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 round-edged 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 quasi-affine varities, projective varities, and quasi-projective 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 (inclusion-reversed) 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 quasi-affines.

LEMMA 3: If X is quasi-affine, then dim X = dim P,X

PROOF:

Since X is quasi-affine, 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)

Combining this with LEMMA 2 gives equality. Dun.

CHOCO 3: Well now the nuts are out in the front. A larger, spikier ball, the sort you could attach a chain to to inspire a more gruesome image. And what does this nut-spudded sphere have to provide? You take a bite, to be greeted by a wafer shell enclosing a milk cream nucleus. Let's just say, you already had the napkin at the ready for this one.

LEMMA 4: If X is a quasi-projective variety, then dim X = dim P,X

PROOF:

Since we need to "reduce" this to an affine case, we're probably going to use the map of 2.2/2.3. Suppose P U0 X without loss of generality. Note two things: U0 X is an open set in X, therefore dense (U0 X = X), and it is isomorphic to some affine variety Y .

Hence

 dim X = dim U0 ∩ X = dim U0 ∩ 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)

(Note that I didn't really have to introduce Y , since U0 X "is" Y , but I did it here just for clarity). Dun.

CHOCO 4: Rocky road, eh? Crude, but chocolate is chocolate. You bite, and pull your face back to detach a piece into your mouth. Oh god, it's pretty tasty. A chewy blend of marshmellow, nuts, and chocolate culminating into a sort of overstimulation for your sweet tooth. This is dangerous. Eat it and forget about it!

Alright, one more:

LEMMA 5: If X is a quasi-projective variety, then dim X = dim P,X

PROOF:

X is an open subset of Z for some projective variety Z. And

 dim X = dim X (2.7b) = dim Z = dim P,Z (LEMMA 4) = dim P,X (LEMMA 1)

Dun.

CHOCO 5: A short, dark rectangular prism with a square base. You flick it with your middle nail and get a tough resistance coupled with a nonhollow sound. Nothing's hiding in here. You bite off a fragment, and chew at the 80% goodness. Easy. You turn the height of the rectangle up towards you and inspect the inside: a solid mini mountain range of dark but slightly lighter brown, clawed in by two pairs of tiny concave buildups on either edge–the remnants of tooth marks. One more bite. Yum.

And that's it for today!