12
votes
2answers
286 views

Group cohomology without G-modules (a.k.a. what does this bar construction compute?)

Without any prior exposure to the cohomology of groups, one might naively proceed by replacing a group by a sort of resolution. For instance, let's take $G = \mathbb{Z}^2$, and "resolve": $$ 0 \to ...
4
votes
4answers
370 views

(Co)homological characterization of homotopy pullbacks

For a commutative square of spaces (of manifolds, or of simplicial sets): $$S=\left(\begin{array}{ccc} A & \to & B \newline \downarrow & & \downarrow \newline C & \to & ...
1
vote
1answer
240 views

Coboundary map on the cochain complex of abelian cosimplicial groups?

Maybe I'm looking at the wrong places, but I can't find a definition of the coboundary map on the cochain complex of abelian cosimplicial groups. What I have in mind is something similar to the ...
2
votes
0answers
179 views

The Mayer-Viertoris exact sequence as a (Zariski) descent spectral sequence.

For certain 'spaces' $U,V$ (they are certain Henselizations of subvarieties) I would like to compute (certain etale) cohomology of $U\cup V$ in terms of the corresponding cohomology of the diagram ...
4
votes
1answer
623 views

Simplicial “universal extensions”, the hammock localization, and Ext

Let $M,B$ be $R$-modules, and suppose we're given an n-extension $E_1\to\dots\to E_n$ of $B$ by $M$, that is, an exact sequence $$0\to M\to E_1\to\dots\to E_n \to B\to 0.$$ A morphism of ...
11
votes
2answers
2k views

Understand Cech Cohomology

I am currently trying to understand Cech cohomology. Five questions arised and I would be glad for help. In what follows $X$ is a topological space. I really like Dugger's and Isaksen's paper ...