ORDER OF THE ARTILLERY, AS OF 20 SEPT:

1. BATTERY OF THE MOUNTAIN: 1 x 36 pdr, 4 x 24 pdr [*]
2. THE BREECHLESS BATTERY: 1 x 44 pdr, 1 x 12 pdr mortar [*]

 
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 heard-a 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

3.15

 

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 k-algebra:

 

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.

0.1 Lemma

If

 

and

 

for

 

then in fact

 

Proof

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.