08 Fakultät Mathematik und Physik
Permanent URI for this collectionhttps://elib.uni-stuttgart.de/handle/11682/9
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Item Open Access Lifting and restricting t‐structures(2022) Marks, Frederik; Zvonareva, AlexandraWe explore the interplay between t-structures in the bounded derived category of finitely presented modules and the unbounded derived category of all modules over a coherent ring 𝐴 using homotopy colimits. More precisely, we show that every intermediate t-structure in 𝐷𝑏(mod(𝐴)) can be lifted to a compactly generated t-structure in 𝐷(Mod(𝐴)), by closing the aisle and the coaisle of the t-structure under directed homotopy colimits. Conversely, we provide necessary and sufficient conditions for a compactly generated t-structure in 𝐷(Mod(𝐴)) to restrict to an intermediate t-structure in 𝐷𝑏(mod(𝐴)), thus describing which t-structures can be obtained via lifting. We apply our results to the special case of HRS t-structures. Finally, we discuss various applications to silting theory in the context of finite dimensional algebras.Item Open Access Interactions between universal localisations, ring epimorphisms and tilting modules(2015) Marks, Frederik; König, Steffen (Prof. Dr.)The aim of this thesis is to study the interaction between universal localisations, ring epimorphisms and (generalised) tilting modules. We show that these concepts are intrinsically connected and that they provide various new applications to representation theory. Universal localisations, as defined by Cohn and Schofield, have recently proven to be useful in tilting theory, a fundamental branch of representation theory. In fact, universal localisations were used to classify tilting modules over some rings and they provide interesting decompositions of the derived module category. However, both the structural properties of universal localisations as well as the nature of the various connections to tilting theory are far from being understood. Satisfying answers are only known in special cases. One way to approach localisations is via ring epimorphisms. These are epimorphisms in the category of all rings that are relevant to study certain abelian subcategories of a given module category. Even though it is well-known that universal localisations yield ring epimorphisms, the question of which epimorphisms arise from universal localisations is still widely open. Chapters 2-4 of this thesis provide some answers to the latter question, on the one hand, by looking at finite localisations over any ring and, on the other hand, by focusing on finite dimensional algebras. In particular, over self-injective algebras a classification of certain ring epimorphisms is accessible. We further focus on correspondences between universal localisations and tilting objects. Explicit bijections are established for certain classes of finite dimensional algebras. Chapters 5-7 of this thesis are dedicated to the new concept of silting modules and its relation to localisations. We begin by developing a general theory of silting modules over any ring. These modules generalise tilting modules as well as support τ-tilting modules over a finite dimensional algebra and they turn out to parametrise diverse structures in the derived module category. Subsequently, we show that minimal silting modules classify all universal localisations over a hereditary ring. Also, in the general setup, we can associate a silting object to every localisation. Thus, silting theory provides an adequate setup to study universal localisations.