I.3.12
5/5/2021
It's 5/5! ... Following a 5 day delay. Sorry for making you wait, reader. You
deserve a chocolate. Actually, you deserve 5 chocolates, one for each day. In fact,
I'm going to split this exercise up into 5 LEMMAs, and after the PROOF of each
LEMMA, I'll give you one CHOCO. Kay?
Since we'll be working with multiple spaces, I'll use the preestablished notation
_{P,Y } to mean "the local ring of Y at P".
LEMMA 1: If X is any variety and U ⊂ X is an open subvariety, then
_{P,U} ≃
_{P,X}
PROOF: obvious.
CHOCO 1: A roundedged cube in a boat–a bitter shell filled with cream
far too sweet. Is this a single bite box? Well, with a mouth so smol as
yours you have to eat but a fourth first, embarassedly wipe off the white
filling from your mouth, then an ambitious third second, chewing a bit too
conspicuously, and then the rest: a cream topped shell fragment. Nom.
Now, the equality holds for affine varieties, thanks to the theorem 3.2(c) they
mentioned:
So now we need to show that it holds for quasiaffine varities, projective
varities, and quasiprojective varieties. But first, let's get "halfway" there.
LEMMA 2: If X is any variety, dim X ≥ dim
_{P,X}.
PROOF:
Given a chain of distinct prime ideals

thanks to last time, we get a corresponding distinct (inclusionreversed) chain of
subvarieties in X. Hence dim X ≥ dim
_{P,X}. Dun.
CHOCO 2: A smol, milk coated ball. Looks simple enough. You bite halfway, and
then become startled as your teeth hits: a nut!? An almond!? Well, you should
have been more careful, ey? You cup your hand below your chin in a vain attempt
to catch the crumbs. In your other hand: a melting nest holding the rest of the
kernel. Oof, this is getting messier than expected. You quickly shove the rest in
your mouth, then grab a napkin to wipe your hands and the edge of your lips.
Okay. Now let's show the equality for all the cases, by reducing them to the affine
case, as the hint suggests. Let's start with quasiaffines.
LEMMA 3: If X is quasiaffine, then dim X = dim
_{P,X}
PROOF:
Since X is quasiaffine, it is an open subset of some affine variety Y . Then
dim _{P,X}  = dim _{P,Y }  (LEMMA 1)  
= dim Y  (3.2(c))  
≥ dim X  (1.10)  
dim X  = dim U_{0} ∩ X  
= dim U_{0} ∩ X  (2.7b)  
= dim Y  
= dim _{P,Y }  (3.2(c))  
= dim _{P,U0∩X}  (Isomorphic varieties have isomorphic local rings)  
= dim _{P,X}  (LEMMA 1)  
dim X  = dim X  (2.7b)  
= dim Z  
= dim _{P,Z}  (LEMMA 4)  
= dim _{P,X}  (LEMMA 1) 