Ihsen Yengui

Two counterexamples about the Nagata and Serre conjecture rings

Abstract We construct two counterexamples to the open questions : is R 〈 n 〉 strong S (resp. catenary) when R ( n ) is ? The first example is a ring R such that R ( n ) is strong S and R 〈 n 〉 is not. The second is a stably strong S -domain R such that for all n ≥1 and n =∞, R ( n ) is catenary and R 〈 n 〉 is not.

On questions related to saturated chains of primes in polynomial rings

Abstract We propose to give positive answers to the open questions: is R(X,Y) strong S when R(X) is strong S? is R stably strong S (resp., universally catenary) when R[X] is strong S (resp., catenary)? in case R is obtained by a (T,I,D) construction. The importance of these results is due to the fact that this type of ring is the principal source of counterexamples. Moreover, we give an answer to the open questions: is R〈X1,…,Xn〉 residually Jaffard (resp., totally Jaffard) when R(X1,…,Xn) is ? We construct a three-dimensional local ring R such that R(X1,…,Xn) is totally Jaffard (and hence, residually Jaffard) whereas R〈X1,…,Xn〉 is not residually Jaffard (and hence, not totally Jaffard).

About the Spectrum of the Rings R(n) and R⟨n⟩

Abstract The purpose of this paper is to find necessary and sufficient conditions for the transfer of the strong S, catenarian, residually Jaffard and totally Jaffard properties between the rings R(1) and R⟨1⟩ for domains R of dimension ≤ 2. Moreover, we give a positive answer to a conjecture of S. Kabbaj in the case of 2-dimensional integrally closed strong S-domains; we prove that if R is a 2-dimensional integrally closed strong S-domain then R[1] is catenarian.

Computing the V-saturation of finitely-generated submodules of V X m where V is a valuation domain

Recently, Lombardi, Quitte and Yengui have given a Grobner-free algorithm which computes the V-saturation of any finitely generated submodule of V X n , where V is a valuation domain. The goal of this paper is to clarify this algorithm, to give precise complexity bounds, and a complete submodule membership test for the saturation. As application, we give precise degree bounds on syzygies over V X .

Syzygies in Polynomial Rings Over Valuation Domains

It is folklore (see for example Theorem 7.3.3 in [68]) that if V is a valuation domain, then V[X 1, . . . ,X k ] (k ∈ ℕ) is coherent: that is, syzygy modules of finitely-generated ideals of V[X 1, . . . ,X k ] are finitely-generated. The proof in the above-mentioned reference relies on a profound and difficult result published in a huge paper by Gruson and Raynaud [75]. There is nevertheless no known general algorithm for this remarkable result, and it seems difficult to compute the syzygy module even for small polynomials.