Oriented tree diagram Lie algebras and their abelian ideals

We introduce oriented tree diagram Lie algebras which are generalized from Xu＇s both upward and downward tree diagram Lie algebras, and study certain numerical invariants of these algebras related to abelian ideals.     

Betti numbers of some monomial ideals

In this paper we compute the graded Betti numbers of certain monomial ideals that are not stable. As a consequence we prove a conjecture, stated by G. Fatabbi, on the graded Betti numbers of two general fat points in

     

Normalization of monomial ideals and Hilbert functions

We study the normalization of a monomial ideal, and show how to compute its Hilbert function (using Ehrhart polynomials) if the ideal is zero dimensional. A positive lower bound for the second coefficient of the Hilbert polynomial is shown.

     

Ring graphs and complete intersection toric ideals

We study the family of graphs whose number of primitive cycles equals its cycle rank. It is shown that this family is precisely the family of ring graphs. Then we study the complete intersection property of toric ideals of bipartite graphs and oriented graphs. An interesting application is that complete intersection toric ideals of bipartite graphs correspond to ring graphs and that these ideals are minimally generated by Gröbner bases. We prove that any graph can be oriented such that its toric ideal is a complete intersection with a universal Gröbner basis determined by the cycles. It turns out that bipartite ring graphs are exactly the bipartite graphs that have complete intersection toric ideals for any orientation.     

Licci ideals and the non-almost complete intersection locus

We extend results of Huneke and Ulrich on the structure of ideals in the linkage class of a complete intersection, for the class of ideals which are linked to a complete intersection in an even number of steps. In particular for such ideals the non-almost complete intersection locus has codimension at most ten.

     

Symbolic powers,Serre conditions and Cohen-Macaulay Rees algebras

Summary LetR be a Cohen-Macaulay ring andI an unmixed ideal of heightg which is generically a complete intersection and satisfiesI ⁽ⁿ⁾=Iⁿ for alln≥1. Under what conditions will the Rees algebra be Cohen-Macaulay or have good depth? A series of partial answers to this question is given, relating the Serre condition (S _r ) of the associated graded ring to the depth of the Rees algebra. A useful device in arguments of this nature is the canonical module of the Rees algebra. By making use of the technique of the fundamental divisor, it is shown that the canonical module has the expected form: ω _R[It] ≅(t(1−t) ^g−2). The third author was partially supported by the NSF This article was processed by the author using theLaTex style filecljour1 from Springer-Verlag.     

Modules and ideals of algebras of associative type

In this paper, we study some properties of algebras of associative type introduced in previous papers of the author. We show that a finite-dimensional algebra of associative type over a field of zero characteristic is homogeneously semisimple if and only if a certain form defined by the trace form is nonsingular. For a subclass of algebras of associative type, it is proved that any module over a semisimple algebra is completely reducible. We also prove that any left homogeneous ideal of a semisimple algebra of associative type is generated by a homogeneous idempotent.     

Rees Algebras of Modules

We study Rees algebras of modules within a fairly general framework.We introduce an approach through the notion of Bourbaki idealsthat allows the use of deformation theory. One can talk aboutthe (essentially unique) generic Bourbaki ideal I(E) of a moduleE which, in many situations, allows one to reduce the natureof the Rees algebra of E to that of its Bourbaki ideal I(E).Properties such as Cohen–Macaulayness, normality and beingof linear type are viewed from this perspective. The known numericalinvariants, such as the analytic spread, the reduction numberand the analytic deviation, of an ideal and its associated algebrasare considered in the case of modules. Corresponding notionsof complete intersection, almost complete intersection and equimultiplemodules are examined in some detail. Special consideration isgiven to certain modules which are fairly ubiquitous becauseinteresting vector bundles appear in this way. For these modulesone is able to estimate the reduction number and other invariantsin terms of the Buchsbaum–Rim multiplicity. 2000 MathematicsSubject Classification 13A30 (primary), 13H10, 13B21 (secondary)     

Diagonal invariant ideals of Toeplitz algebras on discrete groups

Diagonal invariant ideals of Toeplitz algebras defined on discrete groups are introduced and studied. In terms of isometric representations of Toeplitz algebras associated with quasi-ordered groups, a character of a discrete group to be amenable is clarified. It is proved that whenG is Abelian, a closed twosided non-trivial ideal of the Toeplitz algebra defined on a discrete Abelian ordered group is diagonal invariant if and only if it is invariant in the sense of Adji and Murphy, thus a new proof of their result is given.     

Extremal Betti Numbers of Some Cohen-Macaulay Binomial Edge Ideals

We provide the regularity and the Cohen-Macaulay type of binomial edge ideals of Cohen-Macaulay cones,and we show the extremal Betti numbers of some classes of Cohen-Macaulay binomial edge ideals:Cohen-Macaulay bipartite and fan graphs.In addition,we compute the Hilbert-Poincaré series of the binomial edge ideals of some Cohen-Macaulay bipartite graphs.     

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.     

Generic initial ideals of Artinian ideals having Lefschetz properties or the strong Stanley property

For a standard Artinian k-algebra A=R/I, we give equivalent conditions for A to have the weak (or strong) Lefschetz property or the strong Stanley property in terms of the minimal system of generators of gin(I). Using the equivalent condition for the weak Lefschetz property, we show that some graded Betti numbers of gin(I) are determined just by the Hilbert function of I if A has the weak Lefschetz property. Furthermore, for the case that A is a standard Artinian k-algebra of codimension 3, we show that every graded Betti number of gin(I) is determined by the graded Betti numbers of I if A has the weak Lefschetz property. And if A has the strong Lefschetz (respectively Stanley) property, then we show that the minimal system of generators of gin(I) is determined by the graded Betti numbers (respectively by the Hilbert function) of I.     

Graded irreducible representations of Leavitt path algebras: A new type and complete classification

We present a new class of graded irreducible representations of a Leavitt path algebra. This class is new in the sense that its representation space is not isomorphic to any of the existing simple Chen modules. The corresponding graded simple modules complete the list of Chen modules which are graded, creating an exhaustive class: the annihilator of any graded simple module is equal to the annihilator of either a graded Chen module or a module of this new type.Our characterization of graded primitive ideals of a Leavitt path algebra in terms of the properties of the underlying graph is the main tool for proving the completeness of such classification. We also point out a problem with the characterization of primitive ideals of a Leavitt path algebra in Rangaswamy (2013) [15].     

The Castelnuovo regularity of the Rees algebra and the associated graded ring

It is shown that there is a close relationship between the invariants characterizing the homogeneous vanishing of the local cohomology and the Koszul homology of the Rees algebra and the associated graded ring of an ideal. From this it follows that these graded rings share the same Castelnuovo regularity and the same relation type. The main result of this paper is however a simple characterization of the Castenuovo regularity of these graded rings in terms of any reduction of the ideal. This characterization brings new insights into the theory of -sequences.

     

Depth and Stanley depth of the edge ideals of square paths and square cycles

In this paper, we compute depth and Stanley depth for the quotient ring of the edge ideal associated to a square path on n vertices. We also compute depth and Stanley depth for the quotient ring of the edge ideal associated to a square cycle on n vertices, when n≡0,3,4( mod 5), and give tight bounds when n≡1,2( mod 5). We also prove a conjecture of Herzog presented in [5 Herzog, J. (2013). A survey on Stanley depth. In: Bigatti, M. A., Gimenez, P., Sáenz-de-Cabezón, E., eds. Monomial Ideals, Computations and Applications. Lecture Notes in Mathematics, Vol. 2083. Heidelberg: Springer, pp. 3–45. https://arxiv.org/pdf/1702.00781.pdf.[Crossref] , [Google Scholar]], for the edge ideals of square paths and square cycles.     

Flags in zero dimensional complete intersection algebras and indices of real vector fields

We introduce bilinear forms in a flag in a complete intersection local -algebra of dimension 0, related to the Eisenbud–Levine, Khimshiashvili bilinear form. We give a variational interpretation of these forms in terms of Jantzen’s filtration and bilinear forms. We use the signatures of these forms to compute in the real case the constant relating the GSV-index with the signature function of vector fields tangent to an even dimensional hypersurface singularity, one being topologically defined and the other computable with finite dimensional commutative algebra methods. Partially supported by Plan Nacional I+D grant no. MTM2004-07203-C02-02, Spain and CONACYT 40329, México.