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I have been studying the Krull Dimensions of rings and have a couple of questions:

Why is the concept of Krull dimensions important within algebraic geometry?

I have also been proving the result that the Krull dimension of the polynomial ring $K[X_1,...,X_n]$ is equal to $n$ when $K$ is a field. Why is this result in particularly important? In other words, why do we care about this result?

If anyone has any information/references on this, I would find it very interesting.

Note: Not references on the proof as I have been able to complete this!

Thanks,

Johan

user26857
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Martin Sidelzo
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3 Answers3

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In short, Krull dimension is an important concept because it is an intrinsically algebraic definition of dimension which precisely captures our geometric intuition for what the (topological) dimension of a variety/scheme should be.

For example, we associate $\mathbb{A}^n_{\mathbb{C}} = \mathrm{Spec} \,\mathbb{C}[x_1 \ldots x_n]$ with the topological space $\mathbb{C}^n$, although the two are not homeomorphic. The fact that the Krull dimension of $\mathbb{C}[x_1 \ldots x_n]$ is $n$ captures this.

This then allows us to formalize the notions of dimension and co-dimension in such a way that they can be generalized to arbitrary rings (such as $\mathbb Z$) where no geometric intuition is available, and to study them from a geometric point of view. We can take geometric proofs that rely on the concept of dimension, translate them into algebraic language, and then prove the same statements in much greater generality.

A very simple example of this principle is the fact that points in $\mathrm{Spec}\, A$ correspond to prime ideals of $A$, but our geometric intuition for ''points'' is really restricted to the maximal ideals of $A$. The inclusions of these points correspond to maps $A \to A/\mathfrak{m}$, which has Krull dimension 0. Hope this helps.

Suhas Vijaykumar
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Here's some more elementary motivation for the definition of Krull dimension.

A way we can characterize the dimension of vector spaces is as the maximal flag in $V$ (i.e. the longest strictly increasing chain of subspaces). A key property here is that between each vector space, the dimension jumps by only one.

Krull dimension as defined for rings involves a strictly increasing chain of prime ideals. In algebraic geometry, (for decent settings, i.e. integral finite type schemes over a field), an increasing chain of prime ideals corresponds to a strictly increasing chain of varieties of increasing dimension.

In most scenarios, a maximal chain of primes then corresponds to a maximal strictly increasing chain of varieties where each dimension increases by one in the intuitive sense.

E.g. one can think of in $k[x_1,x_2,\ldots,x_n]$ of the sequence of varieties $$ V(x_1,x_2,\ldots,x_n) \subset V(x_1,,x_2,\ldots,x_{n-1}) \subset \cdots \subset V(x_1) $$ which geometrically looks like the origin, a line, a plane, \ldots, a hypersurface. A lot of algebraic geometry is sort of trying to make the leap from the easy land of linear spaces to dealing with spaces defined by systems of polynomials (the roots of them).

Vincent Tran
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The concept of Krull dimension is not only important in algebraic geometry, but in much more generality. In particular, for algebraic number theory and commutative algebra it is very useful and natural. The rings of integers $\mathcal{O}_K$ for a number field $K$ naturally arise in number theory as Noetherian, integrally closed rings with Krull dimension $1$. The most basic example is for $K=\mathbb{Q}$, namely $\dim (\mathbb{Z})=1$. Then $$ \dim \mathcal{O}_K[x_1,\ldots ,x_n]=n+1. $$ More generally, dim $A[x_1,\ldots ,x_n]=\dim (A)+n$ for Noetherian rings $A$.

Dietrich Burde
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