Normal transversality and uniform bounds

For two ideals $I$ and $J$ of a noetherian ring, we characterize, in terms of the vanishing of Tor modules, when the associated graded ring of the sum $I+J$ is isomorphic to the tensor product of the associated graded ring of $I$ and the associated graded ring of $J$. It is shown that the relation type of the tensor product of two standard algebras is bounded above by the maximum of the relation type of each algebra. As a consequence, we deduce a uniform bound for the relation type of maximal ideals of an excellent ring and a classical result of Duncan and O'Carroll on the strong uniform Artin-Rees property.Comment: 12 pages, Late

The relation type of affine algebras and algebraic varieties

We introduce the notion of relation type of an affine algebra and prove that it is well defined by using the Jacobi-Zariski exact sequence of Andr\'e-Quillen homology. In particular, the relation type is an invariant of an affine algebraic variety. Also as a consequence of the invariance, we show that in order to calculate the relation type of an ideal in a polynomial ring one can reduce the problem to trinomial ideals. When the relation type is at least two, the extreme equidimensional components play no role. This leads to the non existence of affine algebras of embedding dimension three and relation type two

The primary components of positive critical binomial ideals

A natural candidate for a generating set of the (necessarily prime) defining ideal of an $n$-dimensional monomial curve, when the ideal is an almost complete intersection, is a full set of $n$ critical binomials. In a somewhat modified and more tractable context, we prove that, when the exponents are all positive, critical binomial ideals in our sense are not even unmixed for $n\geq 4$, whereas for $n\leq 3$ they are unmixed. We further give a complete description of their isolated primary components as the defining ideals of monomial curves with coefficients. This answers an open question on the number of primary components of Herzog-Northcott ideals, which comprise the case $n=3$. Moreover, we find an explicit, concrete description of the irredundant embedded component (for $n\geq 4$) and characterize when the hull of the ideal, i.e., the intersection of its isolated primary components, is prime. Note that these last results are independent of the characteristic of the ground field. Our techniques involve the Eisenbud-Sturmfels theory of binomial ideals and Laurent polynomial rings, together with theory of Smith Normal Form and of Fitting ideals. This gives a more transparent and completely general approach, replacing the theory of multiplicities used previously to treat the particular case $n=3$.Comment: 21 page

Maurras a Catalunya : elements per a un debat

Obra ressenyada: Xavier PLA (ed.), Maurras a Catalunya: elements per a un debat. Barcelona: Quaderns Crema, 2012

Modernitat historiogàfica de Ramon d'Abadal i de Vinyals : a propòsit d'un text inèdit

El autor reconstruye la genealogía intelectual de un texto inédito de Ramón d'Abada1, uno de los historiadores más relevantes de la segunda mitad del siglo XX dentro del ámbito catalan y español. Aunque d'Abadal fue experto en la Alta Edad Media catalana, en este pequeño ensayo -al que dedicó mucho esfuerzv se traza una historia de Cataluña en clave naionalista. A través de los "momentos" cruciales que marcan el desarrollo histórico de Cataluña se desprenden las correspondientes "lecciones".L'autor reconstrueix la genealogia intel·lectual d'un text inedit de Ramon d'Abadal, un dels historiadors més rellevants de la segona mitad del segle XX en l'hbit catala i espanyol. Encara que d'abadal va ser expert en l'Alta Edat Mitjana a Catalunya, en aquest petit assaig ell va tracar tina historia global de Catalunya en clau nacionalista. A través del "moments" crucials que marquen el desenvolupament històric de Catalunya es desprenen les corresponent lliçons.Francesc Vilanova reconstructs the intellectual genealogy of an unpublished text written by Ramon d'Abadal, one of the most oustanding catalan historians in Spain in the second half of the Catalan present century. Even though d'Abadal was an expert in the Early Middle Ages in Catalonia, in this article he traces a more general essay of History of Catalonia