arXiv subjects
Publications and source records attributed to Ngo Viet Trung.
We give explicit formulas for the Bhattacharya function of m-primary complete monomial ideals in two variables in terms of the vertices of the Newton polyhedra or in terms of the decompositions of the ideals as products of simple ideals.
Absolutely superficial sequences was introduced by P. Schenzel in order to study generalized Cohen-Macaulay (resp. Buchsbaum) modules. For an arbitrary local ring, they turned out to be d-sequences. This paper established properties of absolutely superficial sequences with respect to a module. It is shown that they are closely related to other sequences in the theory of generalized Cohen-Macaulay (resp. Buchsbaum) modules. In particular, there is a bounding function for the Hilbert-Samuel function of every parameter ideal such that this bounding function is attained if and only if the ideal is generated by an absolutely superficial sequence.
We present combinatorial characterizations for the associated primes of the second power of squarefree monomial ideals and criteria for this power to have positive depth or depth greater than one.
We introduce the notion of independent sequences with respect to a monomial order by using the least terms of polynomials vanishing at the sequence. Our main result shows that the Krull dimension of a Noetherian ring is equal to the supremum of the length of independent sequences. The proof has led to other notions of independent sequences, which have interesting applications. For example, we can characterize the maximum number of analytically independent elements in an arbitrary ideal of a local ring and that dim B is not greater than dim A if B is a subalgebra of A and A is a (not necessarily finitely generated) subalgebra of a finitely generated algebra over a Noetherian Jacobson ring.
We present criteria for the Cohen-Macaulayness of a monomial ideal in terms of its primary decomposition. These criteria allow us to use tools of graph theory and of linear programming to study the Cohen-Macaulayness of monomial ideals which are intersections of prime ideal powers. We can characterize the Cohen-Macaulayness of the second symbolic power or of all symbolic powers of a Stanley-Reisner ideal in terms of the simplicial complex. These characterizations show that the simplicial complex must be very compact if some symbolic power is Cohen-Macaulay. In particular, all symbolic powers are Cohen-Macaulay if and only if the simplicial complex is a matroid complex. We also prove that the Cohen-Macaulayness can pass from a symbolic power to another symbolic powers in different ways.
We prove that for m > 2, the m-th symbolic power of a Stanley-Reisner ideal is Cohen-Macaulay if and only if the simplicial complex is a matroid. Similarly, the m-th ordinary power is Cohen-Macaulay for some m > 2 if and only if the complex is a complete intersection. These results solve several open questions on the Cohen-Macaulayness of ordinary and symbolic powers of Stanley-Reisner ideals. Moreover, they have interesting consequences on the Cohen-Macaulayness of symbolic powers of facet ideals and cover ideals.
This paper studies properties of simplicial complexes for which the m-th symbolic power of the Stanley-Reisner ideal equals to the m-th ordinary power for a given m > 1. The main results are combinatorial characterizations of such complexes in the two-dimensional case. It turns out that there exist only a finite number of complexes with this property and that these complexes can be described completely. As a consequence we are able to determine all complexes for which the m-th ordinary power of the Stanley-Reisner ideal is Cohen-Macaulay for a given m > 1.
Two-dimensional squarefree monomial ideals can be seen as the Stanley-Reisner ideals of graphs. The main results of this paper are combinatorial characterizations for the Cohen-Macaulayness of ordinary and symbolic powers of such an ideal in terms of the associated graph.
A forest is the clique complex of a strongly chordal graph and a quasi-forest is the clique complex of a chordal graph. Kruskal--Katona type theorems for forests, quasi-forests, pure forests and pure quasi-forests will be presented. In addition, it will be shown that a quasi-forest is shellable if and only if its $h$-vector $(h_0, h_1, h_2, ...)$ satisfies $h_i = 0$ for $i > 1$.
It is proved that all vertex cover algebras of a hypergraph are standard graded if and only if the hypergraph is unimodular. This has interesting consequences on the symbolic powers of monomial ideals.
We prove that the integral closures of the powers of a squarefree monomial ideal I equal the symbolic powers if and only if I is the edge ideal of a Fulkersonian hypergraph.
We introduce and study vertex cover algebras of weighted simplicial complexes. These algebras are special classes of symbolic Rees algebras. We show that symbolic Rees algebras of monomial ideals are finitely generated and that such an algebra is normal and Cohen-Macaulay if the monomial ideal is squarefree. For a simple graph, the vertex cover algebra is generated by elements of degree 2, and it is standard graded if and only if the graph is bipartite. We also give a general upper bound for the maximal degree of the generators of vertex cover algebras.
We establish a uniform bound for the Castelnuovo-Mumford regularity of associated graded rings of parameter ideals in a generalized Cohen-Macaulay ring. As consequences, we obtain uniform bounds for the relation type and the postulation number. Moreover, we show that generalized Cohen-Macaulay rings can be characterized by the existence of such uniform bounds.
The aim of this paper is to characterize simplicial complexes which have standard graded vertex cover algebras. This property has several nice consequences for the squarefree monomial ideals defining these algebras. It turns out that such simplicial complexes are closely related to a range of hypergraphs which generalize bipartite graphs and trees. These relationships allow us to obtain very general results on standard graded vertex cover algebras which cover previous major results on Rees algebras of squarefree monomial ideals.
The main results of this paper interpret mixed volumes of lattice polytopes as mixed multiplicities of ideals and mixed multiplicities of ideals as Samuel's multiplicities. In particular, we can give a purely algebraic proof of Bernstein's theorem which asserts that the number of common zeros of a system of Laurent polynomial equations in the torus is bounded above by the mixed volume of their Newton polytopes.
This paper proves that the Castelnuovo-Mumford regularities of the product and sum of two monomial complete intersection ideals are at most the sum of the regularities of the two ideals, and provides examples showing that these inequalities do not hold for general complete intersections.
The notion of regularity has been used by S. Kleiman in the construction of bounded families of ideals or sheaves with given Hilbert polynomial, a crucial point in the construction of Hilbert or Picard scheme. In a related direction, Kleiman proved that if I is an equidimensional reduced ideal in a polynomial ring S over an algebraically closed field, then the coefficients of the Hilbert polynomial of R = S/I can be bounded by the dimension and the multiplicity of R. Srinivas and Trivedi proved that the corresponding result does not hold for a local domain. However, they proved that there exist a finite number of Hilbert functions for a local Cohen-Macaulay ring of given multiplicity and dimension. The proofs of the above results are very difficult and involve deep results from Algebraic Geometry. The aim of this paper is to introduce a unified approach which gives more general results and easier proofs of the above mentioned results. This approach is based on the fact that a class C of standard graded algebras has a finite number of Hilbert functions if and only if there are upper bounds for the regularity and the embedding dimension of the members of C.
Our focus in this paper is in effective computation of the core core(I) of an ideal I which is defined to be the intersection of all minimal reductions of I. The first main result is a closed formula for the graded core(m) of the maximal graded ideal m of an arbitrary standard graded algebra A over a field k. This formula allows us to study basic properties of the graded core and to construct counter-examples to some open questions on the core of ideals in a local ring. For instance, we can show that in general, core(m \otimes E) \neq core(m)\otimes E, where E is a field extension of k. From this it follows that the equation core(I R') = core(I)R' does not hold for an arbitrary flat local homomorphism R \to R' of Cohen-Macaulay local rings. The second main result proves the formulae core(I)= (J^r:I^r)I = (J^r:I^r)J = J^{r+1}:I^r for any equimultiple ideal I in a Cohen-Macaulay ring R with with characteristic zero residue field, where J is a minimal reduction of I and r is its reduction number. This result has been obtained independently by Polini-Ulrich and Hyry-Smith in the one-dimensional case or when R is a Gorenstein ring. Moreover, we can prove that core(I) = IK, where K is the conductor of R in the blowing-up ring at I.