Francesc Planas-Vilanova

Degree and Algebraic Properties of Lattice and Matrix Ideals

We study the degree of nonhomogeneous lattice ideals over arbitrary fields, and give formulas to compute the degree in terms of the torsion of certain factor groups of

ON THE MODULE OF EFFECTIVE RELATIONS OF A STANDARD ALGEBRA

\mathbb{Z}^s

IDEALS OF HERZOG-NORTHCOTT TYPE

and in terms of relative volumes of lattice polytopes. We also study primary decompositions of lattice ideals over an arbitrary field using the Eisenbud--Sturmfels theory of binomial ideals over algebraically closed fields. We then use these results to study certain families of integer matrices (positive critical binomial (PCB), generalized positive critical binomial (GPCB), critical binomial (CB), and generalized critical binomial (GCB) matrices) and the algebra of their corresponding matrix ideals. In particular, the family of GPCB matrices is shown to be closed under transposition, and previous results for PCB ideals are extended to GPCB ideals. Then, more particularly, we give some applications to the theory of

Rings of weak dimension one and syzygetic ideals

1

On the vanishing and non-rigidity of the André-Quillen (co)homology

-dimensional binomial ideals. If

Vanishing of the André-Quillen homology module H 2(A, B, G(I))

G

Sur l’annulation du deuxième foncteur de (co)homologie d’André-Quillen

is a connected graph, we show as a further application that the order of...

Noncomplete intersection prime ideals in dimension 3

Let A be a commutative ring. We denote by a standard A -algebra a commutative graded A -algebra U =[oplus ] n [ges ]0 U n with U 0 = A and such that U is generated as an A -algebra by the elements of U 1 . Take x a (possibly infinite) set of generators of the A -module U 1 . Let V = A [ t ] be the polynomial ring with as many variables t (of degree one) as x has elements and let f [ratio ] V → U be the graded free presentation of U induced by the x . For n [ges ]2, we will call the module of effective n-relations the A -module E ( U ) n = ker f n / V 1 · ker f n . The minimum positive integer r [ges ]1 such that the effective n -relations are zero for all n [ges ] r +1 is known to be an invariant of U . It is called the relation type of U and is denoted by rt( U ). For an ideal I of A , we define E ( I ) n = E ([Rscr ]( I )) n and rt( I )=rt( [Rscr ] ( I )), where [Rscr ]( I )= [oplus ] n [ges ]0 I n t n ⊂ A [ t ] is the Rees algebra of I . In this paper we give two descriptions of the A -module of effective n -relations. In terms of Andre–Quillen homology we have that E ( U ) n = H 1 ( A , U , A ) n (see 2·3). It turns out that this module does not depend on the chosen [ x ]. In terms of Koszul homology we prove that E ( U ) n = H 1 ([ x ], U ) n (see 2·4). Using these characterizations, we show later some properties on the module of effective n -relations and the relation type of a graded algebra. Our approach has connections with several earlier works on the subject (see [ 2 , 5–7 , 9 , 10 , 13 , 14 ]).

The strong uniform Artin-Rees property in codimension one

This paper takes a new look at ideals generated by 2×2 minors of 2×3 matrices whose entries are powers of three elements not necessarily forming a regular sequence. A special case of this is the ideals determining monomial curves in three-dimensional space, which were studied by Herzog. In the broader context studied here, these ideals are identified as Northcott ideals in the sense of Vasconcelos, and so their liaison properties are displayed. It is shown that they are set-theoretically complete intersections, revisiting the work of Bresinsky and of Valla. Even when the three elements are taken to be variables in a polynomial ring in three variables over a field, this point of view gives a larger class of ideals than just the defining ideals of monomial curves. We then characterize when the ideals in this larger class are prime, we show that they are usually radical and, using the theory of multiplicities, we give upper bounds on the number of their minimal prime ideals, one of these primes being a uniquely determined prime ideal of definition of a monomial curve. Finally, we provide examples of characteristic-dependent minimal prime and primary structures for these ideals.

Integral degree of a ring, reduction numbers and uniform Artin–Rees numbers

We prove that rings of weak dimension one are the rings with all three generated ideals syzygetic This leads to a characterization of these rings in terms of the Andr e Quillen homology Let I be an ideal of a commutative ring A There is a canonical morphism of graded A algebras S I R I from the symmetric algebra of I onto its Rees algebra The ideal I is said to be of linear type if is an isomorphism If S I I is an isomorphism I is said to be syzygetic In C Theorem Costa showed that a domain A is Pr ufer if and only if I is of linear type for every two generated ideal I of A and I is syzygetic for every three generated ideal I of A In this note we show that the preliminary hypothesis that A is a domain can be removed by changing the Pr ufer condition to the condition wd A weak dimension of A one or less Moreover the condition that every two generated ideal of A be of linear type is not necessary Concretely Theorem Let A be a commutative ring The following conditions are equivalent i wd A ii Every ideal of A is of linear type iii Every ideal of A is syzygetic iv Every three generated ideal of A is syzygetic Recall that wd A is the supremum of the at dimensions of all A modules Von Neu mann regular rings are those of weak dimension zero Semihereditary rings i e rings with all its nitely generated ideals projective have weak dimension one or less In fact A is a semihereditary ring if and only if wd A and A is coherent A semihereditary domain is called a Pr ufer ring For a domain A to be Pr ufer is equivalent to wd A see B Mathematics Subject Classi cation Primary F A Secondary D