Brauer p-dimensions of complete discretely valued fields

Abstract: (This is joint work with Bastian Haase) To every central simple algebra A over a field F, one can associate two numerical Brauer class invariants called the index(A) and the period(A). It is well known from that index(A) divides high powers of per(A). The Brauer dimension of a field F is defined to be the least number n such that index(A) divides period(A)^n for every central simple algebra A defined over any finite extension of F. Similarly there exist analogous notions of Brauer-p-dimensions of fields. The 'period-index' questions revolve around bounding the Brauer (p) dimensions of arbitrary fields.

In this talk, we will look at the period-index question over complete discretely valued fields in the so-called 'bad characteristic' case (i.e when the residue field has characteristic p). We will give a flavour of the known results for this question and discuss progress for the cases when the residue fields have small 'p-ranks'. Finally, we will propose a (still open!) conjecture which very precisely relates the Brauer p-dimensions of the complete discretely valued fields to the p-ranks of the residue fields, along with some evidence via a family of examples. The key idea involves working with Kato's filtrations and bounding the symbol length of the second Milnor K group modulo p in a concrete manner, which further relies on the machinery of differentials in characteristic p as developed by Cartier.

number theoryrings and algebras

Audience: researchers in the topic

CMI seminar series

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