Two Definitions of Cohomologies

Asked Jul 29 '19 at 23:32

Active Jul 31 '19 at 15:40

Viewed 90 times

There are two definitions of cohomologies I've found so far: The first (very good to understand) definition is via coboundaries and is very often written as
$$ H^p = {Z^p}/{B^p} ,$$
and the second one is $$ H^p = Ext_g^p .$$ I have no clue why they are identical. If the answer is too long its enough to give a source to read it.

I use it in the context of Lie Algebras, so if there is an easier proof only for Lie Algebras, it's enough for me.

EDIT: i have both definitions and they look pretty similar when u use the right complexe(s). But i still cant figure out why the sequence
$ P_2 \overset{d_1}{\rightarrow} P_1 \overset{d_0}{\rightarrow} P_0 \overset{\epsilon}\rightarrow M $

is exact, the rest is clear now. Tomorrow i will write the exact definitions to this post (I dont have my work at home)

edited Jul 31 '19 at 15:40

asked Jul 29 '19 at 23:32

Padermath

Your first equation should be $H^p = Z^p / B^p$. – Lee Mosher Jul 30 '19 at 01:31
thanks i fixed it – Padermath Jul 30 '19 at 21:48
Ext is the cohomology of a chain complex, so once you define your cochairs and boundary operators, you are in. But I don’t think that is the question you want to ask. – Charlie Frohman Jul 30 '19 at 21:59
2

Somehow what should be $H^p=Z^p/B^p$ is not repaired yet, so far as I can see... – paul garrett Jul 30 '19 at 22:09

Two Definitions of Cohomologies

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