The multiplier ideals of a sum of ideals

We prove that if , are nonzero sheaves of ideals on a complex smooth variety , then for every we have the following relation between the multiplier ideals of , and :

A similar formula holds for the asymptotic multiplier ideals of the sum of two graded systems of ideals.

We use this result to approximate at a given point arbitrary multiplier ideals by multiplier ideals associated to zero dimensional ideals. This is applied to compare the multiplier ideals associated to a scheme in different embeddings.

     

Sequentially Cohen-Macaulay edge ideals

Let be a simple undirected graph on vertices, and let denote its associated edge ideal. We show that all chordal graphs are sequentially Cohen-Macaulay; our proof depends upon showing that the Alexander dual of is componentwise linear. Our result complements Faridi's theorem that the facet ideal of a simplicial tree is sequentially Cohen-Macaulay and implies Herzog, Hibi, and Zheng's theorem that a chordal graph is Cohen-Macaulay if and only if its edge ideal is unmixed. We also characterize the sequentially Cohen-Macaulay cycles and produce some examples of nonchordal sequentially Cohen-Macaulay graphs.

     

Unmixedness and arithmetic properties of matroidal ideals

Let $$R=k[x_1,\ldots ,x_n]$$ be the polynomial ring in n variables over a field k and let I be a matroidal ideal of degree d. In this paper, we study the unmixedness properties and the arithmetical rank of I. Moreover, we show that $$ara(I)=n-d+1$$. This answers the conjecture made by Chiang-Hsieh (Comm Algebra 38:944–952, 2010, Conjecture).     

On the grade of some ideals

An additivity formula is obtained for the grade of an ideal in a residue ring R/I, where I is a perfect ideal. This result is then applied to compute the grade of ideals of linear (inhomogeneous) polynomials. Further results on the homological rigidity of the conormal module I/I² are pointed out. Finally, an elementary proof is given of a result of Buchsbaum concerning the grade of ideals of minors of a matrix.Partially supported a CNPq grantPartially supported by NSF and CNPq grants     

Analytic functionals annihilated by ideals

Let V be a n-dimensional Stein manifold, I be a closed ideal of holomorphic functions on V. It was proved by Roger Gay that, given an analytic functional T such that hT=0 (as a functional) for any h∈I, one can find some (n, n)_compactly supported current , such that for any ϕ ∈Iɛ ^0,0(V) and T(h)= for any h analytic on V. In this paper, we give some explicit construction of Ť in terms of residual currents when I is defined as a complete intersection or is locally Cohen-Macaulay. Moreover, by means of integral representation formulas of the Andersson-Berndtsson-Passare type, we also study the non complete intersection case in order to represent analytic functionals orthogonal to the ideal in terms of currents annihilated (as currents) by some power (less than n) of the local integral closure ofIɛ^0,0.     

Extensions of invariant ideals

An ideai on a setX has the property S(k) iff for every partition ofX into sets of cardinality at leastk there exists a selector inI. We give a solution to a problem of Weglorz [6] proving, in particular, that if G is a group of uncountable, regular cardinalityk then every invariant,k-complete ideal on G can be extended to an invariant,k-complete ideal onG with the property S(k).Presented by Jan Mycielski.     

Reduction numbers and initial ideals

The reduction number of a standard graded algebra is the least integer such that there exists a minimal reduction of the homogeneous maximal ideal of such that . Vasconcelos conjectured that where is the initial ideal of an ideal in a polynomial ring with respect to a term order. The goal of this note is to prove the conjecture.

     

Some remarks on subgroups determined by certain ideals in integral group rings

LetG be a group,ZG the integral group ring ofG andI(G) its augmentation ideal. Subgroups determined by certain ideals ofZG contained inI(G) are identified. For example, whenG=HK, whereH, K are normal subgroups ofG andH∩K⊆ζ(H), then the subgroups ofG determined byI(G)I(H)I(G), andI ³(G)I(H) are obtained. The subgroups of any groupG with normal subgroupH determined by (i)I ²(G)I(H)+I(G)I(H)I(G)+I(H)I²(G), whenH′⊆[H,G,G] and (ii)I(G)I(H)I(G) when degH ²(G/H′, T)≤1, are computed. the subgroup ofG determined byI ⁿ(G)+I(G)I(H) whenH is a normal subgroup ofG withG/H free Abelian is also obtained     

Lifting monomial ideals

We show how to lift any monomial ideal J in n variables to a saturated ideal J of the same codimension in n -+ t variables. We show that I has the same graded Betti numbers as J and we show how to obtain the matrices for the resolution of I. The cohornology of l is described. Making general choices for our lifting, we show that l is the ideal of a reduced union of linear varieties with singularities that are "as small as possible" given the cohomological constraints. The case where J is Artinian is the nicest. In the case of curves we obtain stick figures for l, and in the case of points we obtain certain k-configurations which we can describe in a very precise way.     

Multigraded shifts of matroidal ideals

In this paper, we show that if I is a matroidal ideal, then the ideal generated by the i-th multigraded shifts is also a matroidal ideal for every \(i=0,\ldots ,{\text {proj dim}}(I)\).     

Generalized derivations as Jordan homomorphisms on lie ideals and right ideals

Let R be a prime ring, L a non-central Lie ideal of R and g a non-zero generalized derivation of R. If g acts as a Jordan homomorphism on L, then either g（x） = x for all x ∈ R, or char（R） = 2, R satisfies the standard identity s4（x1, x2, x3, x4）, L is commutative and u2 ∈ Z（R）, for any u C L. We also examine some consequences of this result related to generalized derivations which act as Jordan homomorphisms on the set [I, I], where I is a non-zero right ideal of R.     

Remarks on blowing-up divisorial ideals

In this short paper I describe special situations where a local ring is normally flat along a divisorial prime ideal.This work was supported by C.N.R. (Consiglio Nazionale delle Ricerche)     

Uppers to zero in intersections of prime ideals

Let R be a Noetherian integral domain. The structure of the partially-ordered set of prime ideals of R[z], the polynomial ring in one indeterminate over R, is not fully understood. I demonstrate that if p₁,…,p_n are prime ideals in R[x] with ht(p_i) > 2 and either n = 1 or R is not a Henselian local domain of dimension < 2, then pi D-o-C\pn contains [R] many prime ideals which intersect R at (0). I also show that if R is a Noetherian domain that is not a Henselian local domain and p₁,…,p_n are prime ideals with height > 2 each of which contains a monic polynomial, then their intersection contains [R] many prime ideals meeting R at (0), each containing a monic polynomial.     

Factorization in finite-codimensional ideals of group algebras

Let G be a -compact, locally compact group and I be a closed2-sided ideal with finite codimension in L¹(G). It is shownthat there are a closed left ideal L having a right boundedapproximate identity and a closed right ideal R having a leftbounded approximate identity such that I = L + R. The proofuses ideas from the theory of boundaries of random walks ongroups. 2000 Mathematics Subject Classification: primary 43A20;secondary 42A85, 43A07, 46H10, 46H40, 60B11.     

Mixed multiplicities for arbitrary ideals and generalized Buchsbaum-Rim multiplicities

We introduce first the notion of mixed multiplicities for arbitraryideals in a local d-dimensional Noetherian ring (A,m) which,in some sense, generalizes the concept of mixed multiplicitiesfor m-primary ideals. We also generalize Teissier's productformula for a set of arbitrary ideals and extend the notionof the Buchsbaum-Rim multiplicity (BR-multiplicity) of a submoduleof a free module to the case where the submodule no longer hasfinite colength. For a submodule M of A^p, we introduce a sequenceof multiplicities e^k_BR (M), k = 0, . . . , d + p – 1 whichin the case of an ideal (p = 1) coincides with the multiplicitysequence c0(I,A), . . . , c_d(I,A) defined for an arbitrary idealI of A by Achilles and Manaresi. In the case where M has finitecolength in Ap and is totally decomposable, we prove that ourBR-multiplicity sequence essentially falls into the standardBR-multiplicity of M.     

On the fiber cone of monomial ideals

Archiv der Mathematik - We consider the fiber cone of monomial ideals. It is shown that for monomial ideals $$I\subset K[x,y]$$ of height 2, generated by 3 elements, the fiber cone F(I) of I is a...     

Computation of generalized real radicals of polynomial ideals

For an ordered field (K,T) and an idealI of the polynomial ring , the construction of the generalized real radical ofI is investigated. When (K,T) satisfies some computational requirements, a method of computing is presented. Project supported by the National Natural Science Foundation of China (Grant No. 19661002) and the Climbing Project.     

Acyclic ideals and Koszul complexes

If A is a differential module, then the computation of its homology may frequently be simplified by finding a large acyclic submodule N, for then $\text{H(A)?H(}\text{A}\text{N}\text{)}$ as modules, and hopefully $\text{A}\text{N}$ is more tractable than A. The same idea works if A is a differential algebra, but in that case it is critical to factor out by an acyclic idealI?A, so that $\text{H(A)?H(}\text{A}\text{I}\text{)}$ as algebras. This reduction technique in the classical (ungraded) case is used by Rees [7] and Tate [11], for example. I used a graded version in my thesis [8,9] to study the cohomology of two-stage Postnikov systems. Recently this Acyclic Ideal theorem has been used by Mann, May, Milgram and Sigaard [6] and there has also developed a body of work on the Koszul complex by Józefiak [2,3] and others in which this theorem fits naturally.     

Complemented ideals in the disk algebra

The complemented ideals in the disk algebra are characterized. The projection operators onto the ideals and the complements of the ideals are identified. Research partially supported by a National Academy of Sciences Fellowship to the USSR during the 1977–78 academic year. This author wishes to thank S. Khrushchev, V. Peller, S. Kisljakov, I. Vasoonin, and B. Mitiagin for their many helpful comments and suggestions.     

Extremal Betti numbers of graded ideals

The possible extremal Betti numbers of graded ideals in the polynomial ring K[x₁,…,x_n] in n variables with coefficients in a field K are studied, completing our results in [7]. In case char(K) = 0 we determine, given any integers r < n, the conditions under which there exists a graded ideal I ? K[x₁,…, x_n] with extremal Betti numbers $\beta_{k_{i}k_{i}+\ell_{i}}\ {\rm for}\ i=1,\cdots,r$ . We also treat a similar problem for squarefree lexsegment ideals.