ANOTHER EXERCISE!!!!!!!! YAY. YIPPEE. ARE YOU HYPED? ARE YOU HYPED FOR MORE TWISTS AND TURNS AND CUSSING AND MESSY BUSINESS? Well, TOO BAD. Cause this exercise is just the projective version of an old proposition for affines:



Err, yeah. So (a) (b) and (c) are extremely obvious. And for (d)...err...okay, I'll go into more detail because they were kind of vague:

Let f ∈ , so f^q ∈ α for some q > 0. Which means that any P ∈ Z(α) has to satisfy f^q(P). I.e.

∀P ∈ Z(α) : fq(P)	= 0			(1)
⇐⇒∀P ∈ Z(α) : f(P)	= 0	k is an integral domain.		(2)
⇐⇒f ∈ I(Z(α))				(3)

Since f ∈ was arbitrary, we have ⊂ I(Z(α)) . Done. and the reverse invlusion I(Z(α)) ⊂ is just the homogenous Nullstellensatz (2.1) which is what we need the Z(α)≠∅ condition for. (HOWEVER, according to past research, I believe everything still works out in the case of α = (1). Yes, I am CAREFULLY DEMARCATING these edge cases in like every single post. SPOILER ALERT: it becomes relevant in the next exercise).
Okay, (d) is DONE.

For (e), it's the same proof as the affine case. So we're done.

"BOOOO! BOOOOO! YOU DID LIKE NOTHING."

Indeed, that's my specialty. I can construct a whole post out of a nonpost. Voomp! Out of thin pink! Like a ball of puff. Ey, ey. If y'ain't busy, look busy, eh? I'll waste your time as much as I'd like, tyvm. I'll at least make it entertaining. A tasty waste. "*Gag*". Hey, don't gag right as I'm saying that. It's too deliberate. It's not funny if it's deliberate. Let's try again. On my command: When I say "gag", you gag, alright? 3, 2, 1...

Gag!
"*Gag*"
Gag!
"*Gag*"
Slide to the left!
"*Slide*"
Slide to the right!
"*Slide*"
Cha cha real smooth now.
"*Cha cha*"
Turn it up!
"*Up*"
One hop!
"*Hop*"
Two hops! Two hops!
"*Hop* *Hop*"
Right foot let's stomp!
"*Stomp*"
Criss-cross!
"*Criss-cross*"
Criss-cross!
"*Criss-cross*"
Gag real smooth now.
"*G~aaaaaaaaag*"
Turn it up!
"*Up*"
One hop!
"*Hop*"
Left foot let's stomp!
"*Stomp*"
How low can you go?
"*Low*"
Low to the floor?
"*Lower*"
Now bring it back up.
"*Up*"
One hop this time!
"*Hop*"
Criss-cross!
"*Criss-cross*"
Criss-cross!
"*Criss-cross*"
Now lick my penis real smooth now
"*Gag*"

BRAVO!

(But honestly fuck that song)