We study a new kind of Courant algebroid on Poisson manifolds, which is a variant of the generalized tangent bundle in the sense that the roles of tangent and the cotangent bundle are exchanged. Its symmetry is a semidirect product of β-diffeomorphisms and βtransformations. It is a starting point of an alternative version of the generalized geometry based on the cotangent bundle, such as Dirac ...

Let A be a Banach algebra and M be a Banach left A-module. A linear map δ : M → M is called a generalized derivation if there exists a derivation d : A → A such that δ(ax) = aδ(x) + d(a)x (a ∈ A,x ∈ M). In this paper, we associate a triangular Banach algebra T to Banach A-module M and investigate the relation between generalized derivations on M and derivations on T . In particular, we prove th...

The notion of local cohomology was prominent in the summer school; in this volume it puts in explicit appearances in the papers of Greenlees and Huneke/Taylor, and figures at least peripherally in material related to the paper of Adem. The aim of this short account is to introduce local cohomology and, in a very sketchy way, tie together some of its various manifestations. There are some real d...

Tate local cohomology and Gorenstein theory, which are important generalizations of the classical cohomology, has been investigated. It found that they have such vanishing properties long exact sequences. However, for homology, what about duality? In this paper we concerned with homology homology. first part generalize as study properties, artinianness some sequence modules. We find an artiania...