HI. Since this is supposed to be a daily thing. I finally wrote a messy bash script (using this sample file) to help me generate these posts. Pls no laff. "L0l0l0l You fucking n00b. Instead of 'basename $PWD' you should be using '${PWD##*/}' in order to avoid creating a subprocess and save 0.0000000000001 precious seconds." Yeah, yeah, whatever, fucko. And yes, because the make4ht output has line breaks at very awkward spots, I used some pretty nasty sed commands. I had to learn about BRANCHING, YO.

ANYWAY, let's start the exercise. For part (a), just use yesterday's mess:

n + 1	= dim S(Pn)
	= dim Pn + 1

Subtracting 1 on both sides finishes it off.

That was easy. Hey, the hint says to use yesterday's mess for part (b) too!
1.10 is this guy:
(So, it turns out this is another "projectivization of affine" thing)
So how do we hook up 2.6 and 1.10?
Well, Y being a quasi-projective variety means its an open subset of some projective variety. Actually, by 1.6, Y is precisely that variety (or at least the smallest such variety). Now, the bar notation for closure is going to be a little too ambiguous in this exercise, so let's move everything to the cl notation, and we'll be using this property a lot:

"If A is a subspace of X containing S, then the closure of S computed in A is equal to the intersection of A and the closure of S computed in X: cl_AS = A ∩ cl_XS"

I.e. Let's call X = Y , so we can instead write X = cl_PⁿY . And note that cl_X(Y ) = cl_Pⁿ(Y ) ∩X = X ∩X = X.

. (So our goal in the new notation is to show dim X = dim Y )

Okay. Now let's use the homeomorphism from last time ϕ : X ∩ U₀ → Y ₀. We saw last time that

(1)

So what happens to the closure of Y when intersected with U₀? Well:

clX∩U0(Y ∩ U0)	= clX(Y ∩ U0) ∩ (X ∩ U0)
	= X ∩ X ∩ U0	1.6 again: since Y and U0 are both open in X
	= X ∩ U0

And since ϕ is a homeomorphism, closures commute:

clY 0(ϕ(Y ∩ U0))	= ϕ(clX∩U0(Y ∩ U0))
	= ϕ(X ∩ U0)
	= Y 0

I.e. in "bar" notation:

(2)

And hence, by 1.10:

dim Y 0	= dim ϕ(Y ∩ U0)
	= dim Y ∩ U0	(ϕ is a homeomorphism)

combining this with (1)...

(3)

Holy shit this is getting messy. Let's just apply exercise 1.10a to finish it off.

dim X	= dim Y ∩ U0
	≤ dim Y	1.10a

and also by 1.10a, dim Y ≤ dim X. DONE.

(ACTUALLY, I just realized I could have just used 1.6 again to say dim Y = dim Y ∩ U₀. WHATEVS)

Oh, and since I just realized that neocities allows you to upload txt files, I guess I might as well start linking the corresponding tex files for each post. Here's today's one. ALSO, yes, I need to figure out how to not make the equation descriptions jut out of my post. fffuuuuuuuggggg.