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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.

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up vote 5 down vote accepted

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 $\mathbb C(t)$.

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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 '10 at 19:10
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 '10 at 19:19

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.)

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Thanks, Pete - I knew about the Nullstellensatz but I am indeed interested in infinitely generated algebras. – Miguel Apr 12 '10 at 19:08

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