# Regular in codimension 1

Apologies if this is an obvious question. I've really gotten my head tangled up in knots trying to approach it from the right angle, and I'm not getting anywhere - so I thought I'd ask.

A scheme is said to be regular in codimension 1 if every local ring $\mathcal{O}_x$ of $X$ of dimension one is regular.

Let $X$ be noetherian, integral, separated and regular in codimension 1.

Then every algebraic geometry reference ever says that if a subscheme of $X$ has codimension 1, the local ring of its generic point $\eta$ is of Krull dimension 1 $(*)$.

Nowhere seems to give this any thought, and I can't for the life of me see why it's true. Moreover, I cannot see how the dimension of the local ring of a point, that is, the dimension of a bunch of functions on the point, is in any way at all related to the dimension of the actual space it sits in.

1. Why $(*)$ is true.

2. Why these two things are actually related.

I'd be very grateful. Thanks!

Edit: My commutative algebra has been quite wobbly in recent times, so the more anything of this nature is spelt out, very much the better!

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If Y is a closed irreducible subspace of a scheme X, $y \in Y$ the generic point, then one has always

$$\mathrm{codim}(Y,X) = \dim(\mathscr{O}_{X,y}).$$

The closed irreducible subspaces of $X$ containing $y$ are in bijective correspondence with the closed irreducible subspaces of the local scheme $\mathrm{Spec}(\mathscr{O}_{X,y})$, hence with the prime ideals of $\mathscr{O}_{X,y}$. EGA is a reference that does have details for this (and most other things), see (EGA, IV_2, 5.1.2).

P.S. Happy belated birthday to Grothendieck!

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Thanks for this. For future lost souls: a good reference for this is also Liu Qing's Algebraic geometry, which I discovered after I'd gotten most of the way with EGA. –  Hock Mar 30 '13 at 12:39
Hint: Precisely, the dimension of $\mathcal{O}_{X,x}$ is the codimension of $\overline{\{x\}}$ in $X.$