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Dissertations / Theses on the topic 'Noncommutative derived algebraic geometry'
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Rennie, Adam Charles. "Noncommutative spin geometry." Title page, contents and introduction only, 2001. http://web4.library.adelaide.edu.au/theses/09PH/09phr4163.pdf.
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Lurie, Jacob 1977. "Derived algebraic geometry." Thesis, Massachusetts Institute of Technology, 2004. http://hdl.handle.net/1721.1/30144.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004.Includes bibliographical references (p. 191-193).
The purpose of this document is to establish the foundations for a theory of derived algebraic geometry based upon simplicial commutative rings. We define derived versions of schemes, algebraic spaces, and algebraic stacks. Our main result is a derived analogue of Artin's representability theorem, which provides a precise criteria for the representability of a moduli functor by geometric objects of these types.
by Jacob Lurie.
Ph.D.
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Toledo, Castro Angel Israel. "Espaces de produits tensoriels sur la catégorie dérivée d'une variété." Electronic Thesis or Diss., Université Côte d'Azur, 2023. http://www.theses.fr/2023COAZ4001.
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Dans cette thèse on est intéressé à l'étude des catégories dérivées d'une variété lisse et projective sur un corps. En particulier on étude l'information géométrique et catégorielle d'une variété et sa catégorie dérivée pour mieux comprendre l'ensemble de structures monoïdales qu'on peut munir à la catégorie dérivée. La motivation de ce projet s'inspire en deux théorèmes. L'un c'est le théorème de reconstruction de Bondal-Orlov qu'établisse que la catégorie dérivée d'une variété avec diviseur (anti-)canonique ample est assez pour récupérer la variété. D'un autre côté, on a la construction du spectrum de Balmer qu'utilise le produit tensoriel dérivé pour récupérer un nombre plus grand de variétés à partir de sa catégorie dérivée de complexes parfaits comme une catégorie monoïdale. L'existence de différentes structures monoïdales est par contre garanti par l'existence des variétés avec des catégories dérivées équivalentes. On a pour but alors comprendre quel est le rôle de les produits tensoriels dans l'existence (ou non existence) de ces types de variétés. Les résultats principaux qu'on a obtenu sont : Si X est une variété avec diviseur (anti-)canonique ample, et ⊠ est une structure de catégorie tensoriel triangulée sur Db(X) tel que le spectrum de Balmer Spc(Db(X),⊠) est isomorphe à X, alors pour tous F,G∈Db(X), on a F⊠G≃F⊗G où ⊗ c'est le produit tensoriel dérivée. On utilise le théorème de Morita pour les dg-catégories de Toën pour donner une caractérisation d'une structure tronquée en termes de bimodules sur un produit des dg-algèbres, qu'induisent une structure de catégorie tensoriel triangulée sur la catégorie homotopique. On a étudié la théorie de déformation de ces structures dans le sens de la cohomologie de Davydov-Yetter. On montre qu'il existe une correspondance entre un des groupes de cohomologie et l'ensemble de associateurs dont le produit tensoriel peut s'en déformer. On utilise des techniques à un niveau des catégories triangulées et aussi des perspectives de la théorie des catégories supérieurs comme des dg-catégories et quasi-catégoriesIn this thesis we are interested in studying derived categories of smooth projective varieties over a field. Concretely, we study the geometric and categorical information from the variety and from it's derived category in order to understand the set of monoidal structures one can equip the derived category with. The motivation for this project comes from two theorems. The first is Bondal-Orlov reconstruction theorem which says that the derived category of a variety with ample (anti-)canonical bundle is enough to recover the variety. On the other hand, we have Balmer's spectrum construction which uses the derived tensor product to recover a much larger number of varieties from it's derived category of perfect complexes as a monoidal category. The existence of different monoidal structure is in turn guaranteed by the existence of varieties with equivalent derived categories. We have as a goal then to understand the role of the tensor products in the existence (or not ) of these sort of varieties. The main results we obtained are If X is a variety with ample (anti-)canonical bundle, and ⊠ is a tensor triangulated category on Db(X) such that the Balmer spectrum Spc(Db(X),⊠) is isomorphic to X, then for any F,G∈Db(X) we have F⊠G≃F⊗G where ⊗ is the derived tensor product. We have used Toën's Morita theorem for dg-categories to give a characterization of a truncated structure in terms of bimodules over a product of dg-algebras, which induces a tensor triangulated category at the level of homotopy categories. We studied the deformation theory of these structures in the sense of Davydov-Yetter cohomology, concretely showing that there is a relationship between one of these cohomology groups and the set of associators that the tensor product can deform into. We utilise techniques at the level of triangulated categories and also perspectives from higher category theory like dg-categories and quasi-categories
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Tang, Xin. "Applications of noncommutative algebraic geometry to representation theory /." Search for this dissertation online, 2006. http://wwwlib.umi.com/cr/ksu/main.
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Goetz, Peter D. "The noncommutative algebraic geometry of quantum projective spaces /." view abstract or download file of text, 2003. http://wwwlib.umi.com/cr/uoregon/fullcit?p3102165.
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Thesis (Ph. D.)--University of Oregon, 2003.Typescript. Includes vita and abstract. Includes bibliographical references (leaves 106-108). Also available for download via the World Wide Web; free to University of Oregon users.
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Schelp, Richard Charles. "The standard model and beyond in noncommutative geometry /." Digital version accessible at:, 2000. http://wwwlib.umi.com/cr/utexas/main.
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Solanki, Vinesh. "Zariski structures in noncommutative algebraic geometry and representation theory." Thesis, University of Oxford, 2011. http://ora.ox.ac.uk/objects/uuid:3fa23b75-9b85-4dc2-9ad6-bdb20d61fe45.
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A suitable subcategory of affine Azumaya algebras is defined and a functor from this category to the category of Zariski structures is constructed. The rudiments of a theory of presheaves of topological structures is developed and applied to construct examples of structures at a generic parameter. The category of equivariant algebras is defined and a first-order theory is associated to each object. For those theories satisfying a certain technical condition, uncountable categoricity and quantifier elimination results are established. Models are shown to be Zariski structures and a functor from the category of equivariant algebras to Zariski structures is constructed. The two functors obtained in the thesis are shown to agree on a nontrivial class of algebras.
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Francis, John (John Nathan Kirkpatrick). "Derived algebraic geometry over En̳-rings." Thesis, Massachusetts Institute of Technology, 2008. http://hdl.handle.net/1721.1/43792.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2008.In title on t.p., double underscored "n" appears as subscript.
Includes bibliographical references (p. 55-56).
We develop a theory of less commutative algebraic geometry where the role of commutative rings is assumed by En-rings, that is, rings with multiplication parametrized by configuration spaces of points in Rn. As n increases, these theories converge to the derived algebraic geometry of Tobn-Vezzosi and Lurie. The class of spaces obtained by gluing En-rings form a geometric counterpart to En-categories, which are higher topological variants of braided monoidal categories. These spaces further provide a geometric language for the deformation theory of general E, structures. A version of the cotangent complex governs such deformation theories, and we relate its values to E&-Hochschild cohomology. In the affine case, this establishes a claim made by Kontsevich. Other applications include a geometric description of higher Drinfeld centers of SE-categories, explored in work with Ben-Zvi and Nadler.
by John Francis.
Ph.D.
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Di, Natale Carmelo. "Grassmannians and period mappings in derived algebraic geometry." Thesis, University of Cambridge, 2015. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.709191.
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Melani, Valerio. "Poisson and coisotropic structures in derived algebraic geometry." Thesis, Sorbonne Paris Cité, 2016. http://www.theses.fr/2016USPCC299/document.
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Dans cette thèse, on définit et on étudie les notions de structure de Poisson et coïsotrope sur un champ dérivé, dans le contexte de la géométrie algébrique dérivée. On considère deux présentations différentes de structure de Poisson : la première est purement algébrique, alors que la deuxième est plus géométrique. On montre que les deux approches sont en fait équivalentes. On introduit aussi la notion de structure coïsotrope sur un morphisme de champs dérivés, encore une fois en présentant deux définitions équivalentes : la première est basée sur une généralisation appropriée de l'opérade Swiss-Cheese de Voronov, tandis que la deuxième est formulée en termes de champs de multivecteurs rélatifs. En particulier, on montre que le morphisme identité admet une unique structure coïsotrope ; cela produit une application d'oubli des structures de Poisson n-décalées aux structures de Poisson (n-1)-décalées. On montre aussi que l'intersection de deux morphismes coïsotropes dans un champ de Poisson n-décalée est naturellement equipée d'une structure de Poisson (n-1)-décalée canonique. En outre, on fournit une équivalence entre l'espace de structures coïsotropes non-dégénérées et l'espace des structures Lagrangiennes en géométrie dérivée, introduites dans les travaux de Pantev-Toën-Vaquié-VezzosiIn this thesis, we define and study Poisson and coisotropic structures on derived stacks in the framework of derived algebraic geometry. We consider two possible presentations of Poisson structures of different flavour: the first one is purely algebraic, while the second is more geometric. We show that the two approaches are in fact equivalent. We also introduce the notion of coisotropic structure on a morphism between derived stacks, once again presenting two equivalent definitions: one of them involves an appropriate generalization of the Swiss Cheese operad of Voronov, while the other is expressed in terms of relative polyvector fields. In particular, we show that the identity morphism carries a unique coisotropic structure; in turn, this gives rise to a non-trivial forgetful map from n-shifted Poisson structures to (n-1)-shifted Poisson structures. We also prove that the intersection of two coisotropic morphisms inside a n-shifted Poisson stack is naturally equipped with a canonical (n-1)-shifted Poisson structure. Moreover, we provide an equivalence between the space of non-degenerate coisotropic structures and the space of Lagrangian structures in derived geometry, as introduced in the work of Pantev-Toën-Vaquié-Vezzosi
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DURIGHETTO, Sara. "Classical and Derived Birational Geometry." Doctoral thesis, Università degli studi di Ferrara, 2019. http://hdl.handle.net/11392/2488324.
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In the field of algebraic geometry, the study of birational transforma-
tions and their properties plays a primary role. In this, there are two different
approaches: the classical one due to the Italian school who focuses on the Cremona
group and a modern one which utilizes instruments like derived categories and
semiorthogonal decompositions.
About the Cremona group, that is the group of birational self-morphisms of P^n, we
do not know much in general and we focus on the complex case. We know a set of
generators only in dimension n = 2. Moreover, we do not have a classication of
curves and linear systems in P^2 up to Cremona transformations. Among the known
results there are: irreducible curves and curves with two irreducible components. In
this thesis we approach tha case of a conguration of lines in the projective plane.
The last theorem lists the known contractible configurations.
From a categorical point of view, the semiorthogonal decompositions of the derived
category of a variety provide some useful invariants in the study of the variety.
Following the work of Clemens-Griffiths about the complex cubic threefold, we
want to characterize the obstructions to the rationality of a variety X of dimension
n. The idea is to collect the component of a semiorthogonal decomposition which
are not equivalent to the derived category of a variety of dimension at least n-1. In
this way we defined the so called Griffiths-Kuznetsov component of X. In this thesis
we study the case of surfaces on an arbitrary field, we define that component and
show that it is a birational invariant. It appears clearly that the Griffiths-Kuznetsov
component vanishes only if the surface is rational.Nell'ambito della geometria algebrica, lo studio delle trasformazioni birazionali e delle loro proprietà riveste un ruolo di importanza primaria. In questo, si affiancano l'approccio classico della scuola italiana che si concentra sul gruppo di Cremona e quello più moderno che utilizza strumenti come categorie derivate e decomposizioni semiortogonali. Del gruppo di Cremona Cr_n, cioé il gruppo degli automorfismi birazionali di P^n, in generale non si conosce molto e ci si concentra sul caso complesso. Si conosce un insieme di generatori solo nel caso di dimensione 2. Inoltre non é ancora nota una classicazione tramite trasformazioni di Cremona delle curve e dei sistemi lineari di P^2. Tra i casi noti ci sono: le curve irriducibili e quelle formate da due componenti irriducibili. In questa tesi ci si approccia al caso di una configurazione di d rette nel piano proiettivo. Il teorema finale fornisce condizioni necessarie o sufficienti alla contraibilità. Da un punto di vista categoriale invece, le decomposizioni semiortogonali della cat- egoria derivata di una varietà ci forniscono degli invarianti utili nello studio della varietà. Seguendo l'approccio di Clemens-Griffiths riguardante la cubica complessa di dimensione 3, si vuole caratterizzare le ostruzioni alla razionalità di una varietà X di dimensione n. L'idea è di raccogliere le componenti di una decomposizione ortog- onale che non sono equivalenti a categorie derivate di varietà di dimensione almeno n-1 e in questo modo definire quella che chiamiamo componente di Griffiths- Kuznetsov di X. In questa tesi si studia il caso delle superci geometricamante razionali su un campo arbitrario, si definisce tale componente e si mostra che essa è un invariante birazionale. Si vede anche che la componente di Griffiths-Kuznetsov è nulla solo se la supercie è razionale.
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Bussi, Vittoria. "Derived symplectic structures in generalized Donaldson-Thomas theory and categorification." Thesis, University of Oxford, 2014. http://ora.ox.ac.uk/objects/uuid:54896cc4-b3fa-4d93-9fa9-2a842ad5e4df.
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This thesis presents a series of results obtained in [13, 18, 19, 23{25, 87]. In [19], we prove a Darboux theorem for derived schemes with symplectic forms of degree k < 0, in the sense of [142]. We use this to show that the classical scheme X = t0(X) has the structure of an algebraic d-critical locus, in the sense of Joyce [87]. Then, if (X, s) is an oriented d-critical locus, we prove in [18] that there is a natural perverse sheaf P·X,s on X, and in [25], we construct a natural motive MFX,s, in a certain quotient ring MμX of the μ-equivariant motivic Grothendieck ring MμX, and used in Kontsevich and Soibelman's theory of motivic Donaldson-Thomas invariants [102]. In [13], we obtain similar results for k-shifted symplectic derived Artin stacks. We apply this theory to categorifying Donaldson-Thomas invariants of Calabi-Yau 3-folds, and to categorifying Lagrangian intersections in a complex symplectic manifold using perverse sheaves, and to prove the existence of natural motives on moduli schemes of coherent sheaves on a Calabi-Yau 3-fold equipped with 'orientation data', as required in Kontsevich and Soibelman's motivic Donaldson-Thomas theory [102], and on intersections L??M of oriented Lagrangians L,M in an algebraic symplectic manifold (S,ω). In [23] we show that if (S,ω) is a complex symplectic manifold, and L,M are complex Lagrangians in S, then the intersection X= L??M, as a complex analytic subspace of S, extends naturally to a complex analytic d-critical locus (X, s) in the sense of Joyce [87]. If the canonical bundles KL,KM have square roots K1/2L, K1/2M then (X, s) is oriented, and we provide a direct construction of a perverse sheaf P·L,M on X, which coincides with the one constructed in [18]. In [24] we have a more in depth investigation in generalized Donaldson-Thomas invariants DTα(τ) defined by Joyce and Song [85]. We propose a new algebraic method to extend the theory to algebraically closed fields K of characteristic zero, rather than K = C, and we conjecture the extension of generalized Donaldson-Thomas theory to compactly supported coherent sheaves on noncompact quasi-projective Calabi-Yau 3-folds, and to complexes of coherent sheaves on Calabi-Yau 3-folds.
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Lim, Bronson. "Equivariant Derived Categories Associated to a Sum of Potentials." Thesis, University of Oregon, 2017. http://hdl.handle.net/1794/22628.
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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.
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NOCERA, Guglielmo. "A study of the spherical Hecke category via derived algebraic geometry." Doctoral thesis, Scuola Normale Superiore, 2022. https://hdl.handle.net/11384/125742.
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Khan, Adeel [Verfasser], and Marc [Akademischer Betreuer] Levine. "Motivic homotopy theory in derived algebraic geometry / Adeel Khan. Betreuer: Marc Levine." Duisburg, 2016. http://d-nb.info/1113534451/34.
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Tiger, Norkvist Axel. "Morphisms of real calculi from a geometric and algebraic perspective." Licentiate thesis, Linköpings universitet, Algebra, geometri och diskret matematik, 2021. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-175740.
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Noncommutative geometry has over the past four of decades grown into a rich field of study. Novel ideas and concepts are rapidly being developed, and a notable application of the theory outside of pure mathematics is quantum theory. This thesis will focus on a derivation-based approach to noncommutative geometry using the framework of real calculi, which is a rather direct approach to the subject. Due to their direct nature, real calculi are useful when studying classical concepts in Riemannian geometry and how they may be generalized to a noncommutative setting. This thesis aims to shed light on algebraic aspects of real calculi by introducing a concept of morphisms of real calculi, which enables the study of real calculi on a structural level. In particular, real calculi over matrix algebras are discussed both from an algebraic and a geometric perspective.Morphisms are also interpreted geometrically, giving a way to develop a noncommutative theory of embeddings. As an example, the noncommutative torus is minimally embedded into the noncommutative 3-sphere.Ickekommutativ geometri har under de senaste fyra decennierna blivit ett etablerat forskningsområde inom matematiken. Nya idéer och koncept utvecklas i snabb takt, och en viktig fysikalisk tillämpning av teorin är inom kvantteorin. Denna avhandling kommer att fokusera på ett derivationsbaserat tillvägagångssätt inom ickekommutativ geometri där ramverket real calculi används, vilket är ett relativt direkt sätt att studera ämnet på. Eftersom analogin mellan real calculi och klassisk Riemanngeometri är intuitivt klar så är real calculi användbara när man undersöker hur klassiska koncept inom Riemanngeometri kan generaliseras till en ickekommutativ kontext. Denna avhandling ämnar att klargöra vissa algebraiska aspekter av real calculi genom att introducera morfismer för dessa, vilket möjliggör studiet av real calculi på en strukturell nivå. I synnerhet diskuteras real calculi över matrisalgebror från både ett algebraiskt och ett geometriskt perspektiv. Morfismer tolkas även geometriskt, vilket leder till en ickekommutativ teori för inbäddningar. Som ett exempel blir den ickekommutativa torusen minimalt inbäddad i den ickekommutativa 3-sfären.
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Byrnes, Sean. "Some computational and geometric aspects of generalized Weyl algebras /." [St. Lucia, Qld.], 2004. http://www.library.uq.edu.au/pdfserve.php?image=thesisabs/absthe18765.pdf.
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Calabrese, John. "In the hall of the flop king : two applications of perverse coherent sheaves to Donaldson-Thomas invariants." Thesis, University of Oxford, 2012. http://ora.ox.ac.uk/objects/uuid:b96b2bdd-8c79-4910-8795-f147bc8b2d16.
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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.
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Bach, Samuel. "Formes quadratiques décalées et déformations." Thesis, Montpellier, 2017. http://www.theses.fr/2017MONTS013/document.
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La L-théorie classique d'un anneau commutatif est construite à partir des formes quadratiques sur cet anneau modulo une relation d'équivalence lagrangienne. Nous construisons la L-théorie dérivée, à partir des formes quadratiques $n$-décalées sur un anneau commutatif dérivé. Nous montrons que les formes $n$-décalées qui admettent un lagrangien possèdent une forme standard. Nous montrons des résultats de chirurgie pour la L-théorie dérivée, qui permettent de réduire une forme quadratique décalée en une forme plus simple équivalente. On compare la L-théorie dérivée avec la L-théorie classique.On définit un champ dérivé des formes quadratiques dérivées, et un champ dérivé des lagrangiens dans une forme, qui sont localement algébriques de présentation finie. On calcule les complexes tangents, et on trouve des points lisses. On montre un résultat de rigidité pour la L-théorie : la L-théorie d'un anneau commutatif est isomorphe à celle d'un voisinage hensélien de cet anneau. Enfin, on définit l'algèbre de Clifford d'une forme quadratique n-décalée, qui est une déformation d'une algèbre symétrique en tant qu'E_k-algèbre. On montre un affaiblissement de la propriété d'Azumaya pour ces algèbres, dans le cas d'un décalage nul n=0, qu'on appelle semi-Azumaya. Cette propriété exprime la trivialité de l'homologie de Hochschild du bimodule de SerreThe classical L-theory of a commutative ring is built from the quadratic forms over this ring modulo a lagrangian equivalence relation.We build the derived L-theory from the n-shifted quadratic forms on a derived commutative ring. We show that forms which admit a lagrangian have a standard form. We prove surgery results for this derived L-theory, which allows to reduce shifted quadratic forms to equivalent simpler forms. We compare classical and derived L-theory.We define a derived stack of shifted quadratic forms and a derived stack of lagrangians in a form, which are locally algebraic of finite presentation. We compute tangent complexes and find smooth points. We prove a rigidity result for L-theory : the L-theory of a commutative ring is isomorphic to that of any henselian neighbourhood of this ring.Finally, we define the Clifford algebra of a n-shifted quadratic form, which is a deformation as E_k-algebra of a symmetric algebra. We prove a weakening of the Azumaya property for these algebras, in the case n=0, which we call semi-Azumaya. This property expresses the triviality of the Hochschild homology of the Serre bimodule
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Prabhu-Naik, Nathan. "Tilting bundles and toric Fano varieties." Thesis, University of Bath, 2015. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.690721.
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This thesis constructs tilting bundles obtained from full strong exceptional collections of line bundles on all smooth toric Fano fourfolds. The tilting bundles lead to a large class of explicit Calabi-Yau-5 algebras, obtained as the corresponding rolled-up helix algebra. We provide two different methods to show that a collection of line bundles is full, whilst the strong exceptional condition is checked using the package QuiversToricVarieties for the computer algebra system Macaulay2, written by the author. A database of the full strong exceptional collections can also be found in this package.
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Feyzbakhsh, Soheyla. "Bridgeland stability conditions, stability of the restricted bundle, Brill-Noether theory and Mukai's program." Thesis, University of Edinburgh, 2018. http://hdl.handle.net/1842/31485.
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In [Bri07], Bridgeland introduced the notion of stability conditions on the bounded derived category D(X) of coherent sheaves on an algebraic variety X. This topic is originally inspired by concepts in string theory and mathematical physics and has many interesting applications in algebraic geometry. In the first part of the thesis, we provide a direct proof of an important result in [Bri08, BMS16] which states there is a two dimensional family of weak Bridgeland stability conditions on the bounded derived category D(X) of coherent sheaves on a variety X. As a first application of this result, we prove an effective restriction theorem which provides sufficient conditions on a stable locally free sheaf on a projective variety such that its restriction to a hypersurface remains stable. Secondly, we extend and complete Mukai's program to reconstruct a K3 surface from a curve on that surface. We show that the K3 surface containing the curve can be obtained uniquely as a Fourier-Mukai partner of a suitable Brill-Noether locus of vector bundles on the curve.
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Marangoni, Davide. "On Derived de Rham cohomology." Thesis, Bordeaux, 2020. http://www.theses.fr/2020BORD0095.
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La cohomologie de de Rham dérivée a été introduite par Luc Illusie en 1972, suite à ses travaux sur le complexe cotangent. Cette théorie semble avoir été oubliée jusqu’aux travaux récents de Bhatt et Beilinson, qui ont donné diverses applications, notamment en théorie de Hodge p-adique. D’autre part, la cohomologie de Rham dérivée intervient de manière cruciale dans une conjecture de Flach-Morin sur les valeurs spéciales des fonctions zêta des schémas arithmétiques. Dans cette thèse, on se propose d’étudier et de calculer la cohomologie de de Rham dérivée dans certains casThe derived de Rham complex has been introduced by Illusie in 1972. Its definition relies on the notion of cotangent complex. This theory seems to have been forgot until the recents works by Be˘ılinson and Bhatt, who gave several applications, in particular in p-adic Hodge Theory. On the other hand, the derived de Rham cohomology has a crucial role in a conjecture by Flach-Morin about special values of zeta functions for arithmetic schemes. The aim of this thesis is to study and compute the Hodge completed derived de Rham complex in some cases
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Schmidt, Benjamin. "Stability Conditions on Threefolds and Space Curves." The Ohio State University, 2016. http://rave.ohiolink.edu/etdc/view?acc_num=osu1460542777.
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Lam, Yan Ting. "Calabi-Yau categories and quivers with superpotential." Thesis, University of Oxford, 2014. http://ora.ox.ac.uk/objects/uuid:20e38c16-e8c7-4ed4-85c9-e22ee6f6e467.
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This thesis studies derived equivalences between total spaces of vector bundles and dg-quivers. A dg-quiver is a graded quiver whose path algebra is a dg-algebra. A quiver with superpotential is a dg-quiver whose differential is determined by a "function" Φ. It is known that the bounded derived category of representations of quivers with superpotential with finite dimensional cohomology is a Calabi- Yau triangulated category. Hence quivers with superpotential can be viewed as noncommutative Calabi- Yau manifolds. One might then ask if there are derived equivalences between Calabi-Yau manifolds and quivers with superpotential. In this thesis, we answer this question and, generalizing Bridgeland [15], give a recipe on how to construct such derived equivalences.
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Hennion, Benjamin. "Formal loops spaces and tangent Lie algebras." Thesis, Montpellier, 2015. http://www.theses.fr/2015MONTS160/document.
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L'espace des lacets lisses C(S^1,M) associé à une variété symplectique M se voit doté d'une structure (quasi-)symplectique induite par celle de M.Nous traiterons dans cette thèse d'un analogue algébrique de cet énoncé.Dans leur article, Kapranov et Vasserot ont introduit l'espace des lacets formels associé à un schéma. Il s'agit d'un analogue algébrique à l'espace des lacets lisses.Nous generalisons ici leur construction à des lacets de dimension supérieure. Nous associons à tout schéma X -- pas forcément lisse -- l'espace L^d(X) de ses lacets formels de dimension d.Nous démontrerons que ce dernier admet une structure de schéma (dérivé) de Tate : son espace tangent est de Tate, c'est-à-dire de dimension infinie mais suffisamment structuré pour se soumettre à la dualité.Nous définirons également l'espace B^d(X) des bulles de X, une variante de l'espace des lacets, et nous montrerons que le cas échéant, il hérite de la structure symplectique de X. Notons que ces résultats sont toujours valides dans des cas plus généraux : X peut être un champs d'Artin dérivé.Pour démontrer nos résultats, nous définirons ce que sont les objets de Tate dans une infinie-catégorie C stable et complète par idempotence.Nous prouverons au passage que le spectre de K-théorie non-connective de Tate(C) est équivalent à la suspension de celui de C, donnant une version infini-catégorique d'un résultat de Saito.Dans le dernier chapitre, nous traiterons d'un problème différent. Nous démontrerons l'existence d'une structure d'algèbre de Lie sur le tangent décalé de n'importe quel champ d'Artin dérivé X. Qui plus est, ce tangent agit sur tout quasi-cohérent E, l'action étant donnée par la classe d'Atiyah de E.Ces résultats sont par exemple valides dans le cas d'un schéma X sans hypothèse de lissitéIf M is a symplectic manifold then the space of smooth loops C(S^1,M) inherits of a quasi-symplectic form. We will focus in this thesis on an algebraic analogue of that result.In their article, Kapranov and Vasserot introduced and studied the formal loop space of a scheme X. It is an algebraic version of the space of smooth loops in a differentiable manifold.We generalize their construction to higher dimensional loops. To any scheme X -- not necessarily smooth -- we associate L^d(X), the space of loops of dimension d. We prove it has a structure of (derived) Tate scheme -- ie its tangent is a Tate module: it is infinite dimensional but behaves nicely enough regarding duality.We also define the bubble space B^d(X), a variation of the loop space.We prove that B^d(X) is endowed with a natural symplectic form as soon as X has one.To prove our results, we develop a theory of Tate objects in a stable infinity category C. We also prove that the non-connective K-theory of Tate(C) is the suspension of that of C, giving an infinity categorical version of a result of Saito.The last chapter is aimed at a different problem: we prove there the existence of a Lie structure on the tangent of a derived Artin stack X. Moreover, any quasi-coherent module E on X is endowed with an action of this tangent Lie algebra through the Atiyah class of E. This in particular applies to not necessarily smooth schemes X
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DASTI, LORENZO. "A COMPARISON BETWEEN GEOMETRIC QUASI-FUNCTORS AND FOURIER-MUKAI FUNCTORS." Doctoral thesis, Università degli Studi di Milano, 2023. https://hdl.handle.net/2434/953452.
Full textAbstract:
Nella mia tesi di Dottorato mi sono occupato della relazione tra la più importante classe di funtori tra categorie derivate, ovvero quella dei funtori di Fourier-Mukai, e i quasi-funtori che sono i morfismi nella localizzazione Hqe dalla categoria delle dg categorie rispetto alle quasi-equivalenze.
Più precisamente: siano X e Y due schemi lisci e propri su un campo. Ho definito una biezione esplicita tra la classe di isomorfismo della categoria dei complessi perfetti su X × Y e l’insieme dei morfismi in Hqe tra due dg enhancement (fissati) delle categorie dei complessi perfetti su X e su Y , rispettivamente. Ho mostrato, inoltre, che tale biezione associa al dg-lift di un funtore di Fourier-Mukai la classe di isomorfismo del suo nucleo, dando così risposta affermativa ad una congettura di Toën.In this PhD thesis I have delt with the relationship between the most important class of functors between derived categories, i.e. Fourier-Mukai functors, and quasi-functors which are the morphisms in the localization Hqe of the category of dg categories with respect to quasi-equivalences. To be more precise: let X and Y be two smooth and proper schemes over a field. I have defined an explicit bijection between the isomorphism class of the triangulated category of perfect complexes over X×Y and the set of morphism in Hqe between two (fixed) dg enhancemenst of the categories of perfect complexes over X and over Y , respectively. Moreover, I have showed that this bijection associates to the dg lift of a Fourier-Mukai functor the isomorphism class of its kernel, giving a positive answer to a conjecture of Toën.
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Wallbridge, James. "Higher Tannaka duality." Toulouse 3, 2011. http://thesesups.ups-tlse.fr/1440/.
Full textAbstract:
Dans cette thèse, nous prouvons un théorème de dualité de Tannaka pour les (infini, 1)-catégories. La dualité classique de Tannaka est une dualité entre certains groupes et catégories monoïdales munies d'une structure particulière. La dualité de Tannaka supérieure renvoie, elle, à une dualité entre certains champs en groupes dérivés et certaines (infini, 1)-catégories monoïdales munies d'une structure particulière. Cette dualité supérieure est définie sur les anneaux dérivés et englobe la théorie de dualité classique. Nous comparons la dualité de Tannaka supérieure à la théorie de dualité de Tannaka classique et portons une attention particulière à la dualité de Tannaka sur les corps. Dans ce dernier cas, cette théorie a une relation étroite avec la théorie des types d'homotopie schématique de Toën. Nous décrivons également trois applications de la théorie : les complexes parfaits, les motifs et leur analogue non-commutatif dû à KontsevichIn this thesis we prove a Tannaka duality theorem for (infini, 1)-categories. Classical Tannaka duality is a duality between certain groups and certain monoidal categories endowed with particular structure. Higher Tannaka duality refers to a duality between certain derived group stacks and certain monoidal (infini, 1)-categories endowed with particular structure. This higher duality theorem is defined over derived rings and subsumes the classical statement. We compare the higher Tannaka duality to the classical theory and pay particular attention to higher Tannaka duality over fields. In the later case this theory has a close relationship with the theory of schematic homotopy types of Toën. We also describe three applications of our theory : perfect complexes and that of both motives and its non-commutative analogue due to Kontsevich
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Pippi, Massimo. "Catégories des singularités, factorisations matricielles et cycles évanescents." Thesis, Toulouse 3, 2020. http://www.theses.fr/2020TOU30049.
Full textAbstract:
Le but de cette thèse est d'étudier les dg-catégories de singularités Sing(X, s), associées à des couples (X, s), où X est un schéma et s est une section d'un fibré vectoriel sur X. La dg-catégorie Sing(X, s) est définie comme le noyau du dg foncteur de Sing(X0) vers Sing(X) induit par l'image directe le long de l'inclusion du lieu de zéros (dérivé) X0 de s dans X. Dans une première partie, nous supposons que le fibré vectoriel est trivial de rang n. On démontre alors un théorème de structure pour Sing(X, s) dans le cas où X = Spec(B) est affine. Cet énoncé affirme que tout objet de Sing(X, s) est représenté par un complexe de B-modules concentré dans n+1 degrés. Lorsque n = 1, cet énoncé généralise l'équivalence d'Orlov , qui identifie Sing(X, s) avec la dg-catégorie des factorisations matricielles MF(X, s), au cas où s epsilon OX(X) n'est pas nécessairement plat. Dans une seconde partie, nous étudions la cohomologie l-adique de Sing(X, s) (définie par A. Blanc - M. Robalo - B. Toën and G. Vezzosi), où s est une section globale d'un fibré en droites. Pour cela, on introduit le faisceau l-adique des cycles évanescents invariantes par monodromie. En utilisant un théorème de D. Orlov généralisé par J. Burke et M. Walker, on calcule la réalisation l-adique de Sing(Spec(B), (f1 ,..., fn)) pour (f1 ,..., fn) epsilon Bn. Dans le dernier chapitre, nous introduisons les faisceaux l-adiques des cycles évanescents itérés pour un schéma sur un anneau de valuation discrète de rang 2. On relie ces faisceaux l-adiques à la réalisation l-adique des dg catégories de singularités des fibres prises sur certains sous-schémas fermés de la baseThe aim of this thesis is to study the dg categories of singularities Sing(X, s) of pairs (X, s), where X is a scheme and s is a global section of some vector bundle over X. Sing(X, s) is defined as the kernel of the dg functor from Sing(X0) to Sing(X) induced by the pushforward along the inclusion of the (derived) zero locus X0 of s in X. In the first part, we restrict ourselves to the case where the vector bundle is trivial. We prove a structure theorem for Sing(X, s) when X = Spec(B) is affine. Roughly, it tells us that every object in Sing(X, s) is represented by a complex of B-modules concentrated in n + 1 consecutive degrees (if s epsilon Bn). By specializing to the case n = 1, we generalize Orlov's theorem, which identifies Sing(X, s) with the dg category of matrix factorizations MF(X, s), to the case where s epsilon OX(X) is not flat. In the second part, we study the l-adic cohomology of Sing(X, s) (as defined by A. Blanc - M. Robalo - B. Toën and G. Vezzosi) when s is a global section of a line bundle. In order to do so, we introduce the l-adic sheaf of monodromy-invariant vanishing cycles. Using a theorem of D. Orlov generalized by J. Burke and M. Walker, we compute the l-adic realization of Sing(Spec(B), (f1 ,..., fn)) for (f1 ,..., fn) epsilon Bn. In the last chapter, we introduce the l-adic sheaves of iterated vanishing cycles of a scheme over a discrete valuation ring of rank 2. We relate one of these l-adic sheaves to the l-adic realization of the dg category of singularities of the fiber over a closed subscheme of the base
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Brav, Christopher. "Tilting objects in derived categories of equivariant sheaves." Thesis, 2008. http://hdl.handle.net/1974/1408.
Full textAbstract:
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
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Safronov, Pavel. "Geometry of integrable hierarchies and their dispersionless limits." Thesis, 2014. http://hdl.handle.net/2152/24818.
Full textAbstract:
This thesis describes a geometric approach to integrable systems. In the first part we describe the geometry of Drinfeld--Sokolov integrable hierarchies including the corresponding tau-functions. Motivated by a relation between Drinfeld--Sokolov hierarchies and certain physical partition functions, we define a dispersionless limit of Drinfeld--Sokolov systems. We introduce a class of solutions which we call string solutions and prove that the tau-functions of string solutions satisfy Virasoro constraints generalizing those familiar from two-dimensional quantum gravity. In the second part we explain how procedures of Hamiltonian and quasi-Hamiltonian reductions in symplectic geometry arise naturally in the context of shifted symplectic structures. All constructions that appear in quasi-Hamiltonian reduction have a natural interpretation in terms of the classical Chern-Simons theory that we explain. As an application, we construct a prequantization of character stacks purely locally.text
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Potashnik, Natasha. "Derived Categories of Moduli Spaces of Semistable Pairs over Curves." Thesis, 2016. https://doi.org/10.7916/D8H99542.
Full textAbstract:
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.
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Roy, Arya. "Towards A Stability Condition on the Quintic Threefold." Diss., 2010. http://hdl.handle.net/10161/2976.
Full textAbstract:
In this thesis we try to construct a stability condition on the quintic threefold. We have not succeeded in proving the existence of such a stability condition. However we have constructed a stability condition on a quotient category of projective space that approximates the quintic. We conjecture the existence of a stability condition on the quintic threefold generated by spherical objects and explore some consequences.
Dissertation
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Kratsios, Anastasis. "Bounding The Hochschild Cohomological Dimension." Thèse, 2014. http://hdl.handle.net/1866/12814.
Full textAbstract:
Ce mémoire a deux objectifs principaux. Premièrement de développer et interpréter
les groupes de cohomologie de Hochschild de basse dimension et deuxièmement de
borner la dimension cohomologique des k-algèbres par dessous; montrant que presque
aucune k-algèbre commutative est quasi-libre.The aim of this master’s thesis is two-fold. Firstly to develop and interpret the low dimensional Hochschild cohomology of a k-algebra and secondly to establish a lower bound for the Hochschild cohomological dimension of a k-algebra; showing that nearly no commutative k-algebra is quasi-free.
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Wallbridge, James. "Higher Tannaka duality." Thesis, 2011. http://hdl.handle.net/2440/69436.
Full textAbstract:
In this thesis we prove a Tannaka duality theorem for (∞, 1)-categories. Classical Tannaka duality is a duality between certain groups and certain monoidal categories endowed with particular structure. Higher Tannaka duality refers to a duality between certain derived group stacks and certain monoidal (∞, 1)-categories endowed with particular structure. This higher duality theorem is defined over derived rings and subsumes the classical statement. We compare the higher Tannaka duality to the classical theory and pay particular attention to higher Tannaka duality over fields. In the later case this theory has a close relationship with the theory of schematic homotopy types of Toёn. We also describe three applications of our theory: perfect complexes and that of both motives and its non-commutative analogue due to Kontsevich.Thesis (Ph.D.) -- University of Adelaide, School of Mathematical Sciences, 2011
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Lowrey, Parker Eastin. "Autoequivalences, stability conditions, and n-gons : an example of how stability conditions illuminate the action of autoequivalences associated to derived categories." Thesis, 2010. http://hdl.handle.net/2152/ETD-UT-2010-05-986.
Full textAbstract:
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
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Bruce, Chris. "C*-algebras from actions of congruence monoids." Thesis, 2020. http://hdl.handle.net/1828/11689.
Full textAbstract:
We initiate the study of a new class of semigroup C*-algebras arising from number-theoretic
considerations; namely, we generalize the construction of Cuntz, Deninger,
and Laca by considering the left regular C*-algebras of ax+b-semigroups from actions
of congruence monoids on rings of algebraic integers in number fields. Our motivation
for considering actions of congruence monoids comes from class field theory and work
on Bost–Connes type systems. We give two presentations and a groupoid model for
these algebras, and establish a faithfulness criterion for their representations. We
then explicitly compute the primitive ideal space, give a semigroup crossed product
description of the boundary quotient, and prove that the construction is functorial
in the appropriate sense. These C*-algebras carry canonical time evolutions, so that
our construction also produces a new class of C*-dynamical systems. We classify the
KMS (equilibrium) states for this canonical time evolution, and show that there are
several phase transitions whose complexity depends on properties of a generalized
ideal class group. We compute the type of all high temperature KMS states, and
consider several related C*-dynamical systems.Graduate