I.2.7

2/8/2021

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(P^{n}) | ||

= dim P^{n} + 1 |

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

I.e. Let's call X = Y , so we can instead write X = cl

. (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 ∩ U

| (1) |

So what happens to the closure of Y when intersected with U_{0}? Well:

cl_{X∩U0}(Y ∩ U_{0}) | = cl_{X}(Y ∩ U_{0}) ∩ (X ∩ U_{0}) | |||||

= X ∩ X ∩ U_{0} | 1.6 again: since Y and U_{0} are both open in X | |||||

= X ∩ U_{0} |

And since ϕ is a homeomorphism, closures commute:

cl_{Y 0}(ϕ(Y ∩ U_{0})) | = ϕ(cl_{X∩U0}(Y ∩ U_{0})) | ||

= ϕ(X ∩ U_{0}) | |||

= Y _{0} |

I.e. in "bar" notation:

| (2) |

And hence, by 1.10:

dim Y _{0} | = dim ϕ(Y ∩ U_{0}) | |||||

= dim Y ∩ U_{0} | (ϕ is a homeomorphism) |

| (3) |

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

dim X | = dim Y ∩ U_{0} | |||||

≤ dim Y | 1.10a |

(ACTUALLY, I just realized I could have just used 1.6 again to say dim Y = dim Y ∩ U

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.