Brauer $p$-dimension and Kato's Swan Conductor

We use Kato's Swan conductor to study the Brauer $p$-dimension of fields of characteristic $p>0$. We mainly investigate two types of fields: henselian discretely valued fields and semi-global fields. While investigating the Brauer $p$-dimension of semi-global fields, we use a Gersten-type sequence to analyse the ramification behavior of a Brauer class in a $2$-dimensional regular local ring. Using this result, we give a partial result on the Brauer $p$-dimension of function fields of algebraic curves over $\bar{k}((t))$ with good reduction.