B

 
(Γ notation, for anyone confused) So to be clear, the reason I'm skipping part A is because it's boring. Again, yours truly has to practice divvying up their time wisely, so I'm gonna spend it on stuff where we actually get stuff done, like this:

(1)

We wanna show that it's exact, and uhhh... the thing about "flasqueness".. whose definition is given in 1.16... which I didn't do:



I didn't do it yet because it's 5 parts and that scares me. HOWEVER for now the point that the restriction maps would be surjective is all we need, so we can continue.

Alrighty, so what we've learned in Hartshorned II so far is that it's easier to show exactness on the stalk level. Let's go ahead and define ϕ and ψ and get their stalky equivalents

0.1 Defining ϕ

To create

ϕ : Z0( )

→

 
We need to define, given V ⊂ X,

ϕ(V ) : Z0( )(V )

→ (V )

 
And evidently the LHS is the complicated one, so let's unwrap that baby:

Z0( )(V )	= Γ Z∩V (V, |V )
	= {s ∈ Γ(V, |V )|Supps ⊂ Z ∩ V }
	= {s ∈ Γ(V, |V )|Supps ⊂ Z}	(since Supps ⊂ V trivially)
	= {s ∈ i-1 (V )|Supps ⊂ Z}	(where i : V X the inclusion)
	= {s ∈ lim i(W)⊃V (W)|Supps ⊂ Z}
	= {s ∈ lim W⊃V (W)|Supps ⊂ Z}
	= {s ∈ (V )|Supps ⊂ Z}

 
Okay, that's a bretty good description of the sections over V , and furthermore, it's now clear that

Z0( )(V ) ⊂ (V )

 
so ϕ(V ) can just be the inclusion! And ϕ_P is also the inclusion! EZ.

0.2 Defining ψ

Okay, now we need

ψ :

→ j*( |U)

 
So we define for V ⊂ X open,

ψ(V ) : (V )

→ j*( |U)(V )

 
Unwrapping the RHS...

j*( |U)(V )	= |U(j-1(V ))
	= |U(U ∩ V )
	= j-1 (U ∩ V )	(Kinda did a double take here cuz we used both the direct image and restriction on the same sheaf using the same map lol. trippy af)
	= lim W⊃j(U∩V ) (W)
	= lim W⊃U∩V (W)
	= (U ∩ V )

 
Which makes the most natural definition for ψ(V ) to be the restriction map WHICH, btw, if is flasque, we know is surjective. BUT obviously remember that don't matter cuz we gotta check surjectivity at the stalk level, which i'm about to do, lol

So uh, I'd just like to say that the stalks of j_*( _|U) are pretty weird. Basically the equivalence relation giving rise to (j_*( _|U))_P works like this: To determine if [V,s] = [W,t] (V,W neighborhoods of P), we consider if there's any open set contained in V ∩U and W ∩U where s and t restrict to the same section. The weird part is that V ∩ U and/or W ∩ U might NOT contain P itself, but that's just how the equivalence relation works. "Dude, this is confusing." I KNOW, IT TRIPPED ME UP FOR A WHILE LOL. Just... think about it.

Okay, actually, you know what? I'm a adult. I know I don't act like one on this page, nor do I irl (And honestly, it's extremely embarassing and I'm ashamed of who I am. It's legitimately shocking that I'm continuing to live with myself like this. I'm embarassed to even talk to people and show myself in public because the way I carry myself visibly reveals how mentally underdeveloped I am. People can tell that something is wrong with me just by looking at me. I can feel the weight of all the eyes on me: I feel it on my unkempt hair, submissive smile. I look unnatural. Dissocietal. My mind comes through in my fingers and the way I flinch and tremble at the sight and touch of other people outs me as inhuman. Why am I like this? I love people, I really do, but I don't know how to connect with them. I don't know how to behave around them. I want to be closer to them, and yet all my life I've been so estranged from them. I just want to be able to have a normal conversation with someone. I see people talk, and I start beaming in admiration as they make some hilarious comment and roar in laughter. I want to be like that. I want to be able to make a hilarious comment, or roar in laughter. I want to be a part of the excitement, a part of the moment. But I can't enter into that circle. It doesn't happen. It just doesn't work. My grand entrance makes everything awkward. My comments fall flat. My presence is at best unfelt and at worst a nuisance. Always. No matter how nice and accomodating they are to me, I can't fit in. I don't know how to talk. I've gotten better at it over the years, but not enough. Not enough to actually be an influence on someone. In a groups, everyone takes turns dominating the conversation except me, in some kind of unsaid round robin that I didn't sign up for. One-on-one, people talk about things they've done and experienced that genuinely impress me, and yet all I can respond is "That's cool," and thus sound colder than I actually feel. I sound unimpressed and bored when I'm not. I try to ask followup questions, but I don't know anything about anything. I'm so ignorant of everything–things that normal people would know how to talk about like housing prices uptown and the latest tech–and the babyish questions and comments I offer invariably douses their flame. I'm not able to engage on their level. I can't engage on anyone's level, because my knowledge of everything is so stunted. Then, apparently giving up, they ask me what I've been up to. And here I always turn up a blank. Why am I never prepared for this question? I've been asked it so many times, and yet I always let it kill the conversation. Nothing. I've been up to nothing. Please don't ask me that. I don't want to talk about me. I have nothing. I'm sorry for being a rock. I'm sorry for making everything awkward. I'm sorry, I'm sorry. Why am I like this? I hate it. I want to cry. I want it to end. Please, let me in. Let me in... Please... What? No I'm not going to wear a mask lol suck my co–) but I've been Hartshorned long enough to EXPRESS THIS IN FANCY MATHEMATICAL NOTATION:

(j*( |U))P

= lim V ∋P (V ∩ U)

 
There you go. *Phew*. Now I'd like you to note that in the case that, in the direct limit, if V ∩U is always empty I.E. IF P IS NOT IN THE CLOSURE OF U, then the stalk is 0. Otherwise, ψ_P is just < V,s > [V,s]. Either way, it's surjective if is flasque. IN FACT, if it's flasque, we can add an extra → 0 to (1) and make it a short exact sequence. Well, flasque or not, to finish this exercise off we need to show

0.3 exactness at the middle

that imϕ_P = ker ψ_P. So let's split it up into cases:

0.3.1 P U

In this case, as we explained, (j_*( _|U))_P = 0, so ker ψ_P = _P, and in particular, ∃ a neighborhood V of P such that V ∩ U = ∅. Now let's suppose that < W,s >∈ _P. Refining, if necessary, assume that W ⊂ V . Then

W ∩ U	= ∅
W	⊂ Z
Supps	⊂ Z

 
So s ∈ _Z⁰( )(W), and therefore < W,s >∈ imϕ_P. So _P ⊂ ϕ_P and the reverse inclusion is trivial.

0.3.2 P ∈U

Ok this is the normaller case. Suppose < V,s >∈ imϕ_P. We can assume without loss of generality that its preimage was also < V,s > (derefining–erm, coarsifying if necessary) and therefore that Supps ⊂ Z. Using that WLG, let's write

< V,s >	∈ imϕP
⇔Supps	⊂ Z
⇔∀Q ∈ V : sQ≠0	Q ∈ Z
⇔∀Q ∈ V : sQ≠0	Q X - Z
⇔∀Q ∈ V : sQ≠0	Q U
⇔∀Q ∈ V : sQ≠0	Q U
⇔∀Q ∈ V : Q ∈ U	sQ = 0	(CONTRAPOSITIVE BITCHESSSSSSS)
⇔ρV,V ∩U(s)	= 0
⇔ψ(V )(s) = 0
ψP(< V,s >)	= 0	(C-C-C-C-C-C-C-C-COMBO BREAKER)
⇔ < V,s >	∈ ker ψP

 
ARRRRRRRRGGGGHHHH THAT ONE RUINED THE CHAIN OF ⇔s. WE ONLY GOT ONE SIDE OF THE INCLUSION. NOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO. Okay, don't panic: It's just refining lol; the same way we squeezed out the first ⇔. MR. , I SHALL NOW GRANT YOU YOUR LEFT WING. YOU ARE NOW A ⇔. *Clap clap clap*. And that finishes the exercise.

The next one is a 5 parter so it might take a while for the next post, unless I split it up... Yea, sorry for turning these out so slowly lol. I'm trying to speed it up, but... AGGGHH. 2 more exercises left for this section!!! Almost there!