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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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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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  • $\begingroup$ I like the bit about the codimension being necessarily infinite. Where could I find a proof of that? What kinds of techniques are involved? $\endgroup$
    – Miguel
    Apr 12, 2010 at 19:10
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    $\begingroup$ 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 $\mathbb C$. $\endgroup$ Apr 12, 2010 at 19:19
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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.)

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

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