypergraph, we explicitly compute the regularity of I(H) and show that the graded Betti numbers of I(H) are independent of the ground field. As a consequence, many known results about the graded Betti numbers of forests can now be extended to chordal graphs.

math.AC↗
arXiv2007

Cohen-Macaulay multigraded modules

Let S be a standard N^r-graded algebra over a local ring A, and let M be a finitely generated Z^r-graded S-module. We characterize the Cohen-Macaulayness of M in terms of the vanishing of certain sheaf cohomology modules. As a consequence, we apply our result to study the Cohen-Macaulayness of multi-Rees modules (also called Rees modification). Our work extends previous studies on the Cohen-Macaulayness of multi-Rees algebras.

math.AC↗
arXiv2007

Minimal free resolutions and asymptotic behavior of multigraded regularity

Let S be a standard N^k-graded polynomial ring over a field. Let I be a multigraded homogeneous ideal in S and let M be a finitely generated Z^k-graded S-module. We prove that the resolution regularity, a multigraded variant of Castelnuovo-Mumford regularity, of I^nM is asymptotically a linear function. This shows that the well known Z-graded phenomenon carries to multigraded situation.

math.AC↗
arXiv2007

Multigraded regularity, a*-invariant and the minimal free resolution

In recent years, two different multigraded variants of Castelnuovo-Mumford regularity have been developed, namely multigraded regularity, defined by the vanishing of multigraded pieces of local cohomology modules, and the resolution regularity vector, defined by the multidegrees in a minimal free resolution. In this paper, we study the relationship between multigraded regularity and the resolution regularity vector. Our method is to investigate multigraded variants of the usual a*-invariant. This, in particular, provides an effective approach to examining the vanishing of multigraded pieces of local cohomology modules with respect to different graded irrelevant ideals.

math.AC↗
arXiv2006

Splittable ideals and the resolutions of monomial ideals

We provide a new combinatorial approach to study the minimal free resolutions of edge ideals, that is, quadratic square-free monomial ideals. With this method we can recover most of the known results on resolutions of edge ideals with fuller generality, and at the same time, obtain new results. Past investigations on the resolutions of edge ideals usually reduced the problem to computing the dimensions of reduced homology or Koszul homology groups. Our approach circumvents the highly nontrivial problem of computing the dimensions of these groups and turns the problem into combinatorial questions about the associated simple graph. We also show that our technique extends successfully to the study of graded Betti numbers of arbitrary square-free monomial ideals viewed as facet ideals of simplicial complexes.

math.AC↗
arXiv2005

Toroidalization of generating sequences in dimension two function fields

Let k be an algebraically closed field of characteristic 0 and let K*/K be a finite extension of algebraic function fields of transcendence degree 2 over k. Let v* be a k-valuation of K* with valuation ring V* and let v be the restriction of v* to K. Suppose R --> S is an extension of algebraic regular local rings with quotient fields K and K*, respectively, such that V* dominates S and S dominates R. We prove that there exist sequences of quadratic transforms R --> R' and S --> S' along v* such that S' dominates R' and the map between generating sequences of v and v* in R' and S', respectively, has a toroidal structure. Our result extends the Strong Monomialization theorem of Cutkosky and Piltant.

math.AC↗
arXiv2005

Homology multipliers and the relation type of parameter ideals

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.

math.AC↗
arXiv2004

Asymptotic behaviour of arithmetically Cohen-Macaulay blow-ups

This paper addresses problems related to the existence of arithmetic Macaulayfications of projective schemes. Let Y be the blow-up of a projective scheme X = Proj R along the ideal sheaf of I \subset R. It is known that there are embeddings Y \cong Proj k[(I^e)_c] for c \ge d(I)e + 1, where d(I) denotes the maximal generating degree of I, and that there exists a Cohen-Macaulay ring of the form k[(I^e)_c] if and only if H^0(Y,O_Y) = k, H^i(Y,O_Y) = 0 for i = 1,...,dim Y-1, Y is equidimensional and Cohen-Macaulay. Cutkosky and Herzog asked when there is a linear bound on c and e ensuring that k[(I^e)_c] is a Cohen-Macaulay ring. We obtain a surprising compelte answer to this question, namely, that under the above conditions, there are well determined invariants a and b such that k[(I^e)_c] is Cohen-Macaulay for all c > d(I)e + a and e > b. Our approach is based on recent results on the asymptotic linearity of the Castelnuovo-Mumford regularity of ideal powers. We also investigate the existence of a Cohen-Macaulay Rees algebra of the form R[(I^e)_ct] (which provides an arithmetic Macaulayfication for X). If R has negative a*-invariant, we prove that such a Cohen-Macaulay Rees algebra exists if and only if f_*O_Y = O_X, R^i f*O_Y = 0 for i > 0, Y is equidimensional and Cohen-Macaulay. Especially, these conditions imply the Cohen-Macaulayness of R[(I^e)_ct] for all c > d(I)e + a and e > b. The above results can be applied to obtain several new classes of Cohen-Macaulay algebras.

math.AC↗
arXiv2004

Asymptotic behaviour of the length of local cohomology

Let k be a field of characteristic 0, R = k[x_1, ..., x_d] be a polynomial ring, and m its maximal homogeneous ideal. Let I be a homogeneous ideal in R. In this paper we investigate asymptotic behaviour of the quotient between the length of local cohomology group H^0_m(R/I^n) and n^d. We show that this quantity always has a limit as n goes to infinity. We also give an example for which the limit is irrational; in particular, this proves that the length of H^0_m(R/I^n) is not asymptotically a polynomial in n.

math.AC↗
arXiv2003

Powers of ample divisors and syzygies of projective varieties

Suppose X is a projective variety, which needs not be smooth, and L an ample divisor on X. We show that there are integers c and b such that for any nonnegative integer p, L^d is normally generated and embeds X as a variety who defining ideal has linear syzygies upto the p-th step (i.e. L^d has property N_p) for all d >= cp + b.

math.AG↗
arXiv2003

The regularity of points in multi-projective spaces

Let I = p_1^{m_1} \cap ... \cap p_s^{m_s} be the defining ideal of a scheme of fat points in P^{n_1} x ... x P^{n_k} with support in generic position. When all the m_i's are 1, we explicitly calculate the Castelnuovo-Mumford regularity of I. In general, if at least one m_i >= 2, we give an upper bound for the regularity of I, which extends the result of Catalisano, Trung and Valla to the multi-projective case.

math.AC↗
arXiv2003

Projective embeddings of projective schemes blown up at subschemes

Let X be a nonsingular arithmetically Cohen-Macaulay projective scheme, Z a nonsingular subscheme of X. Let π: Y --> X be the blowup of X along the ideal sheaf of Z, E_0 the pull-back of a general hyperplane in X and E the exceptional divisor. In this paper, we study projective embeddings of Y given by the divisor tE_0 - eE. We give explicit values of d and δsuch that for all e > 0 and t > ed + δ, these embeddings is projectively normal and arithmetically Cohen-Macaulay. We also study the regularity of the ideal sheaf and syzygies of these embeddings. When X is a surface and Z is a 0-dimensional subscheme of X, we show that these embeddings possess N_p property for t >> e.

math.AG↗