Unramified extensions October 20, 2009Posted by Akhil Mathew in algebra, algebraic number theory, number theory.
Tags: discrete valuation rings, Nakayama's lemma, unramified extensions
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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 . 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, 2009Posted by Akhil Mathew in algebra, representation theory.
Tags: algebra, finite-dimensional vector spaces, Nakayama's lemma, representation theory, simple modules
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I had a post a few days back on why simple representations of algebras over a field 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: is our ground field, which we no longer assume algebraically closed (thanks to a comment in the previous post), is a -algebra, its center. We assume is a finitely generated ring over , so in particular Noetherian: each ideal of is finitely generated.
Theorem 1 (Dixmier, Quillen) If is a finite -module, then any simple -module is a finite-dimensional -vector space.