Francesc Planas-Vilanova — Page 2

arXiv1999

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.

math.AC↗

arXiv1999

The strong uniform Artin-Rees property in codimension one

The purpose of this paper is to prove the following theorem of uniform Artin-Rees properties: Let $A$ be an excellent (in fact J-2) ring and let $N\subset M$ be two finitely generated $A$-modules such that ${\rm dim}(M/N)\leq 1$. Then there exists an integer $s\geq 1$ such that, for all integers $n\geq s$ and for all ideals $I$ of $A$, $I^{n}M\cap N=I^{n-s}(I^{s}M\cap N)$.

math.AC↗