The Department of Mathematics and Statistics invites students, faculty and staff to attend a talk given by Dr. Basil Nanayakkara on Friday, May 19th, 2023 from 2:00 pm to 3:00 pm in Mackenzie Chown J-block room 404. The talk is entitled Crossed product algebras and Galois cohomology
Given a Galois extension K/k with Galois group G and a 2-cocycle f : G × G → K∗ , we will construct a k-algebra A = (K/k, f) called a crossedproduct algebra. We will show that A is central over k and simple, and that K is a self-centralizing subfield of A. Thus, A determines an element in the relative Brauer group Br(K/k) of the extension K/k. The similarity class [A] of A depends only on the cohomology class [f] of f. Therefore, the map [f] 7→ [A] from H2 (G, K∗ ) to Br(K/k) is well-defined. It can be shown that this map is a group isomorphism, giving the relation between Galois cohomology and the theory of Brauer groups. We will proceed at a pace comfortable for everyone without paying attention to the time. If time runs out, we will complete the remainder in a future talk.