Graded Derived Equivalences

We consider the equivalences of derived categories of graded rings over different groups. A Morita type equivalence is established between two graded algebras with different group gradings. The results obtained here give a better understanding of the equivalences of derived categories of graded rings.

Mapping the messenger: Exploring the disinformation of QAnon

We analyzed message board content originating with the online persona “Q,” leader of the right-wing conspiracy community known as QAnon. We qualitatively placed all of Q’s messages into one of five qualitatively derived categories: allusion to hidden knowledge, undermining institutions and individuals, inspirational, administration and security, and call to action. Further analysis of how these categories are used by Q over time illustrates how the messaging evolved. Specifically, later Q messaging focused less on hidden knowledge and conspiratorial thinking and more on politics relative to earlier messaging. We also note what Q does not include in messages: very few direct calls to action are made to the QAnon community and no specific, direct calls for violent action. Implications and future directions of research are discussed.

Fitting ideals of p-ramified Iwasawa modules over totally real fields

We completely calculate the Fitting ideal of the classical p-ramified Iwasawa module for any abelian extension K/k of totally real fields, using the shifted Fitting ideals recently developed by the second author. This generalizes former results where we had to assume that only p-adic places may ramify in K/k. One of the important ingredients is the computation of some complexes in appropriate derived categories.

Homological Theory of Representations

Modern developments in representation theory rely heavily on homological methods. This book for advanced graduate students and researchers introduces these methods from their foundations up and discusses several landmark results that illustrate their power and beauty. Categorical foundations include abelian and derived categories, with an emphasis on localisation, spectra, and purity. The representation theoretic focus is on module categories of Artin algebras, with discussions of the representation theory of finite groups and finite quivers. Also covered are Gorenstein and quasi-hereditary algebras, including Schur algebras, which model polynomial representations of general linear groups, and the Morita theory of derived categories via tilting objects. The final part is devoted to a systematic introduction to the theory of purity for locally finitely presented categories, covering pure-injectives, definable subcategories, and Ziegler spectra. With its clear, detailed exposition of important topics in modern representation theory, many of which were unavailable in one volume until now, it deserves a place in every representation theorist's library.