Irreducible weight modules over the twisted Schrödinger-Virasoro algebra

It is shown that the support of an irreducible weight module over the SchrSdinger-Virasoro Lie algebra with an infinite-dimensional weight space coincides with the weight lattice, and all nontrivial weight spaces of such a module are infinite-dimensional. As a by-product, it is obtained that every simple weight module over Lie algebra of this type with a nontrivial finite-dimensional weight space is a Harish-Chandra module.     

Vanishing of the top local cohomology modules over Noetherian rings

Let R be a (not necessarily local) Noetherian ring and M a finitely generated R-module of finite dimension d. Let be an ideal of R and denote the intersection of all prime ideals . It is shown that

A generalization of the finiteness problem in local cohomology modules

Let a be an ideal of a commutative Noetherian ring R with non-zero identity and let N be a weakly Laskerian R-module and M be a finitely generated R-module. Let t be a non-negative integer. It is shown that if H _a ⁱ (N) is a weakly Laskerian R-module for all i < t, then Hom _R (R/a, H _a ^t (M, N)) is weakly Laskerian R-module. Also, we prove that Ext _R ⁱ (R/a, H _a ^t )) is weakly Laskerian R-module for all i = 0, 1. In particular, if Supp _R (H _a ⁱ (N)) is a finite set for all i < t, then Ext _R ⁱ (R/a, H _a ^t (N)) is weakly Laskerian R-module for all i = 0, 1.     

Holonomic -modules and positive characteristic

We discuss a hypothetical correspondence between holonomic -modules on an algebraic variety X defined over a field of zero characteristic, and certain families of Lagrangian subvarieties in the cotangent bundle to X. The correspondence is based on the reduction to positive characteristic. This article is based on the 5th Takagi Lectures that the author delivered at the University of Tokyo on October 4 and 5, 2008.     

Hopf modules,Frobenius functors and (one-sided) Hopf algebras

We investigate the property of being Frobenius for some functors strictly related with Hopf modules over a bialgebra and how this property reflects on the latter. In particular, we characterize one-sided Hopf algebras with anti-(co)multiplicative one-sided antipode as those for which the free Hopf module functor is Frobenius. As a by-product, this leads us to relate the property of being an FH-algebra (in the sense of Pareigis) for a given bialgebra with the property of being Frobenius for certain naturally associated functors.     

Transformation d'halphen

We give a modern version of the following result of Halphen: “degrees and classes of successive evolutes of a plane algebraic curve form two arithmetic progressions of the same ratio after a sufficient number of iterations”. The iteration of this transform provides a desingularisationprocess (branch by branch) of plane curves, different to the Nash transform and the Noether process. To prove these two points,we introduce systematically Puiseux's exponents, semi-groups of exponents and use elimination and duality formulae.     

On Lie Algebra Crossed Modules

Abstract

The goal of this article is to construct a crossed module representing the cocycle 〈[,],〉 generating H ³(;?) for a simple complex Lie algebra .     

A Version of the Weak Krull–Schmidt Theorem for Infinite Direct Sums of Uniserial Modules

We show a version of the weak Krull–Schmidt theorem concerning infinite families of uniserial modules.