On Clifford A-algebras and a framework for localization of A-modules

Abstract

Nowadays, Clifford A-algebras are hot areas of research due to their applicability to di erent disci- plines; capacity to create relationship between quadratic and linear A-morphisms; and relation to tensor A-algebras. In this work, we investigate the commutative property of the Cli ord functor on sheaves of Cli ord algebras, the natural ltration of Cli ord A-algebras, and localization of vector sheaves; out of which two papers are extracted for publication [43], [44]. To present the thesis in a coherent way, we organize the thesis in ve chapters. Chapter 1 is a part where relevant classical results are reviewed. Chapter 2 covers basic results on Cli ord A-algebras of quadratic A-modules (which are of course results obtained by Prof. PP Ntumba [42]). In Chapter 3, we discuss the commutativity of the Cli ord functor Cl and the algebra extension functor (through the tensor product) of the ground algebra sheaf A of a quadratic A-module (E; q). We also observe the existence of an isomorphism between the functors Sð€€€1 and (Sð€€€1A) {. As a particular case, we show the commutativity of the Cli ord functor Cl and the localization functor Sð€€€1. A discussion about the localization of A-modules at prime ideal subsheaves and at subsheaves induced by maximal ideals is also included. In Chapter 4, we study two main A-isomorphisms of Cli ord A-algebras: the main involution and the anti-involution A-isomorphisms, which split Cli ord A-algebras into even sub-A-algebras and sub-A-modules of odd products. Next, we give a de nition for the natural ltration of Cli ord A- algebras and show that for every A-algebra sheaf E, endowed with a regular ltration, one obtains a new graded A-algebra sheaf, denoted Gr(E), which turns out to be A-isomorphic to E. A conclusive remark and list of research topics that can be addressed in connection with this research do appear in Chapter 5.