Dimension of varieties

This semester I am continuing my course on algebraic geometry. I wanted to list here the steps I used to get a useful dimension theory for varieties so that the next time I teach I can look it up:

  1. Prove going up for finite ring maps (done last semester).
  2. For a finite surjective morphism of schemes X —> Y you prove that dim(X) = dim(Y) using going up and the fact that the fibres are discrete.
  3. Prove the Krull Hauptidealsatz: In a Noetherian ring a prime minimal over a principal ideal has height at most 1. For a proof see [E, page 232].
  4. Generalize to longer sequences: In a Noetherian ring a prime minimal over (f_1, …, f_r) has height at most r. For a proof see [E, page 233].
  5. If A is a Noetherian local ring and x ∈ m_A then dim(A/xA) ∈ {dim(A), dim(A) – 1} and is equal to dim(A) – 1 if and only if x is not contained in any of the minimal prime ideals of A. In particular if x is a nonzero divisor then dim(A/xA) = dim(A) – 1.
  6. Prove that if A is a Noetherian local ring, then dim(A) is equal to the minimal number of elements generating an ideal of definition.
  7. If Z is irreducible closed in a Noetherian scheme X show that codim(Z, X) is the dimension of O_{X, ξ} where ξ is the generic point of Z.
  8. A closed subvariety Z of an affine variety X has codimension 1 if and only if it is an irreducible component of V(f) for some nonzero f ∈ Γ(X, O_X).
  9. Prove Noether normalization.
  10. If Z is a closed subvariety of X of codimension 1 show that trdeg_k k(Z) = trdeg_k k(X) – 1. This you do using Tate’s argument which you can find in Mumford’s red book: Namely you first do a Zariski shrinking to get to the situation where Z = V(f). Then you choose a finite dominant map Π : X —> A^d_k by Noether normalization. Then you let g = Nm(f) and you show that V(g) = Π(V(f)). Hence k(Z) is a finite extension of k(V(g)) and it is easy to show that k(V(g)) has transcendence degree d – 1.

At this point you know that if you have ANY maximal chain of irreducible subvarieties {x} = X_0 ⊂ X_1 ⊂ X_2 ⊂ … ⊂ X_d = X, then the transcendence degree drops by exactly 1 in each step. Therefore we see that not only is the dimension equal to the transcendence degree of the function field, but also each maximal chain has the same length. This implies that dim(Z) + codim(Z, X) = dim(X) for any irreducible closed subvariety Z and in particular it implies that dim(O_{X, x}) = dim(X) for each closed point x ∈ X.

Let me know if I neglected to mention a “biggish” step in the outline above.

What is missing in this account of the theory is the link between dimension of a Noetherian local ring A and the degree of the Hilbert polynomial of the graded algebra Gr_{m_A}(A). Which is just so cool! Oh well, you can’t do everything…

[E] Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry.