II.3.15
4/1/2022
ORDER OF THE ARTILLERY, AS OF 20 SEPT:
YO REP, , my little geometers shall
show you that they can SHOOT. LOAD EM UP, BOYS. While their doing their
thang, lemme pull you aside for a bit. Lissen, ysaw my little manifesto right? Ntch
Ntch ntch. Yall know these ”leaders” aint hittin it outta the park. They didnt
even have a park. But I made one back up at Ollioules. I’m quick n nimble like
dat. I can set up a thingy just like THAT, and BOOM BOOM BOOM, there she
blows, yknow, EXCEPT: I can’t QUITE show you the full power of our
cannons and aim cause yknow we’re kinda low on ammunition so the
boysll have to hold their horses for a good shot, but look: Lemme pull on
your lapels and lean into you, we’re at kissing distance now, and I’ll tell
you with a twinkle in my eye that the very leader of the Alps that so
violently massacred the port cities en route here does not have the wit nor
determination for this siege. You know what that twinkle means right?
Slick. Slit. His. Throat. We dont have time for halfassery here. Hood
on the other end’s paging for reinforcements, and unlike our batches of
boys, theyve got discipline. Mr Reprpeepreprpeapperpsanetnateive: I have
two hands and NOT ENOUGH AMMO. Look at the algebra yonder.
I need me some red hot shot if I wanna take out somma those suckas.
Evariste Galois? Never hearda him. All’s in my heart is bread n Marat. I’m not
into theory. I’m into just getting down n dirty with the machinery, ysee. That’s
how i got two batteries up here on the coast so fast. I GET THE JOB DONE,
MR REP. Set the foundation now, worry about the details later. SHOOT FIRST,
AIM LATER, I SAY. But in order to be generous in our volume of fire, we err
need MORE AMMO. Look, it’s not me that’s dragging my feet, it’s everyone else.
Everyone else is just so. Damn. Slow. I can’t STAND it up here. Nah,
I’m always on my feet. THE FIRST DERIVATIVE OF A SEPERABLE
POLYNOMIAL ISNT ZERO. A SEPERABLE ELEMENT IN AN EXTENSION
IS ALGEBRAIC OVER A SEPARABLE MINIMAL POLYNOMIAL.
SEPARABLE CLOSURE IS THE EXTENSION WHOSE SET IS THE
SEPERABLE ELEMENTS IN THE ALGEBRAIC CLOSURE. That’s all I need.
No Frobenius morphism OooOoOO aAAAAaaaAa BS. No dungamental theorems
or w.e. No CERTAINTY, no CONTEXT. ALL I GOT IN MY HANDS
IS AN EMPTY HEART. NO AMMO. NOTHING TO SHOOT. JUST
CRUMBS TO THROW AT THE PIDGEONS. JUST A LIL FINGER TO
PREEN THE PIDGEY. I’ve got feminine legs and feminine arms, but my
feminine hands are as dirty as dung. I work with the Earth. These earthwork
fortifications ysee on these two batteries were build by these hands, right
alongside the boys, and YES: it’s dubiously patched up with earthfilled
barrels. I STAY UP TOO LATE AND I WAKE UP TOO EARLY. But I
have two batteries up. Fibers on the run. I’m doing a great job. And
QWHROQHWROHWQROHWQORHOWQHROWQHROHWQROHQRWOHWQROWQOHYWQOTHYOWQQTWQTWQHTOWQHTPWQHTPWQHTPWQHUTPWQTPWQUTPWQUTPWQHTRPWQHUTPWQQQQQQQQQQQQQHTPWQHTPWQHTWQPTPWQHTQPWWQT!PPJPJWQPRJWQPJRWQPRJWQQWJRPWQJRPQJWPJRWQPJRWQPJRPWQRWQJUTPUTPWQJUTPWQQJTPWQJTPWQJUTPQTWQUTPWQTPWQTJPWQHTPWQJUTPWQWJUTPQWJUPUTJPWQTJUPWQJUTPWQTJUPWQTJUPWQJUTPWQTPWQTJUPJTQWPJUPQJUPTWQJUPWQTJUPWQTJUPQWJUTPTWJPWQWUTPRWUQPRUPJRWQJRQPJRWQJRWQOJRWQOJROWQJROJROWQJJQWTOJQWOOWQJRKRWQKWQKRWQOKRWQOKRWQOKRWQKRWQKOKRWQOKRWQOJWROKJROWQJROWQJRKOWQJRQOWJROWQKROWQJRWQOJORWQJROWQKJRWQOJRWQOJROWQKJROWQJJOQWJROWQWRKWQPRKWQPRPWQRPWQRPWQRWQRWQWQWQJTPWQJUPWQITPWQUTPWQRUPWQUTPWQUTPWQIQIRPWQUTPWQITPWQIRPWQWQIRPWQUTOQWJPJFPNCQNC:PNJWQFJWFPQWJRPWQJRPWQJRPJUTPQPQUTPWQJUTPWQJPWQJTUPWQJUTPWQTJUPWQITPWQUTPWQRUPWQRJUPWQRPRWQIPWQRKIPRWQRWQIPWQRKIPWRQKIPWQRKIPQWRKIPWQRKPRWQJPRWQKRPWQKRPWQJRPWQJRPWQJRWQPKRPWQKRPWQKRPRKWQKPWQKTPWQRWQPURWQJUPQWTTJPWQJTWQPJTPWQJPWQJTJWQTPJQWTPJWPJPQWKPKEPWQKEPWQKEPWQIEPWQEIPWQIEPWQIEPQWEPQWKQQWRWQRWQTWQOTJWQOJTWQPOJRPWQJRWOQRWQRWQPRWQRORPWQRQJUTPQWJUTPWQUROWQJUTPWQIUTPWQUPRWQURWQRWQTWQIJPRWQIJPRWQJIPIJRPWQRPOWQRWQIRPWQJUTPWQIJRPWQRWQUPWQUTPWQRIPWQJUTPWQJUTPWQUURPWQJRPWQJPURPQWURPQWURPQWURPQIRPIWQRPIRKPKPWJRPQWJRPWQJPWQJPWQJRPWQRJPWQKJRPWQJRPWQJRPWQJRPWQJRPWQJTPWQJPWQJTPWQWQP.
I’m perfectly fine Mr Representative. In fact, I’m the only one sane one here,
amongst the leadership. *Laughs as the represants run away like pussies from all
the kicked up dust from the sudden onfire* HAHAHAHAHA. Well dont worry
you get used to it. Or we do. Cause we’re the ones out here doing the hard work
yknow. But man, wed be able to fire back a bit if we had a bit more ammo you
know. Well anyway thats why im not too concered about Gribeauval or Galois or
Grothendieck or whoever. I just have to do the thing assigned to me here and
READY FOR SOME TARGET×PRACTICE⊗? Watch this, Mr. Rep. WE ARE
NOW COMFORTABLE WITH FIBERS BOYS, LOAD EM UP. Now, the initial
instinct of an algebraic geometer here, is, ofc, can we assume X is affine? The
answer this time is, hmm, not really. If we assume we can just break up X into an
affine cover and prove irreducibility on each open affine, then we would want the
union of irreducible open sets to be irreducible, which isnt necessarily the case.
And there’s nothing special about this configuration here that gives us any leeway
here.
However....
Do note the following chain of inclusions

Which induces a morphism of affine schemes

And thus, now, by the universal property of products (YEP, I JUST DID THAT.
I JUST INVOKED A GOD DAMN UNIVERSAL PROPERTY. WE HAVE THE
HIGH GROUND NOW: A BIRDS EYE VIEW OF THE CITY, AND) the
following commutative diagram:
The important thing is that unique morphism induced:

Now, because the image of an irreducible set is irreducible, and furthermore,
the closure of an irreducible set is irreducible, then showing that f is
dominant (i.e. its image is dense in the range) proves (i)
(ii). Also,
since dense subsets of irreducible sets are irreducible, showing that f is
dominant would also prove (ii)
(i). So we are very, very, very
tempted to show that f is dominant. And furthermore, according to old
notes, one case where this happens on affines is when the corresponding
ring morphism is injective. Goal: We want f to be INJECTIVE!
MARK THE ∃! There is only ONE morphism that works here. But let’s take a
look a back at this field inclusion:

Now let’s say SpecA is an open affine of X (one in an open cover). Let me tensor
the above:
 (1) 
WHEW, REP. WE LOST THE HOOK, BUT THAT’S
OKAY. THERE’S NO NEED TO PANIC. I HAVE A PLAN, REPPY.
DONT WORRY. EVERYTHING WILL BE JUUUUUUUSSSSTT FINE.
WATCH MY BOYS AT WORK. WATCH EM.
If I Spec the above, I get

Now, gluing the f_{A} together yields a morphism that satisfies the ∃! of the
universal property above, hence the f must locally manifest as f_{A}. Locally, it’s
natural. And since injective maps glue into injective maps, we can reduce to
showing f_{A} is injective. So we do reduce to affines (but to show injectivity rather
than irreducibility)
So we want to show that the map (1) just described is injective

OHHHHHHHHH GOD I REALLY WISH I HAD THE
HOOK NOW. BUT THAT’S OKAY. WE’LL GET IT BACK. WE’LL FIND
OUR WAY.
Because, you see, X is finite type over k. Hah! I came prepared, Mr.
Reprereperpeprperpepesentative. That means I can assume A is a finitely
generated kalgebra:

Which turns (1) into

But the natural structure on these folks is just

.......No hook yet?
Well the quotient might mess with injectivity, might it? What I need to show is
that only 0 maps to 0. I.e.
If

and

for

then in fact

Now, listen Mr. Rep. As you can see, I’m in quite a pinch. NO AMMO and all.
Boy oh boy would I like some AMMO. I’ve got a lot of retired coast guardsmen
resting in chairs around the area, and boy oh boy would i like to go up to them
with yall by my side. Theyd shiver at the mighty Terror and crawl to the park Ive
got up at Ollioules to repair those old iron suckers. Threaten their children: Mr.
Rep. Threaten to kill the children. Hold a gun at the wife, and if she’s
pregnant bayonet her in the stomach. Yall are rough and tough soon as you
capture a city, but you gotta extort from the living too. You gotta show
you still mean business. I need those lazy fencesitters to come out here
and do work. I dont wanna have to advocate for these things, but i got
no choice if we’ve got no AMMO. God I want some goddamn AMMO.
Screw it. Kill the children. Kill the women. Kill the men. Kill em all. Just
get the ammo for me. That’s all anyone really needs, right? Ammo. If
everyone had some ammo, no one would need to shoot. That’s how i feel. I
really really really wish I could get some ammo right now. Ammo, ammo,
ammo. God I am hungry for ammo. I’d strangle a child for some ammo.
Spinny balls. Red hot shot. Appleshot. Grapeshot. Bananashot. Canisters. I
have to wake up early tomorrow, and i’m always running on too little
sleep. I’m in a constant manic panic going through the day. my mind
is all over the place and hyperattentive. I’m just whoooooooooo. The
least i could have is a little ammo. Ammo ammo ammo ammo ammo
ammo ammo ammo ammo ammo. Ahhh, ammo. To sweeten the day.
That’s all a parvenu needs. Come onnnnnn. Gimme the ammo. GIVE ME
THE AMMO, REPRESENTATIVE. COME ONNNN MR REP. USE
YOUR NAME AND GIVE ME SOME AMMO. COME ONNN, MR
REPRESENTATIVE. YOU FFFFFFFFFFFFFFUCKER. GIVE ME
THE AMMO. GIVE ME SOMETHING TO WORK WITH HERE. Ahh,
ammo. Ammo. Ammo. Boy do I want some ammo in my life. And by
the way: April Fools! This exercise is left to the reader. Hahahahahaha.