Multiplicative Tate Spectral Sequences

Abstract

Many important results in mathematics deal with the question of figuring out what objects are ‘the same’. One way of rigorously dealing with this is through the notion of isomorphic objects. However, it is sometimes better to consider a weaker form of ‘sameness’, known as homotopy equivalence, which allow for more flexibility. Although homotopy theory is historically intertwined with fields of mathematics that appeal to our spatial imagination, the concept of a homotopy appears under various guises in other areas as well, and homotopy theoretical generalisations of classical algebra have recently had an upswing in popularity. The papers included in this thesis deal with spectral sequences, which can roughly be understood as computational tools that are able to process large amounts of homotopical information. In particular, this thesis deals with various constructions of the Tate spectral sequence, which is a spectral sequence giving us information on the so-called Tate construction. In my thesis, I construct multiplicative Tate spectral sequences in a larger generality than what was known before. This is motivated by the study of an invariant in homotopical algebra known as topological Hochschild homology, which has been shown to have important connections to arithmetic questions.