Well, that didn't turn out so well, hahahahahaha. Yes, I got slightly overconfident there and genuinely thought I could solve it in time, and ended up embarassing myself. Traumatic. Ahh, yes, I can still hear the Jeopardy music. (cue the part at 2:58 when I wrote "OKAY. I GOT IT.").

Well, I did figure it out later that night... although I'm posting this like 3 days late because I got lazy. Woops. And thus, we are at the VERY FIRST ERRATA OF HARTSHORNED II. WOOOOOOOO *CLAP CLAP CLAP CLAP CLAP*. It ain't Hartshorned without some good old fashioned errata posts. Remember those?






"WTF"

Anyway, since I already framed the problem in the last post, I can just pick up from there. That's right; the errata motto: IF YOU DROP THE BALL, PICK IT BACK UP. Here's how the story goes. We can actually use initiality of Z again!!! there's a unique morphism.

Z → X(U ∩ V )

 
And hence if you localize, the induced local morphisms

ϕP* : Z ϕ(P)	→ AP
ψP* : Z ψ(P)	→ BP

 
are unique. so that Z_ϕ(P) ≃ Z_ψ(P) ϕ(P) = ψ(P). DONE.



"...It took you 3 days to upload that?"



Well, excuuuuuse me princess reader. I'll (try to) upload a post everyday this weekend to make up for it: FRIDAY, SATURDAY, SUNDAY. Are ya happy now, chump? Ready to get your head BASHED with Hartshorned this weekend? Prepare your brain for some very fancy ero guro.