Academic literature on the topic 'Tensor triangulated categories'

Various authors classified the thick triangulated ⊗-subcategories of the category of compact objects for appropriate compactly generated tensor triangulated categories by using supports of objects. In this paper we introduce R-supports for ring objects, showing that these completely determine the thick triangulated ⊗-subcategories. R-supports give a general framework for the celebrated classification theorems by Benson–Carlson–Rickard–Friedlander–Pevtsova, Hopkins–Smith and Hopkins–Neeman–Thomason.

It is known that the thick tensor-ideal subcategories in a tensor triangulated cate-gory can be classified via its prime ideal spectrum.We use this to provide new proofs of two well-known classifications theorems:that of the thick tensor-closed triangulated subcategories of the stable category ofmodules over a finite group algebra, and that of the thick triangulated subcategoriesof the derived category of perfect complexes over a commutative Noetherian ring.

This elaborate consists of a detailed presentation of the construction introduced for the first time by Paul Balmer and aimed to define a locally ringed space associated to a given tensor triangulated category, the so called spectrum of the category. The focus of this thesis is the case of the tensor triangulated category of perfect complexes on a noetherian scheme X, the full triangulated subcategory of the derived category of sheaves of modules consisting of complexes of sheaves locally quasi-isomorphic to complexes of locally free sheaves. This category inherits the structure of derived tensor product of complexes of sheaves of modules, becoming a tensor triangulated category. Its spectrum is a scheme isomorphic to X, providing a powerful reconstruction result.

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-catégories
In 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

This volume contains eight research papers inspired by the 2019 'Equivariant Topology and Derived Algebra' conference, held at the Norwegian University of Science and Technology, Trondheim in honour of Professor J. P. C. Greenlees' 60th birthday. These papers, written by experts in the field, are intended to introduce complex topics from equivariant topology and derived algebra while also presenting novel research. As such this book is suitable for new researchers in the area and provides an excellent reference for established researchers. The inter-connected topics of the volume include: algebraic models for rational equivariant spectra; dualities and fracture theorems in chromatic homotopy theory; duality and stratification in tensor triangulated geometry; Mackey functors, Tambara functors and connections to axiomatic representation theory; homotopy limits and monoidal Bousfield localization of model categories.