I.3.9

4/27/2021

Hello, Hartshorners. Last time I made a joke about the exercise being short (...yes, that was the joke, initially), but turns out: this one is even tinier. I ain't complaining. The only other gag would be to do the whole exercise in a tiny font. Hahahaha Err. STOP. DON'T SHRINK ME. I CHECKED THE BOTTLE. IT DIDN'T SAY POISON. AHHHHHH, I'M SHUTTING UP LIKE A TELESCOPE. Now i'm a smol holeinmyheart :<

Well, I guess you gotta do what you gotta do. Big or smol, the hart's gotta be shorned. As they say,3.4 tells us that X Y , so we just have to show that S(X) S(Y ). Now S(X) = k[r,s], and S(Y )? Well, instead of making the same mistake that I did in 3.1c , I'm going to actually look back at previous problems, and realize that in 3.1c itself, I discuss how Y is actually Z(xz - y2). Hence, S(Y ) = k[x,y,z](xz - y2)

So we want to show k[x,y,z](xz -y2) k[r,s]. Suppose for contradiction that they were isomorphic. then we would have a surjective morphism ϕ : k[x,y,z] k[r,s] whose kernel is (xz - y2). So let's say this morphism sends

 x f y g z h

Then we must have

 ϕ(xz - y2) = 0 fh - g2 = 0 fh = g2

Now, my little Hartshorners (since we are, indeed, little at the moment), this is one of those lovely points in time where as I am writing the post, I realize that I have a logic gap. I wanted to conclude that since k[r,s] is a UFD, we must have f = g = h... but this only applies if f,g,h were irreducible.

Errr... let's pass over to the quotient. ϕ induces an isomorphism ψ : k[r,s] k[x,y,z](xz - y2). Now, uh, my little reader. If you can believe that x,y,z are still irreducible in the quotient, then by virtue of being an isomorphism, so do f,g,h.... Post so small that no one will notice THAT I'M LEAVING THIS AS AN EXERCISE TO THE READER, MUAHAHAHAHAHAHA... so... f = g = h, rite. Now the contradiction comes up. that would mean that ψ(f) = ψ(g) = ψ(h) which implies that x = y = z, clearly a contradiction. Dun.

Did it work? OMG IT'S WORKING? WEEEEEEEEEEEEEEEEEEEEEEEEE EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE (as long as your browser supports it) oh no. oh no no no. I just had the idea ever, and I'm going to try it, and you're going to watch me fail. LiterallyHifumi, I'm about to use your educational post for a failed meme attempt. Here goes nothing.

This is supposed to be the cap of the O but God knows where it will go

Oh, almost forgot. Here's the actual confession: Everytime I post, I think "Well, this is the one." "This is going to be the one that makes my readers turn against me"

Well, then, surely it's going to be this failed LOL that does me in. The failed "LOL" will be [cont]
[cont] the last straw for you, eh? Yes, behold: the final insult... I could murder your kid, steal your waifu, cut off your arm. But this failed "LOL" will be the [cont]

[cont] will be the one that got away--I mean the crime that went too far. Well, you want an "LOL"???
THEN LAUGH, READER. LAUGH AT THIS. LOL AT THIS LOL LOLOLOLOLOLOLOLO