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Title: Categorical and geometric aspects of noncommutative algebras : mutations, tails and perversities
Author: Vitoria, Jorge
ISNI:       0000 0004 2705 0219
Awarding Body: University of Warwick
Current Institution: University of Warwick
Date of Award: 2010
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This thesis concerns some interactions between algebraic geometry and noncommutative algebra in a categorical language. This interplay allows noncommutative constructions of geometric motivation and we explore their structure. In chapters 1 and 2 we survey the main ideas, contextualising this area and introducing the main concepts and results used later in the thesis. These include Morita theory for derived categories, tilting t-structures with respect to torsion theories and generalities on noncommutative projective geometry. Chapter 3 is devoted to prove that, under certain conditions, mutations for quivers with potentials induce derived equivalences on the corresponding Jacobian algebras. We give examples of such Jacobian algebras and show how they occur naturally in geometry. In chapter 4 we turn our attention to to a class of skew-polynomial algebras and explore ways of classifying their noncommutative projective geometry, studying graded Morita equivalences, point varieties and birational equivalences. Finally, chapter 5 contains an algebraic description of perverse coherent t-structures for the derived category of coherent sheaves on a complex projective variety. Furthermore we define analogous structures in adequate noncommutative settings.
Supervisor: Not available Sponsor: Fundação para a Ciência e a Tecnologia (FCT) (SFRH/BD/28268/2006)
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA Mathematics