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Unramified extensions October 20, 2009

Posted by Akhil Mathew in algebra, algebraic number theory, number theory.
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As is likely the case with many math bloggers, I’ve been looking quite a bit at MO and haven’t updated on some of the previous series in a while.

Back to ANT. Today, we tackle the case {e=1}. We work in the local case where all our DVRs are complete, and all our residue fields are perfect (e.g. finite) (EDIT: I don’t think this works out in the non-local case). I’ll just state these assumptions at the outset. Then, unramified extensions can be described fairly explicitly. (more…)

Why simple modules are often finite-dimensional II July 22, 2009

Posted by Akhil Mathew in algebra, representation theory.
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I had a post a few days back on why simple representations of algebras over a field {k} which are finitely generated over their centers are always finite-dimensional, where I covered some of the basic ideas, without actually finishing the proof; that is the purpose of this post.

So, let’s review the notation: {k} is our ground field, which we no longer assume algebraically closed (thanks to a comment in the previous post), {A} is a {k}-algebra, {Z} its center. We assume {Z} is a finitely generated ring over {k}, so in particular Noetherian: each ideal of {Z} is finitely generated.

Theorem 1 (Dixmier, Quillen) If {A} is a finite {Z}-module, then any simple {A}-module is a finite-dimensional {k}-vector space.

(more…)

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