SGA Over 9000: Séminaire d'Geometric Approach

« first day (45 days earlier) ← previous day next day → last day (113 days later) »

Narcissus jewel

00:09

@BalarkaSen Is there a reason why you linked this, instead of this?

Balarka Sen

Ah no not really

I forgot Hatcher added it as chapter 5 to AT

Narcissus jewel

Right, just making sure

Balarka Sen

That seems to be the updated version tho

Narcissus jewel

The other one said 'last updated 2004' or something, and then I saw CH5.

Balarka Sen

Yeah

Ch5 is the right thing

Daminark

00:34

I guess I may as well continue my feeble attempts at working through Brown

So contractibility obv implies acyclicity because you'll be homotopy equivalent to the trivial chain complex

Daminark

01:06

Trailed off, honestly over the break I'm not gonna have time to do much because my parents keep calling

Okay so a short exact sequence $0\to A \to B \to C \to 0$ splits if $B = A\oplus C$

So if we take $0 \to Z_{n+1} \to C_{n+1} \to Z_n \to 0$

So if $id = dh + hd$, we restrict $h$ to $Z$, which is just $dh$

Narcissus jewel

@Daminark It splits if there are maps going in the opposite direction, such that going into $B$ and then back out again is the identity on $A$ or $C$, but what you've written is equivalent in an abelian category

Daminark

Ah aight, thanks!

Narcissus jewel

No idea how often we'll leave abelian cats

Also the result I refer to is en.wikipedia.org/wiki/Splitting_lemma

Daminark

01:22

For chain complexes we'll prob stick around in abelian cats

Maybe when we actually get to the group cohomology, we'll have to leave

But yeah alright so, if we write $I = dh + hd$, we know $I\restriction_Z = dh\restriction_Z$

So $h\restriction_Z$ splits $d$

Narcissus jewel

I guess homotopy doesn't make sense outside of an additive category, so we'll always keep that shrugs

Daminark

(when you think about it as $C_{n+1} \to Z_n$)

Now, the other direction of this theorem says that if $C$ has 0 homology and every such exact sequence splits, then $C$ is contractible

@Narcissus does it not?

Narcissus jewel

@Daminark We want atleast a zero object in the category, so that mapping in and out of it induces a zero morphism (so we can say $d^2=0$)

Next, we constantly use the group structure of the hom classes

Daminark

Wait do you need that for homotopy though? I could see homotopy making sense where homology doesn't

Narcissus jewel

Well the latter we need group structure atleast

(27.5 hours awake atm so I might say strange things, just gotta stay awake to teach in 1.5 hours)

Daminark

01:31

Jeez, what are you teaching?

Narcissus jewel

Calculus II :P

Daminark

(Also drink coffee soon if you haven't already, otherwise that won't be fun)

Narcissus jewel

I've had so many coffees they are losing effectiveness unfortunately :P. I missed the last bus last night, and was trapped here.

We do want abelian group structure on $[C,C']$ right?

Daminark

For homology I definitely see us needing that, I just didn't realize that any category for which there was a homotopy category had to give rise to homology

So yeah

(I'm presuming that group cohomology is gonna correspond nicely to ordinary cohomology, since if it was really a different cohomology theory, I wouldn't think chain complexes would be of relevance, in which case we'll prob stay in abelcats)

Daminark

01:49

Okay maybe I should read homological algebra at least concurrently

MatheinBoulomenos

02:08

@Daminark group cohomology does correspond to ordinary cohomology, but chain complexes come up when dealing with any abelian category, even when it's not ordinary cohomology

Daminark

02:21

I see

7 hours later…

Balarka Sen

08:55

@Daminark Right acyclic implies contractible too.

$\mathbf{Ch}_\bullet$ is a super convenient category

Homotopy and homology are the same

@Daminark $n$-th group cohomology of a discrete group $G$ is $H^n_{singular}(K(G, 1))$

3 hours later…

Balarka Sen

11:48

room topic changed to SGA Over 9000: Séminaire d'Geometric Approach: For potential Grothendiecks (no tags)

room topic changed to SGA Over 9000: Séminaire d'Geometric Approach: For potential Grothendiecks. When the Fields committee look for us, we'll be too busy doing the communist revolution. (no tags)

That's more like it.

4 hours later…

Balarka Sen

16:07

Hey @Daminark, Michael here

3 hours later…

Daminark

18:46

Amazing

Balarka Sen

19:44

sup