The purpose of this thesis is to give a new construction for central extensions of certain classes of infinite dimensional Lie algebras which include multiloop Lie algebras as motivating examples. The key idea of this construction is to view multiloop Lie algebras as twisted forms. This perspective provides a beautiful bridge between infinite dimensional Lie theory and descent theory and is cru...

The theory of local homology modules was initiated by Matlis in 1974. It is a dual version of the theory of local cohomology modules. Mohammadi and Divaani-Aazar (2012) studied the connection between local homology and Gorenstein flat modules by using Gorenstein flat resolutions. In this paper, we introduce generalized local homology modules for complexes and we give several ways for computing ...

The behavior of units in a tensor product of rings is studied, as one factor varies. For example, let k be an algebraically closed field. Let A and B be reduced rings containing k, having connected spectra. Let u ∈ A ⊗k B be a unit. Then u = a ⊗ b for some units a ∈ A and b ∈ B. Here is a deeper consequence, stated for simplicity in the affine case only. Let k be a field, and let φ : R → S be a...

We define a notion of flexible functor from a category X to the category of topological spaces (or other similar categories). This is a generalization of the notion of functor, where the condition of compatibility with composition is replaced by the data of homotopies, homotopies between the homotopies, and so on. When X is a site, we define a sheaf condition and show that a flexible functor (t...

Narrow, inextensible, and naturally flat ribbons have some special and interesting phenomena under isometric deformations. Although a ribbon has a shape between rod and shell, directly applying the geometric representation designed for them imposes a challenge to faithfully reproduce interesting behaviors. We thus parameterize the ribbon surface as a developable ruled surface along its centerli...

In this article, we present $GPW$-flatness property of acts over monoids, which is a generalization of principal weak flatness. We say that a right $S$-act $A_{S}$ is $GPW$-flat if for every $s in S$, there exists a natural number $n = n_ {(s, A_{S})} in mathbb{N}$ such that the functor $A_{S} otimes {}_{S}- $ preserves the embedding of the principal left ideal ${}_{S}(Ss^n)$ into ${}_{S}S$. We...