Gaetana Restuccia

ON THE NORMALITY OF MONOMIAL IDEALS OF MIXED PRODUCTS

Let R = K[x, y] be a polynomial ring in two disjoint sets of variables x, y over a field K. We study ideals of mixed products L = IkJr + IsJt such that k + r = s + t, where Ik (resp. Jr ) denotes the ideal of R generated by the square-free monomials of degree k (resp. r) in the x (resp. y ) variables. Our main result is a characterization of when a given ideal L of mixed products is normal. *Partially supported by CONACyT grant 27931E and SNI, México.

Geometric and combinatorial aspects of commutative algebra

On free complete intersections on the number of ideals of finite colength realizable sequences linked to the Hilbert function of a 0-dimensional projective scheme lexsegment ideals in the exterior algebra on the equations defining toric projective varieties KRS and determinantal ideals the Koszul complex in projective dimension 1 Grobner bases as characteristic sets threefolds with degenerate secant variety - on a theorem of G. Scorza the unirationality of all conic bundles implies the unirationality of the quartic threefold Gaussian ideals and the Dedekind-Mertens lemma depth formulas for certain graded modules associated to a filtration - a survey extremal Betti numbers of lexsegment ideals local monomialization isomorphism of complexes and lifts Weil divisors on rational normal scrolls families of Wronksian correspondences a note on Hodges postulation formula for Schubert varieties unimodular rows and subintegral extensions on commutative FGS-rings with ascending condition on annihilators Cohen-Macaulay F-injective homomorphisms valuations of ideals, evolutions and the vanishing of cohomology groups on integral schemes of vector fields Robert Rings and Dutta multiplicities Hilbert functions of squarefree Veronese rings on the conductor of a surface at a point whose projectivized tangent cone is a generic union of lines lifting problems for codimension two subvarieties in Pn border cases rank two bundles and reflexive sheaves on P and corresponding curves - an overview on the structure of ext groups of strongly stable ideals an introduction to tight closure Eisenbud-Goto inequality on Stanley-Reisner rings.

q-Torsion freeness of symmetric powers

Letq be an integer ≥1, we study theq-torsion freeness of the symmetric powers of a module of projective dimension ≥2, by using the approximation complex of the module.

Integrable derivations in rings of unequal characteristic

In this paper we will investigate integrability of derivations in rings of unequal characteristic. Generally speaking this case is more difficult than the case of characteristic p which was studied in [4], New interesting phenomena arise, cf. e.g. Theorem 1 and Example 3. Although we obtain useful sufficient conditions for integrability only in very restricted cases, we also give several examples to illuminate the situation.

On the Symmetric Algebra of the First Syzygy of a Graded Maximal Ideal

Let Syz1(𝔪) be the first syzygy of the graded maximal ideal 𝔪 of a polynomial ring K[x1,…, xn] over a field K. Using the theory of s-sequences, the dimension and depth of the symmetric algebra Sym(Syz1(𝔪)) are calculated. As a conclusion, Sym(Syz1(𝔪)) is not Cohen–Macaulay for any n ≥ 4.

Integrable derivations and formal groups in unequal characteristic

In this paper we study the link between formal group of dimension one and integrable derivations of a ring of unequal characteristic.

ON THE SYMMETRIC ALGEBRA OF SYZYGY MODULES OF MONOMIAL IDEALS

We consider the symmetric algebra of the first syzygy module of a monomial ideal generated by an s -sequence. We introduce on that algebra an admissible term order which allows us to compute its algebraic invariants.

On the set of coactions of an Hopf algebra on K-algebras

For a K-Hopf algebra H, for a K-algebra A, K a field, we study when the set of coactions of H on A is closed with respect to the product of two coactions.

Monomial orders in the vast world of mathematics

Algebraic models of phenomena arising from different fields utilize the polynomial ring P in a finite number of variables, with coefficients in a field K (of characteristic zero). The study of phenomena does not always require to consider a total order on the multiplicative set of monomials. However, in the last years,this is the more attractive direction. Moreover, the classic lexicographic order has been substituted by other orders, sometimes more powerful. Examples of total orders different from the lexicographic order can be found in recent papers of algebraic combinatoric and commutative computational algebra. Algebras of Veronese type K[L] (subrings of polynomial rings) are present in a lot of models like graphs, generalized graphs, transportation problems, economy. Segre products of type Veronese algebras are employed to study bipartite graphs, mixed bipartite generalized graphs, transportation problems and to study varieties of Veronese type and Segre varieties in algebraic geometry. The crucial points are the determination and the study of the toric ideal I of K[L] that we would like to be generated from binomials of the lower possible degree(degree two). From the geometric point of view this means that our algebraic variety is an intersection of quadrics.The order on the monomials needs to determine a Grobner basis G of I since the structure of G reflects strong properties of K[L] (the binomials of degree two of G produce very good properties of the algebra K[L], like Koszul, strongly Koszul).The interest to have a Grobner basis of low degree answers to the request to have S-couples of the same degree of equations, in the not linear case . The linear case is in fact completely known (theory of circuits). Our problems are the following: 1. Research of monomial orders that produce Grobner bases of the minimum possible degree for the ideal I. 2. Determination of the Grobner basis of toric ideals, not necessarily of degree two. We will examine monomial orders in a polynomial ring with multi indexed variables. They appear in the presentation of so called ”subrings of mixed products”. They come from ideals of mixed products studied in [4] and they can be associated to some special complete generalized bipartite graphs.

Actions of formal groups on special quotients of algebras

Let k be a field of characteristic p > 0 and let F be a one dimensional commutative formal group over k. The endomorphisms of a k-algebra A that defines an action of F on A when A is isomorphic to the quotient B/pB, with B torsion free Zalgebra, are studied.