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To assuage my conscience over an unsourced statement in a paper I'm writing:

I am looking for an example of a commutative algebra over the complex numbers having a maximal ideal of codimension >1, or a statement of inexistence.

flag	asked Apr 12 2010 at 18:59 Miguel 91●5

Mariano Suárez-Alvarez · Accepted Answer · 2010-04-12 19:03:18Z UTC

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If the codimension is finite, then there is no such thing. If the codimension can be infinite, then yes, because there are infinite dimensional complex division algebras which are simple, like $\mathbb C(t)$.

answered Apr 12 2010 at 19:03

Mariano Suárez-Alvarez
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I like the bit about the codimension being necessarily infinite. Where could I find a proof of that? What kinds of techniques are involved? – Miguel Apr 12 2010 at 19:10

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Miguel, if a maximal ideal $\mathfrak m$ has finite codimension in your $\mathbb C$-algebra $A$, then $A/\mathfrak m$ is a finite dimensional commutative $\mathbb C$-algebra which is simple. It is therefore a finite dimensional division $\mathbb C$-algebra, and it must then be of dimension $1$, as it is in fact an agebraic field extension of $\mathbb C$. – Mariano Suárez-Alvarez Apr 12 2010 at 19:19

Pete L. Clark · Answer 2 · 2010-04-12 19:04:12Z UTC

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By codimension you just mean as a $\mathbb{C}$-vector space? Take the rational function field $\mathbb{C}(t)$.

(Note: by the Nullstellensatz, it is not possible to do so with a finitely generated $\mathbb{C}$-algebra.)

answered Apr 12 2010 at 19:04

Pete L. Clark
39.6k●3●102●226

Thanks, Pete - I knew about the Nullstellensatz but I am indeed interested in infinitely generated algebras. – Miguel Apr 12 2010 at 19:08

Maximal ideal of codimension >1

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