We construct classical tilting objects in derived categories of equivariant sheaves on quasi-projective varieties, which give equivalences with derived categories of modules over algebras. Our applications include a conceptual explanation of the importance of the McKay quiver associated to a representation of a finite group G and the development of a McKay correspondence for the cotangent bundle of the projective line. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2008-09-04 14:42:25.099

We construct a semi-orthogonal decomposition for the equivariant derived category of a hypersurface associated to the sum of two potentials. More specifically, if $f,g$ are two homogeneous poynomials of degree $d$ defining smooth Calabi-Yau or general type hypersurfaces in $\mathbb{P}^n$, we construct a semi-orthogonal decomposition of $D[V(f\oplus g)/\mu_d]$. Moreover, every component of the semi-orthogonal decomposition is explicitly given by Fourier-Mukai functors.

The context of this thesis is derived categories in algebraic geometry and geo- metric quotients. Specifically, we prove the embedding of the derived category of a smooth curve of genus greater than one into the derived category of the moduli space of semistable pairs over the curve. We also describe closed cover conditions under which the composition of a pullback and a pushforward induces a fully faithful functor. To prove our main result, we give an exposition of how to think of general Geometric Invariant Theory quotients as quotients by the multiplicative group.

L'objectif de cette thèse est d'étudier les propriétés de variétés toriques particulières. La thèse est divisée en trois parties, les deux premières étant fortement liées. Dans la première partie, nous étudions des variétés algébriques associées aux processus de Markov sur les arbres. A chaque processus de Markov sur un arbre on peut associer une variété algébrique. Motivé par la biologie, nous nous concentrons sur les processus de Markov dé finis par une action de groupe. Nous étudions les conditions pour que la variété obtenue soit torique. Nous donnons un résultat où les variétés obtenues sont normales, ainsi que des exemples où elles ne le sont pas. L'une des principales méthodes que nous utilisons est la généralisation des notions de prises et de réseaux introduites dans [BW07] à des groupes abéliens arbitraires. Dans notre contexte, les réseaux forment un groupe qui agit sur la variété. Par ailleurs, l'espace ambiant de lavariété est la représentation régulière de ce groupe. Le principal problème ouvert que nous essayons de résoudre dans cette partie est une conjecture de Sturmfels et Sullivant [SS05, Conjecture 2] indiquant que le schéma a fine associé au modèle 3-Kimura estdé fini par un idéal engendré en degré 4. Notre meilleur résultat dit que le schéma projectif associé peut être dé fini par un idéal engendré en degré 4. Avec Maria Donten -Bury, nous proposons une méthode pour engendrer l'idéal associé à la variété pour tous les modèles. Nous montrons que notre méthode fonctionne pour de nombreux modèles ainsi que pour les arbres si et seulement si la conjecture de Sturmfels et Sullivant est vraie. Nous présentons quelques applications, par exemple au problème d'identi abilité en biologie. La deuxième partie concerne les variétés algébriques associées aux graphes trivalents pour le modèle de Jukes-Cantor binaire. Il s'agit d'un travail en commun avec Weronika Buczyńska, Jarosław Buczyński et Kaie Kubjas. La variété associée á un graphe peut être représentéevpar un semi-groupe gradué. Nous étudions les liens entre les propriétés du graphe et le semigroupe. Le théorème principal borne le degré en lequel le semi-groupe est engendré par le premier nombre de Betti du graphe, plus un. Dans la dernière partie, nous étudions la structure de la catégorie dérivée des faisceaux cohérents des variétés toriques lisses. Dans un travail commun avec Michał Lasoń [LM11], nous construisons une collection fortement exceptionnelle complète de fi brés en droites pour une grande classe de variétés toriques complètes lisses dont le nombre de Picard est égal á trois. De nombreuses questions concernant le type de collections auxquelles on peut s'attendre sur les variétés toriques de certains types sont encore ouvertes. A ce titre, nous prouvons que Pn éclaté en deux points ne possède pas de collection fortement exceptionnelle complète de fibrés en droites pour n assez grand. Ceci fournit une collection infi nie de contre-exemples à la conjecture de King. Le premier contre-exemple est dû à Hille et Perling [HP06]. Récemment, des contre-exemples ont également été trouvés par E mov [E ] dans le cadre des variétés de Fano. Nous allons travailler sur le corps des nombres complexes C. Toutes les variétés considérées sont des variétés algébriques dans le sens de [Har77]. / The aim of this thesis is to investigate the properties of special toric varieties. The thesis is divided into three parts. The first two of them are strongly related to each other.In the fi rst, main part we study algebraic varieties associated to Markov processes on trees. To each Markov process on a tree one can associate an algebraic variety. Motivated by biology, we focus on Markov processes de fined by a group action. We investigate underwhich conditions the obtained variety is toric. We provide conditions ensuring that the obtained varieties are normal, as well as give examples when they are not. One of the main tools we use is the generalization of the notions of sockets and networks introduced in [BW07] to arbitrary abelian groups. In our setting the networks form a group, that acts on the variety. Moreover the ambient space of the variety is the regular representation of this group. The main open problem that we address in this part is a conjecture of Sturmfels and Sullivant [SS05, Conjecture 2] stating that the afi ne scheme associated to the 3-Kimura model is de fined by an ideal generated in degree 4. Our strongest result states that the associated projective scheme can be generated in degree 4. Together with Maria Donten -Bury we also propose a method for generating the ideal defi ning the variety for any model. We prove that our method works for many models and trees if and only if the conjecture of Sturmfels and Sullivant holds. We present some applications, for example to theidenti ability problem in biology. The second part concerns algebraic varieties associated to trivalent graphs for the binary Jukes-Cantor model. It is a joint work with Weronika Buczyńska, Jarosław Buczyński and Kaie Kubjas. In case of the graph, the associated variety can be represented by a graded semigroup. We investigate the connections between properties of the graph and the semigroup. The main theorem bounds the degree in which the semigroup is generated by the first Betti number of the graph plus one. Due to connections with the first part much of the terminology that we use is either a specialization or generalization of previous de finitions. From the one hand, as we are working with graphs with possible loops the notions of leaves, nodes and valency are more subtile than for trees. From the other hand, as we are dealing only with the binary Jukes-Cantor model, sockets and networks have got a very special form. In the last part we study the structure of the derived category of coherent sheaves for smooth toric varieties. As a result of a joint work with Michał Lasoń [LM11] we construct a full, strongly exceptional collection of line bundles for a large class of smooth, complete toric varieties with Picard number three. Many questions concerning what kind of collections should be expected on toric varieties of certain types are still open. As a contribution we prove that Pn blown up in two points does not have a full, strongly exceptional collection of line bundles for n large enough. This provides an in finite collection of counterexamples to King's conjecture. The first such counterexample is due to Hille andPerling [HP06]. Recently also counterexamples in the Fano case were found by E mov [E ].

vi, 85 p. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. / We define and study a gluing procedure for Bridgeland stability conditions in the situation where a triangulated category has a semiorthogonal decomposition. As one application, we construct an open, contractible subset U in the stability manifold of the derived category [Special characters omitted.] of [Special characters omitted.] -equivariant coherent sheaves on a smooth curve X , associated with a degree 2 map X [arrow right] Y , where Y is another curve. In the case where X is an elliptic curve we construct an open, connected subset in the stability manifold using exceptional collections containing the subset U . We also give a new proof of the constructibility of exceptional collections on [Special characters omitted.] . This dissertation contains previously unpublished co-authored material. / Committee in charge: Alexander Polishchuk, Chairperson, Mathematics; Daniel Dugger, Member, Mathematics; Victor Ostrik, Member, Mathematics; Brad Shelton, Member, Mathematics; Michael Kellman, Outside Member, Chemistry

We develop a theory of unbounded derived categories of quasi-coherent sheaves on algebraic stacks. In particular, we show that these categories are compactly generated by perfect complexes for stacks that either have finite stabilizers or are local quotient stacks. We also extend Toën and Antieau–Gepner’s results on derived Azumaya algebras and compact generation of sheaves on linear categories from derived schemes to derived Deligne–Mumford stacks. These are all consequences of our main theorem: compact generation of a presheaf of triangulated categories on an algebraic stack is local for the quasi-finite flat topology.

Understanding the action of an autoequivalence on a triangulated category is generally a very difficult problem. If one can find a stability condition for which the autoequivalence is "compatible", one can explicitly write down the action of this autoequivalence. In turn, the now understood autoequivalence can provide ways of extracting geometric information from the stability condition. In this thesis, we elaborate on what it means for an autoequivalence and stability condition to be "compatibile" and derive a sufficiency criterion. / text

The common view of brand choice in consumer marketing is that brands compete against each other within a specified product category. For example, different coffee brands are compared and evaluated by consumers and the most preferred brand is selected. Is this the only adequate way of demonstrating how consumers make brand choices? This thesis challenges the common view on brand choice and brand choice determinants in consumer markets on several accounts. First, brand choice is made in goal-derived categories (e.g., "foods to eat while on a diet"), and research on goal-derived categories and consumption goals suggests that consumers often choose between brands from different product categories. For example, a consumer may choose between brands of coffee, tea, and soft drinks to fulfill a consumption goal. Second, there is the question of complementarity. Are brands always chosen one by one? We argue in this thesis that consumers often choose brand constellations from complementary product categories in goal-derived categories (e.g., hamburgers and soft drinks when on a short lunch break). The thesis consists of four articles based on empirical studies. The articles cover single-brand choice and brand constellation choice in goal-derived categories, and the use of goal-derived categories by marketing practitioners. The general conclusion is that consumers evaluate more aspects than just brand-related ones when they choose brands in goal-derived categories. Product category associations (i.e., how typical a product category is perceived in a goal-derived category) are a more important determinant of brand (constellation) choice than brand associations. Also, in brand constellation choice, complementarity (i.e., perceived fit between brands) is more strongly related to brand choice than attitude towards individual brands. / <p>Diss. Stockholm : Handelshögsk., 2003</p>

This thesis consists of three parts. In the first part we look at Hopf algebras. We classify pointed rank one Hopf algebras over fields of prime characteristic which are generated as algebras by the first term of the coradical filtration. These Hopf algebras were classified by Radford and Krop for fields of characteristic zero. We obtain three types of Hopf algebras presented by generators and relations. The third type is new and has not previously appeared in literature. The second part of this thesis deals with Auslander-Reiten theory of finitedimensional algebras over fields. We consider G-transitive algebras and develop necessary conditions for them to have Auslander-Reiten components with Euclidean tree class. Thereby a result in [F3, 4.6] is corrected and generalized. We apply these results to G-transitive blocks of the universal enveloping algebras of restricted p-Lie algebras. Finally we deduce a condition for a smash product of a local basic algebra &Lambda; with a commutative semi-simple group algebra to have components with Euclidean tree class, in terms of the components of the Auslander-Reiten quiver of &Lambda;. In the last part we introduce and analyze Auslander-Reiten components for the bounded derived category of a finite-dimensional algebra. We classify derived categories whose Auslander-Reiten quiver has either a finite stable component or a stable component with finite Dynkin tree class or a bounded stable component. Their Auslander-Reiten quiver is determined. We use these results to show that certain algebras are piecewise hereditary. Also a necessary condition for the existence of components of Euclidean tree class is deduced. We determine components that contain shift periodic complexes.

This thesis contains two main results. The first is a comparison formula for the Donaldson-Thomas invariants of two (complex, smooth and projective) Calabi-Yau threefolds related by a flop; the second is a proof of the projective case of the Crepant Resolution Conjecture for Donaldson-Thomas invariants, as stated by Bryan, Cadman and Young. Both results rely on Bridgeland’s category of perverse coherent sheaves, which is the heart of a t-structure in the derived category of the given Calabi-Yau variety. The first formula is a consequence of various identities in an appropriate motivic Hall algebra followed by an implementation of the integration morphism (using the technology of Joyce and Song). Our proof of the crepant resolution conjecture is a quick and elegant application of the first formula in the context of the derived McKay correspondence of Bridgeland, King and Reid. The first chapter is introductory and is followed by two chapters of background material. The last two chapters are devoted to the proofs of the main results.