The lastest version of the notes on the github repository has a variety of changes to and additions. Since the changes are somewhat scattered, I thought it might be useful to summarize them in a post:
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chapter 2, now titled idempotents was started a while back, and contains some of the preliminary material that went into the proof that the the identification of part of the Brauer group with the second cohomology group is compatible with the group structures on both sides. the bulk of this remains to be typed up.
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section 4.2, is a starting point for some generalized Galois theory, in case we need it later. I’m not totally convinced yet that we will, but I started writing it down in case.
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much of chapter 4 is simply a list of topics, many from the last couple lectures which still need to be inserted. Most of the details are available, however in [Jackson’s Notes][jackson].
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chapter 5 is likely where we’ll start today. it covers the concept of the corestriction of a central simple algebra, which will be a useful construction to have later when we start to think about “involutions of the 2nd kind.”
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chapter 5 also contains the beginnings of a discussion of primary decomposition of algebras.
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proposition 3.2.12 was added which shows that field extensions induce homomorphisms on the respective Brauer groups.
see you in class later today! My plan is to talk a little bit about algebras of small degree and segway into involutions.