The height of ideals and regular sequences

Summary We prove that given a Noetherian ringR and a finitely generatedR-inoduleM, there exists a finite set of prime idealsΛ inR such that the depth of an arbitrary idealI onM is determined by the height ofI modulo each of the primes inΛ. As an application we answer a question raised by the second author and V. Srinivas concerning m-adic approximations of regular sequences in a local ring. This article was processed by the author using the Springer-Verlag T_EX P Jourlg macro package 1991.     

On the normality of the rings of Schubert varieties

We prove that the cone over a Schubert variety inG/P (P being a maximal parabolic subgroup of classical type) is normal by exhibiting a 2-regular sequence inR(w) (the homogeneous coordinate ring of the Schubert varietyX(w) inG/P under the canonical protective embeddingG/P ⊂→ (p (H° G/P,L)),L being the ample generator of (PicG/P), which vanishes on the singular locus ofX(w). We also prove the surjectivity ofH° (G/Q, L) H° (X(w), L), whereQ is a classical parabolic subgroup (not necessarily maximal) ofG andL is an ample line bundle onG/Q.     

On cubic graphs which are irreducible for nonorientable surfaces

Let ∑?_n denote the nonorientable surface of nonorientable genus n. A graph G is irreducible for ∑?_n if G does not embed in ∑?_n but any proper subgraph does embed. Let I₃∑?_n) denote the set of cubic irreducible graphs for ∑?_n. This note provesTheorem. I₃(∑?_n) is finite for eachn.     

Large-time solutions of the two-phase Stefan problem with delay

The paper is devoted to questions of an analysis of the large-time solutions to a class of two-phase Stefan problems with delay in the source terms of the governing equations. The case of one space dimension is considered. Results on the asymptotic behaviour of the solutions at t → ∞ are established.