Ths following collection of comments is taken (essentially verbatim and entirely out of context) from my lectures in the first semester Algebriac Geometry course during Fall 1998. Thanks are due to Yvonne Shashoua for typing them up and sending them to me. I can remember saying most of them, so I suspect that even the ones I don't remember are fairly accurate....
k is always a field because that's what I call fields.
I guess it's a good idea to name something alpha for the singular point of a curve that looks like an alpha.
``Normal'' and ``regular'' are words that mathematicians like to use.
I know that Q means the rationals, but I will be careful not to put a line through my Q so it's okay.
I guess I should say, "Assume J is a relevant ideal," but nobody ever says that.
I hate to call a function parentheses "()"
... but that's too much terminology, so we won't talk about that.
This is the theorem with the ugliest name in the universe. I don't like either of the terms "valuative" or "separatedness," but that's what it's called so we are stuck with it.
We sheafify this even though sheafify isn't a word.
I don't think geometrize is a word either, but that's okay.
Let A stand for anything; that's why it's an A.
It should probably be a capital B because it's a really big thing, but I don't care.
By some accident of history, this is called the first exact sequence.
I can't make all of the notation historically consistent.
Capital K is an overloaded symbol, but we are stuck with it.
We will switch from F and G to L and M to make the notation more confusing.
I can almost spell "nice."
NISF doesn't pronounce as well as NISR, but we will deal with it.
I know I should write that down but I don't feel like it.
This is my favorite proof of all time.
I love this phrase: "Unique up to unique isomorphism."
Not that I care about that, but it's a nice thing to think about every once in a while.
I don't have to do this, but I can't resist doing it.
I'm doing something I hate to do.
Some of you will learn to hate the word "category." Some of you already hate the word "category."
I hate writing too many subscripts.
You expect, or at least hope; it depends on how optimistic you are.
... which I didn't feel like writing down, and I still don't feel like writing down, so I won't.
This is like the stupidest construction in the universe. It's amazing anything comes out of this.
I'm not proving this because I'm getting tired of teaching commutative algebra.
Life is tough.
-This is a quiz on topology.
-Does that mean I get an A in the class?
-Yes. You passed the one quiz.
But I'll leave that as an exercise to anybody with the determination to actually do it.
I'm going to leave the details to the masochistic reader.
There are lots of technical details, all of which are disgusting.
The same proof that I didn't do two times before shows that this is a topology.
By omitting lots of technical details that I don't care about.
Go look up the proof and tell me next time.
I always think about leaving this an an exercise, which I may do if I don't figure it out soon.
We will leave it as an exercise until one of us figures it out.
If it's a theorem, I'm not going to prove it.
I'll leave the verification of this for the privacy of your own room.
I'm not going to prove that; I bet you're not surprised.
Here is a very deep fact.
We make the following stupid observation: These are two different curves.
I don't want to write that down because it looks stupid that way.
Proof in six characters.
Proof by handwaving.
This is the same equation with lots of zeros thrown in so it looks fancier.
We get something that even my daughter in sixth grade now knows is a group.
Who cares about non-separated schemes anyway?
That's the problem. There's gazoodles of those.
It's a PID and a UFD and all that wonderful stuff.
As I have told you approximately a million times this semester, topology matters little.
Almost any nice property you can think of gets screwed up.
That's the difference between algebra and geometry; in geometry we don't write the equations; we just wave our hands at the pictures.
Hartshorne's text is called Algebraic Geometry without any additional verbiage either in the title or anywhere else.
We won't worry about that because algebraic geometers never have.
Why study algebraic geometry? It's good for your soul.
There's a tradition of algebraic geometers doing things because they work.
I'm teaching 241 at the same time, and we just did Stokes Theorem, so that should be clear to them. I didn't tell them that, but I was tempted.
Writing the polynomial this way is the difference between algebraists and analysts. Analysts write the coefficients in the other order because they think they're all power series.
With the usual topology on R instead of the bogus one we have been using.
I mean the usual topology on R in the sense that the Math 141 students would know about if we taught them what topology was.
There are logicians here. They can see it's an "or" so this must be true.
That's the problem with trying to do both geometry and number theory. All of the letters are overloaded.
Since the "algebraic" comes first in "algebraic geometry."
Because algebraic structures are good! If you don't believe that, you don't belong in this course; you should be in real manifolds or something.
This is not what we are going to do, even though this is what we are going to do.
"Proj" is not short for anything.
The fundamental nature of sheafhood...
What I'm trying to say, awkwardly construed, is...
If f and f' have property diddly-wah then fxf' also has property diddly-wah.
It's not a trick question.
There's nothing up my sleeves.
It's supposed to be thought of in a literary context as foreshadowing.
The ring is sitting there waiting for me to recover it.
Schemes weren't complicated enough before, so now they are also functors.
That's the affine tilde and that's the projective tilde, but you can't tell because they look the same.
This proof doesn't work.
-Is this obvious?
-Yes... Would you like me to expand on that?
-What does complete mean?
-There ain't nothing missing.
... as the chalk crumbles in my hand.
Let's see if we can find an eraser and a way to deal with that question.
You need more blackboard to prove this.
This involves more chalk than the other way.
I'm not going to make it with one-and-a-half pieces of chalk, but I'll
I did make it with one-and-a-half pieces of chalk.
Let's continue to move right-to-left across the board to confuse the issue.
I believe in Zorn's Lemma.
I believe in number theory as much as I believe in complex analysis.
-How did you get that?
-By cheating.... I remember thinking about this at one point.
Have I proved that?... Probably not, but it's true.
Never memorize a formula if you can find a way to rederive it.
2b or not 2b? That's what I get for doing elementary computations.
This is the promise that I'm holding out to get us through the next sea of definitions.
Is this god-awful criterion useful for anything?
Since J vanishes on the set where J vanishes.
There either is or is not content in that statement.
We need something more local than local.
Twice unique is unique.
It's a sheaf because it's a sheaf.
It's linear because it's linear.
The notation is always awful. It basically means whatever it's supposed to in the given context.
The real proof would just be to write "obvious."
If you look at it, you realize that there are no words in that proof.
It's just the x-y plane minus the part that you can draw.
Because we pretend we are Newton doing calculus.
Limits don't exist. Limits are a fiction invented by analysts.
Derivatives are purely symbol manipulation on polynomials. That's how thousands of first semester students get through calculus.
Yes, I know this is an algebra class, but let's differentiate anyway.
Epsilon is the number that's so small that its square is zero even though it's not zero. You can do most of calculus that way. Physicists do it that way all the time.
As any Calc 140 student will tell you, differentiation is purely a formal operation on polynomials having nothing to do with taking limits.
Today I'm going to show you how to take the derivative of seven.
This is all anybody needs to talk about derviatives. The epsilons and limits are a myth invented by analysis to keep themselves employed.
This is all formal symbol manipulation, algebra at its finest.
We will do the proof first, then decide what the theorem should be. That's the secret most people don't actually know. You do the proof first, then write the theorem.
A fact is a theorem we won't prove yet.
Because the rule is that I can't run over two classes in a row, only every other class.
I'll do another aside that I didn't have planned but I don't care.
I want to start by answering a question from last time, but first you get to hear a story.
This proves that it is possible to do an infinite amount of commutative algebra. So I will do a little more commutative algebra, but eventually I'll stop.
Well, I always put propositions into definitions, so I guess I can put definitions into lemmas.
As is my tendency, we will start someplace else entirely.
I should stop for a second and catch my breath.
We will literally make that comment parenthetical by putting big parentheses around it.
-I can never rememember if zeros has an e.
-So if you write it both ways, one must be right.
-But the whole thing is necessarily wrong.
We could get really deep into the category theory if we wanted to, but we don't want to.
So my handwriting is really lousy, and that t was really an x on its side.
Which I'm not going to do because it's too painful.
There's no sense in trying to divide by zero, because it will just hurt.
-What are you proving now?
-I'm just fiddling.
See, I knew this was a mistake to do, but it's a worthwhile mistake.
We will write "Recall," which may be a euphemism.
That's the last time I want to be that careful on sheaves.
-Not bad, huh?
-I can't do that in my notes.
-Oh, just hit the "print screen" button.
-Which of these random things should I do first?
-Are we going to cast lots?
-We could do that.
Someone should have yelled ten minutes ago.
I'll pretend to prove this.
Let's pretend some more, since it's a pretend proof anyway.
This will be the second-to-last definition... Well, that depends on how you count.
I don't know what the definition of a quasicoherent lecture is, but I know what a quasicoherent sheaf is.
As we continue in my little adventure.
However many remarks I'm up to.
One can say more words that that, but this one won't right now.
I didn't tell you that before, but it just slipped out.
... which I have been doing with reckless abandon.
This is the ``thirty second'' version.
"Is" is a very amorphous verb in English, so I can get away with this.
This is almost a summary of the previous class.
And if something doesn't cancel, then we did it wrong.
I'm going to "Remark" because we have run out of semester.
It's always nice to state one fancy theorem as an advertisement for next semester.