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

*Posted by Akhil Mathew in algebra, representation theory.*

Tags: algebra, finite-dimensional vector spaces, Nakayama's lemma, representation theory, simple modules

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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.

We know from the previous post that a simple representation of is just for a maximal left ideal; then is a left -ideal, hence a two-sided -ideal. Although was maximal, we don’t necessarily have maximal in . So choose to be maximal and containing .

Claim 1.

First of all, is a left ideal in , because consists of central elements. Now, if , we would have by maximality

In other words, we are viewing as a -module.

Now we use:

Lemma 2 (Nakayama)Let be a commutative ring, be an ideal, and a finitely generated -module such that

then there exists such that .

The proof uses the Cayley-Hamilton theorem, and is given here. It’s essentially equivalent to other versions of Nakayama’s lemma.

So now, back to the claim. In (1), by Nakayama, we would get an element such that , i.e. , or . But then , contradiction. We’ve thus proved our claim.

Now, onto the theorem itself: since , we can consider

as both a -module and a -module. As a -module it is finitely generated by assumption, so as a -module it is finitely generated.

But we can now invoke the following:

Theorem 3 (Generalized Nullstellensatz)Let be a field, a finitely generated commutative ring over , and a maximal ideal. Then is a finite extension of .

When is algebraically closed, any finite extension of is just itself, so this becomes a result from our prevoius post.

So, we see that is a finite-dimensional -vector space for a finite extension of . Thus is a finite-dimensional -vector space.

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