This thesis consists of two main parts. The first part addresses period-index problems and symbollength problems of the ?∞-torsion part of Brauer groups of henselian discretely valued fields with residue fields of characteristic ? > 0. In Chapter 1, we provide an overview of the background, progress, and motivation behind period-index problems of Brauer groups and, more generally, Kato’s groups. Chapter 2 recalls some properties of Brauer groups, especially the ?-torsion part. In Chapter 3, we survey Kato’s unit group filtration and Kato’s Swan conductors, which are the main tools in this research area. We investigate the symbol length problems of certain groups related to absolute logarithmic differential forms over fields of characteristic ? > 0. This symbol length problem plays an important role in the period-index problems of Kato’s groups. Chapter 4 presents a systematic investigation of period-index problems of the ?-torsion part of Brauer groups of henselian discretely valued fields with residue fields of characteristic ? > 0. We provide positive support for Chipchakov’s conjecture on this topic. Assuming a conjecture on the symbol length, we offer a complete proof of Chipchakov’s conjecture on the Brauer ?-dimension of henselian discretely valued fields. We also generalize this idea to investigate the symbol length problem of higher Kato’s groups, yielding results on the splitting dimension problems. In Chapter 5, we use Kato’s Swan conductor to investigate the period-index problem of the ?-torsion part of Brauer groups of semiglobal fields. Semiglobal fields are intermediate entities between local fields and global fields. Using patching methods, we reduce the period-index problems to two types: period-index problems of henselian discretely valued fields and quotient fields of a complete local ring of Krull dimension 2. To study the second type, we employ a Gersten-type exact sequence of logarithmic de Rham cohomology with support, analogous to the Artin-Mumford ramification sequence. Both sequences are derived from the Bloch-Ogus spectral sequence. We compute the logarithmic de Rham cohomology with support and their connecting morphisms in this context. Using these computations, we obtain partial results on the period-index problem of semiglobal fields in characteristic ? > 0.
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