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In my algebraic geometry class, the dimension of an affine variety $X=V(I)$ was defined as the supremum of the length of chains of prime ideals in the coordinate ring $R=k[x_1,\ldots,x_n]/\sqrt{I}$, or, the Krull dimension of $R$. A few minutes later, we were given the definition in terms of transcendence degree: the dimension of $X$, if $X$ is integral, is the transcendence degree of $K(R)$ over $k$. I have also seen dimension defined in terms of the degree of Hilbert polynomials (Theorem C, page 225 in Eisenbud).

My question: which of these came first and why? Why were the other definitions developed? What did they allow us to do more easily or effectively that previous definitions did not?

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Search this - encyclopediaofmath.org/index.php/Commutative_algebra - page for the word "dimension" - it containts a lot of interesting commutative algebra history. In particular, it seems like dimension were originally defined as the transcendence degree of $K(R)/k$. –  Fredrik Meyer Sep 30 '12 at 15:30
    
I think you're right: "Beginning with Kronecker and Lasker's dimension, defined then as the transcendence degree of the quotient ring corresponding to a prime ideal in a ring of polynomials, the modern combinatorial definition of dimension was subsequently proposed by W. Krull." –  Derek Allums Sep 30 '12 at 15:39
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Actually, if you have Eisenbud at hand, you can read chapter 8, which gives a short history of dimension in algebraic geometry. –  Andrew Sep 30 '12 at 16:42
    
@Andrew Well this is embarrassing: I cited Eisenbud in my question but didn't think to turn back a few pages. Thanks. If you convert this to an answer I'll accept it. –  Derek Allums Oct 1 '12 at 17:36

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Chapter 8 of Eisenbud has a short history of dimension in algebraic geometry, even giving axioms for a theory of dimension. The historical order seems to be transcendence degree (think meromorphic functions on a Riemann surface), Krull dimension, then Hilbert functions. In particular, Eisenbud mentions that, though one might suspect differently, the most powerful computation method today is via the Hilbert function, through applications of Groebner bases.

I also tracked down a quote that first came to mind regarding the Zariski tangent space, from Eisenbud-Harris, p.28: "[...] this was some years after Krull had introduced the notion of a regular local ring to generalize the properties of polynomials rings, one of the rare cases in which the algebraists beat the geometers to a fundamental geometric notion."

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