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.
Why simple modules are often finite-dimensional, I July 19, 2009Posted by Akhil Mathew in algebra.
Tags: algebra, Hilbert basis theorem, Nullstellensatz, polynomials, simple modules
Today I want to talk (partially) about a general fact, that first came up as a side remark in the context of my project, and which Dustin Clausen, David Speyer, and I worked out a few days ago. It was a useful bit of algebra for me to think about.
Theorem 1 Let be an associative algebra with identity over an algebraically closed field ; suppose the center is a finitely generated ring over , and is a finitely generated -module. Then: all simple -modules are finite-dimensional -vector spaces.
We’ll get to this after discussing a few other facts about rings, interesting in their own right.