704 research outputs found
Evaluations of initial ideals and Castelnuovo-Mumford regularity
This paper characterizes the Castelnuovo-Mumford regularity by evaluating the
initial ideal with respect to the reverse lexicographic order
Integral closures of monomial ideals and Fulkersonian hypergraphs
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.Comment: 5 page
Constructive characterization of the reduction numbers
We present a constructive description of minimal reductions with a given
reduction number. This description has interesting consequences on the minimal
reduction number, the big reduction number, and the core of an ideal. In
particular, it helps solve a conjecture of Vasconcelos on the relationship
between reduction numbers and initial ideals.Comment: 15 page
Absolutely superficial sequences
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
Grobner bases, local cohomology and reduction number
D. Bayer and M. Stillman showed that Grobner bases can be used to compute the
Castelnuovo-Mumford regularity, which is a measure for the vanishing of graded
local cohomology modules. The aim of this paper is to show that the same method
can be applied to study other cohomological invariants as well as the reduction
number
Positivity of mixed multiplicities
This paper studies mixed multiplicities of an arbitrary standard bigraded
algebra and mixed multiplicities of two ideals I, J in a local ring (A,m),
where I is an m-primary ideal and J an arbitrary ideal. The main results are
criteria for their positivity which can be used to compute them effectively. We
also show that the range of positive mixed multiplicities of a bigraded algebra
is rigid if the algebra satisfies the first chain condition and is connected in
codimension one and that this range is always rigid for mixed multiplicities of
ideals. These results can be used to study the mu-invariants of analytic
hypersurfaces, the degree of rational varieties obtained by blowing-up
projective spaces, and the degree of the Stuckrad-Vogel cycles in intersection
theory.Comment: 25 page
On the core of ideals
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.Comment: 20 pages, to appear in Compositio Mat
Mixed multiplicities of ideals versus mixed volumes of polytopes
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.Comment: 19 pages, to appear in Trans. Amer. Math. So
Depth and regularity of powers of sums of ideals
Given arbitrary homogeneous ideals and in polynomial rings and
over a field , we investigate the depth and the Castelnuovo-Mumford
regularity of powers of the sum in in terms of those of
and . Our results can be used to study the behavior of the depth and
regularity functions of powers of an ideal. For instance, we show that such a
depth function can take as its values any infinite non-increasing sequence of
non-negative integers.Comment: 19 pages; to appear in Math.
Krull dimension and Monomial Orders
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.Comment: This is a revised version of the submitted manuscrip
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