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.
If anyone could answer
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Why $(*)$ is true.
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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!
Best Answer
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!